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%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: paper6.dvi %%Pages: 64 %%PageOrder: Ascend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSCommandLine: dvips paper6 -o paper6.ps %DVIPSParameters: dpi=300, comments removed %DVIPSSource: TeX output 1997.09.08:0036 %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}B /TR{translate}N /isls false N /vsize 11 72 mul N /hsize 8.5 72 mul N /landplus90{false}def /@rigin{isls{[0 landplus90{1 -1}{-1 1} ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[matrix currentmatrix{dup dup round sub abs 0.00001 lt{round}if} forall round exch round exch]setmatrix}N /@landscape{/isls true N}B /@manualfeed{statusdict /manualfeed true put}B /@copies{/#copies X}B /FMat[1 0 0 -1 0 0]N /FBB[0 0 0 0]N /nn 0 N 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b(pap)q(er)f(is)h(organized)g(as)f(follo)o (ws.)35 b(Section)22 b(2)e(in)o(tro)q(duces)h(some)f(of)g(the)g (mathematical)h(mac)o(hinery)g(and)-75 31 y(notation)16 b(w)o(e)g(will)h(use.)23 b(In)17 b Fm(x)p Fs(3)f(w)o(e)g(in)o(tro)q (duce)h(the)f(quan)o(tum)g(T)l(uring)h(Mac)o(hine)f(as)g(a)g(natural)g (extension)h(of)e(classical)-75 87 y(probabilistic)k(T)l(uring)f(Mac)o (hines.)26 b(W)l(e)18 b(also)f(sho)o(w)f(that)h(quan)o(tum)f(T)l(uring) i(Mac)o(hines)g(need)g(not)f(b)q(e)g(sp)q(eci\014ed)j(with)-75 144 y(an)g(unreasonable)g(amoun)o(t)f(of)g(precision.)35 b(In)20 b Fm(x)q Fs(4)f(and)h Fm(x)p Fs(5)f(w)o(e)h(demonstrate)f(the)h (basic)g(constructions)g(whic)o(h)g(will)-75 200 y(allo)o(w)d(us)g(to)f (build)j(up)f(large,)f(complicated)h(quan)o(tum)f(T)l(uring)g(Mac)o (hines)h(in)f(subsequen)o(t)h(sections.)26 b(Man)o(y)16 b(actions)-75 257 y(whic)o(h)g(are)f(quite)h(easy)f(for)f(classical)j (mac)o(hines,)f(suc)o(h)f(as)g(completing)h(a)f(partially)i(sp)q 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264 y Fk(i)945 257 y Fm(i)18 b Fs(and)g(the)h(\\ro)o(w")e(of)h Fl(U)23 b Fs(with)c(index)h Fl(i)e Fs(is)h(the)f(dual)i(v)o(ector)-75 313 y Fm(h)p Fl(\022)-36 320 y Fk(i)-22 313 y Fm(j)p Fl(U)5 b Fs(.)-4 387 y(F)l(or)18 b(a)g(Hilb)q(ert)j(space)e Fm(H)p Fs(,)h Fm(fj)p Fl(\022)536 394 y Fk(i)550 387 y Fm(i)o(g)590 394 y Fk(i)p Fy(2)p Fk(I)664 387 y Fs(is)g(a)e Fr(Hilb)n(ert)h(sp)n(ac) n(e)g(b)n(asis)e Fs(for)i Fm(H)g Fs(if)h(it)f(is)g(a)g(maximal)g(set)f (of)h(orthonormal)-75 444 y(v)o(ectors)c(in)j Fm(H)p Fs(.)23 b(Ev)o(ery)16 b(v)o(ector)g Fm(j)p Fl(\036)p Fm(i)e(2)g(H)j Fs(can)g(b)q(e)g(expressed)g(as)f(the)g(limit)i(of)d(a)h (sequence)i(of)e(v)o(ectors,)f(eac)o(h)h(of)g(whic)o(h)-75 500 y(can)f(b)q(e)h(expressed)g(as)f(a)g(\014nite)h(linear)g(com)o (bination)g(of)f(basis)h(v)o(ectors.)-4 574 y(Giv)o(en)i(a)g(linear)h (op)q(erator)f Fl(U)23 b Fs(in)c(an)f(inner)h(pro)q(duct)g(space,)g(if) f(there)h(is)f(a)g(linear)i(op)q(erator)d Fl(U)1699 558 y Fy(\003)1737 574 y Fs(whic)o(h)i(satis\014es)-75 631 y Fm(h)p Fl(U)-21 614 y Fy(\003)-2 631 y Fl(\036)p Fm(j)p Fl( )r Fm(i)h Fs(=)g Fm(h)p Fl(\036)p Fm(j)p Fl(U)5 b( )r Fm(i)19 b Fs(for)g(all)i Fl(\036;)8 b( )r Fs(,)19 b(then)h Fl(U)724 614 y Fy(\003)764 631 y Fs(is)g(called)i(the)e Fr(adjoint)g Fs(or)f Fr(Hermitian)i(c)n(onjugate)e Fs(of)h Fl(U)5 b Fs(.)34 b(If)20 b(a)g(linear)-75 687 y(op)q(erator)f(in)i(an)e (inner)i(pro)q(duct)g(space)f(has)g(an)f(adjoin)o(t,)i(it)f(is)g (unique.)36 b(The)20 b(adjoin)o(t)g(of)f(a)h(linear)h(op)q(erator)e(in) h(a)-75 744 y(Hilb)q(ert)h(space)e(or)f(in)i(a)f(\014nite)h (dimensional)h(inner)g(pro)q(duct)e(space)g(alw)o(a)o(ys)g(exists.)32 b(It)19 b(is)g(easy)g(to)g(see)g(that)f(if)i(the)-75 800 y(adjoin)o(ts)e(of)f Fl(U)185 807 y Fj(1)223 800 y Fs(and)h Fl(U)345 807 y Fj(2)383 800 y Fs(exist)g(then)g(\()p Fl(U)648 807 y Fj(1)679 800 y Fs(+)13 b Fl(U)758 807 y Fj(2)777 800 y Fs(\))795 784 y Fy(\003)832 800 y Fs(=)18 b Fl(U)921 784 y Fy(\003)916 812 y Fj(1)952 800 y 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y(represen)o(ted)i(b)o(y)f(the)h(conjugate)f (transp)q(ose)g(of)g Fl(U)5 b Fs(.)36 b(So,)22 b(in)f(Dirac's)f (notation)g(w)o(e)g(ha)o(v)o(e)h(the)f(con)o(v)o(enien)o(t)h(iden)o (tit)o(y)-75 1100 y Fm(h)p Fl(\036)p Fm(j)o Fl(U)18 1084 y Fy(\003)38 1100 y Fm(j)p Fl( )r Fm(i)11 b Fs(=)i(\()p Fm(h)p Fl( )r Fm(j)n Fl(U)5 b Fm(j)p Fl(\036)p Fm(i)p Fs(\))351 1084 y Fy(\003)370 1100 y Fs(.)-4 1174 y(Recall)16 b(that)d(if)i(the)g(inner-pro)q(duct)h(space)e Fl(V)24 b Fs(is)15 b(the)g(tensor)f(pro)q(duct)g(of)g(t)o(w)o(o)f(inner-pro)q (duct)j(spaces)f Fl(V)1821 1181 y Fj(1)1840 1174 y Fl(;)8 b(V)1888 1181 y Fj(2)1907 1174 y Fs(,)14 b(then)-75 1231 y(for)k(eac)o(h)h(pair)h(of)e(v)o(ectors)g Fm(j)p Fl(\036)455 1238 y Fj(1)475 1231 y Fm(i)g(2)i Fl(V)588 1238 y Fj(1)607 1231 y Fl(;)8 b Fm(j)p Fl(\036)668 1238 y Fj(2)687 1231 y Fm(i)18 b(2)h Fl(V)799 1238 y Fj(2)838 1231 y Fs(there)g(is)g(an)g (asso)q(ciated)g(tensor)g(pro)q(duct)g Fm(j)p Fl(\036)1649 1238 y Fj(1)1668 1231 y Fm(i)13 b(\012)g(j)p Fl(\036)1787 1238 y Fj(2)1806 1231 y Fm(i)19 b Fs(in)h Fl(V)9 b Fs(.)31 b(In)-75 1287 y(Dirac's)15 b(notation,)f(w)o(e)h(denote)h Fm(j)p Fl(\036)527 1294 y Fj(1)546 1287 y Fm(i)10 b(\012)g(j)p Fl(\036)659 1294 y Fj(2)679 1287 y Fm(i)15 b Fs(as)f Fm(j)p Fl(\036)807 1294 y Fj(1)827 1287 y Fm(i)o(j)p Fl(\036)884 1294 y Fj(2)904 1287 y Fm(i)p Fs(.)-4 1361 y(The)k Fr(norm)g Fs(of)g Fl(U)23 b Fs(is)c(de\014ned)g(as)f Fm(k)p Fl(U)5 b Fm(k)17 b Fs(=)h(sup)811 1372 y Fy(kj)p Fk(x)p Fy(i)o(k)p Fj(=1)945 1361 y Fm(k)p Fl(U)5 b Fm(j)p Fl(x)p Fm(i)o(k)o Fs(.)29 b(A)18 b(linear)i(op)q(erator)d(is)i(called)g Fr(b)n(ounde)n(d)f Fs(if)h Fm(k)p Fl(U)5 b Fm(k)17 b Fs(is)-75 1418 y(\014nite.)k(W)l(e)15 b(will)i(freely)f(use)f(the)h (follo)o(wing)g(standard)e(facts)h(ab)q(out)g(b)q(ounded)i(linear)f(op) q(erators:)436 1516 y(If)f Fl(U)517 1497 y Fy(\003)552 1516 y Fs(exists)g(then)h Fm(k)p Fl(U)840 1497 y Fy(\003)859 1516 y Fm(k)42 b Fs(=)f Fm(k)p Fl(U)5 b Fm(k)882 b Fs(\(1\))735 1585 y Fm(k)p Fl(U)789 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1452 y Fk(M)986 1445 y Fs(ma)o(y)g(b)q(e)i(represen)o(ted)f(b)o(y)g (the)g(\(coun)o(table)g(dimensional\))-75 1501 y(\\square")13 b(matrix)h(with)h(columns)g(and)f(ro)o(ws)f(indexed)j(b)o(y)e (con\014gurations)g(where)h(the)f(matrix)g(elemen)o(t)h(from)e(column) -75 1558 y Fl(c)h Fs(and)g(ro)o(w)e Fl(c)151 1541 y Fy(0)177 1558 y Fs(giv)o(es)i(the)g(amplitude)h(with)f(whic)o(h)h (con\014guration)f Fl(c)f Fs(leads)i(to)e(con\014guration)h Fl(c)1576 1541 y Fy(0)1602 1558 y Fs(in)g(a)g(single)h(step)f(of)f Fl(M)5 b Fs(.)-4 1632 y(F)l(or)11 b(con)o(v)o(enience,)j(w)o(e)d(will)j (o)o(v)o(erload)d(notation)h(and)g(use)g(the)g(expression)h Fl(\016)r Fs(\()p Fl(p;)8 b(\033)o(;)g(\034)s(;)g(q)r(;)f(d)o Fs(\))i(to)i(denote)h(the)h(amplitude)-75 1688 y(in)j Fl(\016)r Fs(\()p Fl(p;)8 b(\033)r Fs(\))13 b(of)i Fm(j)p Fl(\034)5 b Fm(i)o(j)p Fl(q)r Fm(i)o(j)p Fl(d)p Fm(i)o Fs(.)-4 1763 y(The)12 b(next)f(de\014nition)j(pro)o(vides)e(an)g (extremely)g(imp)q(ortan)o(t)g(condition)h(that)e(QTMs)g(m)o(ust)h 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y Fi(P)1093 -14 y Fk(i)1115 -26 y Fl(\013)1144 -19 y Fk(i)1158 -26 y Fr(c)1179 -19 y Fk(i)1210 -26 y Fr(is)i(observe)n(d)f(or)i(me)n(asur)n(e)n(d,)f(c)n(on\014gur)n(ation)f (c)2008 -19 y Fk(i)-75 31 y Fr(is)g(se)n(en)f(with)h(pr)n(ob)n(ability) g Fm(j)p Fl(\013)p Fm(j)446 10 y Fj(2)466 31 y Fr(.)k(Mor)n(e)n(over,)c (the)h(sup)n(erp)n(osition)f(of)g Fl(M)21 b Fr(is)16 b(up)n(date)n(d)h(to)g Fl( )1484 14 y Fy(0)1507 31 y Fs(=)c Fr(c)1576 38 y Fk(i)1590 31 y Fr(.)-4 105 y(We)k(may)h(also)f(p) n(erform)h(a)g Fs(partial)e(measuremen)o(t)p Fr(,)i(say)f(only)g(on)g (the)g(\014rst)g(c)n(el)r(l)g(of)g(the)h(tap)n(e.)24 b(In)16 b(this)i(c)n(ase,)f(sup-)-75 161 y(p)n(ose)f(that)i(the)f (\014rst)f(c)n(el)r(l)f(may)i(c)n(ontain)f(the)h(values)f Fs(0)h Fr(or)g Fs(1)p Fr(,)f(and)h(supp)n(ose)f(the)h(sup)n(erp)n (osition)f(was)h Fl( )e Fs(=)1784 129 y Fi(P)1828 173 y Fk(i)1850 161 y Fl(\013)p Fs(0)1902 168 y Fk(i)1916 161 y Fr(c)s Fs(0)1963 168 y Fk(i)1987 161 y Fs(+)-75 186 y Fi(P)-31 229 y Fk(i)-10 218 y Fl(\013)p Fs(1)42 225 y Fk(i)57 218 y Fr(c)s Fs(1)104 225 y Fk(i)117 218 y Fr(,)i(wher)n(e)h(the)g(c)s Fs(0)402 225 y Fk(i)432 218 y Fr(ar)n(e)g(those)f(c)n(on\014gur)n(ations)f(that)i(have)g(a)g Fs(0)f Fr(in)g(the)g(\014rst)g(c)n(el)r(l,)g(and)g(c)s Fs(1)1664 225 y Fk(i)1695 218 y Fr(ar)n(e)g(those)h(c)n(on\014g-)-75 274 y(ur)n(ations)h(that)h(have)g(a)f Fs(1)g Fr(in)g(the)h(\014rst)f(c) n(el)r(l.)29 b(Me)n(asuring)18 b(the)i(\014rst)e(c)n(el)r(l)h(r)n (esults)f(in)h Fl(P)6 b(r)q Fs([0])17 b(=)1604 242 y Fi(P)1648 286 y Fk(i)1669 274 y Fm(j)p Fl(\013)p Fs(0)1734 281 y Fk(i)1748 274 y Fm(j)1761 253 y Fj(2)1780 274 y Fr(.)30 b(Mor)n(e)n(over,)-75 331 y(if)19 b(a)g Fs(0)f Fr(is)h(observe)n(d,)g(the)g(new)f(sup)n(erp)n(osition)h(is)f(given)g (by)1047 313 y Fj(1)p 996 320 120 2 v 996 325 a Fm(p)p 1034 325 82 2 v 32 x Fk(P)5 b(r)q Fj([0])1128 299 y Fi(P)1172 342 y Fk(i)1193 331 y Fl(\013)p Fs(0)1245 338 y Fk(i)1260 331 y Fr(c)s Fs(0)1307 338 y Fk(i)1320 331 y Fr(.)29 b(i.e.)f(the)19 b(p)n(art)g(of)g(the)h(sup)n(erp)n(osition)-75 405 y(c)n(onsistent)14 b(with)j(the)g(answer,)f(with)g(amplitudes)h(sc) n(ale)n(d)e(to)i(give)f(a)g(unit)g(ve)n(ctor.)-4 502 y Fs(Note)g(that)g(the)h(w)o(ell-formedness)h(condition)f(on)g(a)f(QTM) h(simply)h(sa)o(ys)e(that)g(the)h(the)f(time)h(ev)o(olution)h(op)q (erator)-75 558 y(of)c(a)h(QTM)f(m)o(ust)g(satisfy)g(the)h(condition)h (that)e(in)h(eac)o(h)g(successiv)o(e)h(sup)q(erp)q(osition,)g(the)f (sum)g(of)f(the)g(probabilities)j(of)-75 615 y(all)f(p)q(ossible)h (con\014gurations)e(m)o(ust)g(b)q(e)h(1.)-4 689 y(Notice)d(that)f(a)g (QTM)g(di\013ers)h(from)f(a)h(classical)h(TM)e(in)h(that)f(the)h (\\user")f(has)h(decisions)h(b)q(ey)o(ond)g(just)e(c)o(ho)q(osing)h(an) -75 745 y(input.)20 b(A)14 b(priori)h(it)f(is)g(not)g(clear)g(whether)g (m)o(ultiple)i(observ)m(ations)e(migh)o(t)f(increase)i(the)f(p)q(o)o(w) o(er)f(of)h(QTMs)f(This)h(p)q(oin)o(t)-75 802 y(is)k(discussed)g(in)g 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1537 y Fr(b)n(e)g(a)g(distribution)g(such)h(that)f(observing)g Fl(M)861 1521 y Fy(0)891 1537 y Fr(on)g(input)h Fl(x)f Fr(after)g Fl(f)5 b Fs(\()p Fl(T)h Fs(\))19 b Fr(steps)g(pr)n(o)n(duc)n (es)f(a)i(sample)f(fr)n(om)g Fm(D)1996 1521 y Fy(0)2008 1537 y Fr(.)-75 1594 y(Then)c(we)i(say)f(that)h Fl(M)333 1577 y Fy(0)361 1594 y Fr(simulates)f Fl(M)21 b Fr(with)c(ac)n(cur)n (acy)f Fl(\017)h Fr(if)f Fm(jD)11 b(\000)g(D)1131 1577 y Fy(0)1143 1594 y Fm(j)h(\024)h Fl(\017)p Fr(.)-4 1690 y Fs(W)l(e)k(will)h(sometimes)f(\014nd)h(it)f(con)o(v)o(enien)o(t)h(to) e(measure)h(the)g(accuracy)g(of)g(a)f(sim)o(ulation)i(b)o(y)f (calculating)i(the)e(Eu-)-75 1747 y(clidean)h(distance)e(b)q(et)o(w)o (een)g(the)g(target)e(sup)q(erp)q(osition)k(and)e(the)g(sup)q(erp)q (osition)h(ac)o(hiev)o(ed)g(b)o(y)e(the)h(sim)o(ulation.)23 b(The)-75 1803 y(follo)o(wing)e(sho)o(ws)f(that)f(the)i(v)m(ariation)g (distance)g(b)q(et)o(w)o(een)g(the)f(resulting)i(distributions)g(is)f (at)e(most)h(4)g(times)h(this)-75 1860 y(Euclidean)c(distance.)-75 1957 y Fn(Lemma)g(3.2.6)23 b Fr(L)n(et)14 b Fl(\036;)8 b( )13 b Fm(2)g(S)18 b Fr(such)d(that)h Fm(k)p Fl(\036)p Fm(k)c Fs(=)h Fm(k)p Fl( )r Fm(k)e Fs(=)i(1)p Fr(,)i(and)h Fm(k)p Fl(\036)9 b Fm(\000)i Fl( )r Fm(k)g(\024)i Fl(\017)p Fr(.)21 b(Then)15 b(the)g(total)g(variation)h(distanc)n(e)-75 2013 y(b)n(etwe)n(en)f(the)i(pr)n(ob)n(ability)f(distributions)g(r)n (esulting)f(fr)n(om)h(me)n(asur)n(ements)f(of)i Fl(\036)f Fr(and)g Fl( )i Fr(is)e(at)g(most)h Fs(4)p Fl(\017)p Fr(.)-75 2110 y Fn(Pro)q(of.)38 b Fs(Let)21 b Fl(\036)i Fs(=)300 2078 y Fi(P)344 2121 y Fk(i)365 2110 y Fl(\013)394 2117 y Fk(i)408 2110 y Fm(j)p Fl(i)p Fm(i)e Fs(and)g Fl( )j Fs(=)682 2078 y Fi(P)726 2121 y Fk(i)747 2110 y Fl(\014)773 2117 y Fk(i)787 2110 y Fm(j)p Fl(i)p Fm(i)o Fs(.)38 b(Observing)22 b Fl(\036)f Fs(giv)o(es)h(eac)o(h)f Fm(j)p Fl(i)p Fm(i)f Fs(with)i(probabilit)o(y)g Fm(j)p Fl(\013)1837 2117 y Fk(i)1851 2110 y Fm(j)1863 2089 y Fj(2)1883 2110 y Fs(,)g(while)-75 2166 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Fj(2)1354 2459 y Fs(+)d(2)p Fm(k)p Fl(\013)p Fm(k)o(k)p Fl(\015)s Fm(k)i(\024)h Fl(\017)1647 2440 y Fj(2)1677 2459 y Fs(+)d(2)p Fl(\017)-75 2572 y Fs(Finally)l(,)17 b(note)e(that)f(since)i(w)o(e)f(ha)o(v)o(e)g(unit)h (sup)q(erp)q(ositions,)h(w)o(e)d(m)o(ust)h(ha)o(v)o(e)g Fl(\017)e Fm(\024)g Fs(2.)610 b Fd(2)951 2779 y Fs(10)p eop %%Page: 11 11 11 10 bop -75 -26 a Fh(3.3)56 b(Quan)n(tum)18 b(computing)f(as)i(a)g (unitary)f(transformation)-75 78 y Fs(In)g(the)g(preceding)h(sections,) g(w)o(e)e(in)o(tro)q(duced)i(QTMs)e(as)h(an)f(extension)i(of)e(the)h (notion)g(of)f(probabilistic)j(TMs.)27 b(W)l(e)-75 134 y(stated)15 b(there)g(that)g(a)g(QTM)g(is)h(w)o(ell-formed)h(if)e(it)h (preserv)o(es)g(the)f(norm)g(of)g(the)h(sup)q(erp)q(ositions.)22 b(In)16 b(this)g(section,)f(w)o(e)-75 191 y(explore)k(a)g(di\013eren)o (t,)g(and)g(extremely)g(useful,)h(alternativ)o(e)f(view)g(of)f(QTMs:)27 b(in)19 b(terms)f(of)g(prop)q(erties)i(of)e(the)h(time)-75 247 y(ev)o(olution)e(op)q(erator.)k(W)l(e)16 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Fs(to)g Fm(H)p Fs(.)25 b(Let)16 b Fl(x;)8 b(y)16 b Fm(2)f(H)p Fs(.)25 b(Then)16 b(there)h(are)f(sequences)i Fm(f)p Fl(x)1404 376 y Fk(n)1427 369 y Fm(g)p Fl(;)8 b Fm(f)p Fl(y)1516 376 y Fk(n)1538 369 y Fm(g)15 b(2)g(S)k Fs(suc)o(h)e(that)f Fl(x)1898 376 y Fk(n)1936 369 y Fm(!)f Fl(x)-75 426 y Fs(and)i Fl(y)37 433 y Fk(n)76 426 y Fm(!)e Fl(y)r Fs(.)25 b(Moreo)o(v)o(er,)15 b(for)h(all)i Fl(n)p Fs(,)f(\()p Fl(U)5 b(x)685 433 y Fk(n)708 426 y Fl(;)j(y)751 433 y Fk(n)774 426 y Fs(\))15 b(=)g(\()p Fl(x)901 433 y Fk(n)924 426 y Fl(;)8 b(U)981 409 y Fy(\003)1000 426 y Fl(y)1022 433 y Fk(n)1046 426 y Fs(\).)24 b(T)l(aking)17 b(limits,)h(w)o(e)f(get)f(that)g(\()1667 414 y(^)1659 426 y Fl(U)5 b(x;)j(y)r Fs(\))14 b(=)h(\()p Fl(x;)1929 414 y Fs(^)1913 426 y Fl(U)1949 413 y Fy(\003)1968 426 y Fl(y)r Fs(\),)-75 482 y(as)g(desired.)-4 557 y(As)j(an)g(aside,)h (w)o(e)f(should)h(brie\015y)h(men)o(tion)e(that)g(another)g(resolution) h(of)e(this)i(issue)g(is)g(ac)o(hiev)o(ed)g(b)o(y)f(follo)o(wing)-75 613 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(could)i(not)e(b)q(e)h(reac)o(hed)-75 969 y(from)g(an)o(y)h(other)g (con\014guration.)-4 1044 y Fc(Theorem)i(A.0.12)e Fr(A)h(QTM)f(is)h (wel)r(l-forme)n(d)h(i\013)f(its)g(time)g(evolution)g(op)n(er)n(ator)h (is)f(unitary.)-75 1183 y Fh(3.4)56 b(Precision)17 b(required)g(in)i(a) g(QTM)-75 1287 y Fs(One)d(imp)q(ortan)o(t)g(concern)g(is)g(whether)g (QTMs)f(are)h(really)g(analog)g(devices,)g(since)h(they)f(in)o(v)o(olv) o(e)g(complex)h(transition)-75 1343 y(amplitudes.)k(The)15 b(issue)g(here)g(is)h(ho)o(w)e(accurately)h(these)g(transition)f (amplitudes)i(m)o(ust)e(b)q(e)i(sp)q(eci\014ed)g(to)e(ensure)i(that)-75 1399 y(the)d(correctness)h(of)e(the)i(computation)f(is)h(not)f (compromised.)19 b(In)14 b(an)g(earlier)g(v)o(ersion)f(of)g(this)h(pap) q(er,)g(w)o(e)f(sho)o(w)o(ed)g(that)-75 1456 y Fl(T)-42 1439 y Fk(O)q Fj(\(1\))47 1456 y Fs(bits)i(of)e(precision)j(are)e (su\016cien)o(t)h(to)f(correctly)g(carry)g(out)g Fl(T)20 b Fs(steps)14 b(of)g(computation)g(to)f(within)j(accuracy)e Fl(\017)h 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Fm(j)p Fl(\036)748 2619 y Fj(0)767 2612 y Fm(i)10 b Fs(+)g Fl(U)876 2593 y Fk(T)904 2612 y Fm(j)p Fl( )947 2619 y Fj(0)966 2612 y Fm(i)f Fs(+)i Fl(U)1075 2593 y Fk(T)5 b Fy(\000)p Fj(1)1148 2612 y Fm(j)p Fl( )1191 2619 y Fj(1)1209 2612 y Fm(i)10 b Fs(+)h Fm(\001)d(\001)g(\001)g Fs(+)i Fm(j)p Fl( )1434 2619 y Fk(T)1461 2612 y Fm(i)951 2779 y Fs(12)p eop %%Page: 13 13 13 12 bop -75 -26 a Fs(The)15 b(theorem)g(follo)o(ws)h(b)o(y)f(the)g (triangle)h(inequalit)o(y)h(since)f Fl(U)k Fs(is)c(unitary)f(and)h Fm(kj)p Fl( )1387 -19 y Fk(i)1400 -26 y Fm(i)o(k)d(\024)g Fl(\017)p Fs(.)456 b Fd(2)-75 143 y Fn(De\014nition)19 b(3.4.2)j Fr(We)17 b(say)g(that)g(QTMs)f Fl(M)21 b Fr(and)c Fl(M)903 126 y Fy(0)931 143 y Fr(ar)n(e)g Fl(\017)p Fs(-close)h Fr(if)f(they)g(have)g(the)g(same)f(state)h(set)f(and)h(alphab)n(et)-75 199 y(and)i(if)h(the)f(di\013er)n(enc)n(e)g(b)n(etwe)n(en)f(e)n(ach)h (p)n(air)h(of)g(c)n(orr)n(esp)n(onding)e(tr)n(ansition)g(amplitudes)i (has)f(magnitude)h(at)g(most)g Fl(\017)p Fr(.)-75 256 y(Note)c(that)h Fl(M)22 b Fr(and)16 b Fl(M)330 239 y Fy(0)358 256 y Fr(c)n(an)f(b)n(e)h Fl(\017)p Fr(-close)g(even)g(if)g (one)g(or)h(b)n(oth)f(ar)n(e)g(not)h(wel)r(l-forme)n(d.)-4 368 y Fs(The)k(follo)o(wing)g(theorem)f(sho)o(ws)g(that)g(t)o(w)o(o)g (QTMs)g(whic)o(h)i(are)e(close)i(in)f(the)g(ab)q(o)o(v)o(e)f(sense)h (giv)o(e)g(rise)h(to)e(time)-75 425 y(ev)o(olution)g(op)q(erators)e (whic)o(h)i(are)f(close)h(to)f(eac)o(h)g(other,)g(ev)o(en)h(if)g(the)f (QTMs)g(are)g(not)g(w)o(ell-formed.)32 b(As)20 b(a)e(simple)-75 481 y(consequence,)23 b(the)e(time)g(ev)o(olution)g(op)q(erator)f(of)g (a)g(QTM)g(is)i(alw)o(a)o(ys)d(b)q(ounded,)k(ev)o(en)e(if)g(the)g(QTM)g (is)g(not)f(w)o(ell-)-75 538 y(formed.)-75 650 y Fn(Theorem)d(3.4.3)22 b Fr(If)17 b(QTMs)f Fl(M)23 b Fr(and)17 b Fl(M)675 633 y Fy(0)705 650 y Fr(with)h(alphab)n(et)f Fs(\006)h Fr(and)f(state)g (set)h Fl(Q)f Fr(ar)n(e)g Fl(\017)p Fr(-close,)h(then)g(the)f(di\013er) n(enc)n(e)g(in)-75 706 y(their)f(time)g(evolutions)f(has)g(norm)h(at)g (most)g Fs(2)f Fr(c)n(ar)n(d)p Fs(\(\006\))g Fr(c)n(ar)n(d)p Fs(\()p Fl(Q)p Fs(\))o Fl(\017)p Fr(.)21 b(Mor)n(e)n(over,)16 b(this)f(statement)g(holds)h(even)f(if)g(one)h(or)-75 763 y(b)n(oth)g(of)h(the)f(machines)g(ar)n(e)g(not)h(wel)r(l-forme)n (d.)-75 875 y Fn(Pro)q(of.)22 b Fs(Let)16 b(QTMs)f Fl(M)21 b Fs(and)16 b Fl(M)517 858 y Fy(0)545 875 y Fs(with)g(alphab)q(et)h (\006)f(and)g(state)f(set)g Fl(Q)h Fs(b)q(e)h(giv)o(en)f(whic)o(h)h (are)e Fl(\017)p Fs(-close.)23 b(Let)17 b Fl(U)j Fs(b)q(e)d(the)-75 931 y(time)f(ev)o(olution)g(of)e Fl(M)5 b Fs(,)15 b(and)g(let)h Fl(U)544 915 y Fy(0)571 931 y Fs(b)q(e)g(the)f(time)h(ev)o(olution)g (of)e Fl(M)1112 915 y Fy(0)1124 931 y Fs(.)-4 1006 y(No)o(w,)i (consider)j(an)o(y)e(unit)h(length)g(sup)q(erp)q(osition)h(of)e (con\014gurations)g Fm(j)p Fl(\036)p Fm(i)f Fs(=)1379 973 y Fi(P)1423 1017 y Fk(j)1448 1006 y Fl(\013)1477 1013 y Fk(j)1496 1006 y Fm(j)p Fl(c)1529 1013 y Fk(j)1546 1006 y Fm(i)p Fs(.)26 b(Then)18 b(w)o(e)f(can)h(express)-75 1062 y(the)d(di\013erence)i(in)f(the)f(mac)o(hines')h(op)q(eration)f (on)g Fm(j)p Fl(\036)p Fm(i)g Fs(as)f(follo)o(ws.)521 1199 y Fl(U)5 b Fm(j)p Fl(\036)p Fm(i)10 b(\000)g Fl(U)706 1180 y Fy(0)718 1199 y Fm(j)p Fl(\036)p Fm(i)i Fs(=)836 1159 y Fi(X)858 1250 y Fk(j)903 1115 y Fi(0)903 1190 y(@)963 1159 y(X)940 1253 y Fk(i)p Fy(2)p Fk(P)5 b Fj(\()p Fk(j)r Fj(\))1046 1199 y Fs(\()p Fl(\025)1091 1206 y Fk(i;j)1141 1199 y Fm(\000)10 b Fl(\025)1213 1180 y Fy(0)1213 1210 y Fk(i;j)1253 1199 y Fs(\))p Fl(\013)1300 1206 y Fk(i)1313 1115 y Fi(1)1313 1190 y(A)1357 1199 y Fm(j)p Fl(c)1390 1206 y Fk(j)1408 1199 y Fm(i)-75 1340 y Fs(where)15 b Fl(P)6 b Fs(\()p Fl(j)s Fs(\))14 b(is)h(the)g(set)f(of)h Fl(i)f Fs(suc)o(h)h(that)f(con\014guration)h Fl(c)933 1347 y Fk(i)962 1340 y Fs(can)f(lead)i(to)e Fl(c)1215 1347 y Fk(j)1248 1340 y Fs(in)h(a)g(single)h(step)e(of)h Fl(M)k Fs(or)c Fl(M)1779 1324 y Fy(0)1790 1340 y Fs(,)g(and)g(where)-75 1397 y Fl(\025)-48 1404 y Fk(i;j)7 1397 y Fs(and)g Fl(\025)122 1380 y Fy(0)122 1409 y Fk(i;j)177 1397 y Fs(are)g(the)g(amplitudes)i (with)e(whic)o(h)h Fl(c)817 1404 y Fk(i)846 1397 y Fs(leads)g(to)f Fl(c)1036 1404 y Fk(j)1069 1397 y Fs(in)h Fl(M)k Fs(and)15 b Fl(M)1323 1380 y Fy(0)1335 1397 y Fs(.)-4 1471 y(Applying)i(the)f (triangle)g(inequalit)o(y)h(and)f(the)g(fact)f(that)g(the)h(square)f (of)g(the)h(sum)g(of)f Fl(n)h Fs(reals)g(is)g(at)f(most)g Fl(n)h Fs(times)-75 1527 y(the)f(sum)g(of)g(their)h(squares,)f(w)o(e)g (ha)o(v)o(e)297 1633 y Fm(k)p Fl(U)5 b Fm(j)p Fl(\036)p Fm(i)k(\000)h Fl(U)504 1616 y Fy(0)516 1633 y Fm(j)p Fl(\036)p Fm(i)o(k)596 1611 y Fj(2)657 1633 y Fs(=)734 1601 y Fi(P)778 1644 y Fk(j)804 1584 y Fi(\014)804 1609 y(\014)804 1634 y(\014)818 1601 y(P)861 1644 y Fk(i)p Fy(2)p Fk(P)5 b Fj(\()p Fk(j)r Fj(\))970 1633 y Fs(\()p Fl(\025)1015 1640 y Fk(i;j)1064 1633 y Fm(\000)11 b Fl(\025)1137 1616 y Fy(0)1137 1645 y Fk(i;j)1176 1633 y Fs(\))p Fl(\013)1223 1640 y Fk(i)1237 1584 y Fi(\014)1237 1609 y(\014)1237 1634 y(\014)1251 1597 y Fj(2)657 1716 y Fm(\024)734 1684 y Fi(P)778 1727 y Fk(j)804 1716 y Fs(2)k(card\(\006\))f(card\()p Fl(Q)p Fs(\))1176 1684 y Fi(P)1220 1727 y Fk(i)p Fy(2)p Fk(P)5 b Fj(\()p Fk(j)r Fj(\))1336 1667 y Fi(\014)1336 1692 y(\014)1336 1717 y(\014)1350 1716 y Fs(\()p Fl(\025)1395 1723 y Fk(i;j)1444 1716 y Fm(\000)10 b Fl(\025)1516 1699 y Fy(0)1516 1728 y Fk(i;j)1556 1716 y Fs(\))p Fl(\013)1603 1723 y Fk(i)1617 1667 y Fi(\014)1617 1692 y(\014)1617 1717 y(\014)1631 1680 y Fj(2)-4 1834 y Fs(Then)15 b(since)i Fl(M)j Fs(and)15 b Fl(M)427 1818 y Fy(0)454 1834 y Fs(are)g Fl(\017)p Fs(-close,)h(w)o(e)f(ha)o(v)o(e)193 1908 y Fi(P)237 1951 y Fk(j)262 1940 y Fs(2)g(card\(\006\))g(card\()p Fl(Q)p Fs(\))634 1908 y Fi(P)678 1951 y Fk(i)p Fy(2)p Fk(P)5 b Fj(\()p Fk(j)r Fj(\))828 1891 y Fi(\014)828 1916 y(\014)828 1941 y(\014)842 1940 y Fs(\()p Fl(\025)887 1947 y Fk(i;j)937 1940 y Fm(\000)10 b Fl(\025)1009 1923 y Fy(0)1009 1952 y Fk(i;j)1049 1940 y Fs(\))p Fl(\013)1096 1947 y Fk(i)1109 1891 y Fi(\014)1109 1916 y(\014)1109 1941 y(\014)1123 1904 y Fj(2)751 2023 y Fs(=)42 b(2)15 b(card\(\006\))g(card\()p Fl(Q)p Fs(\))1200 1991 y Fi(P)1244 2034 y Fk(j)1270 1991 y Fi(P)1314 2034 y Fk(i)p Fy(2)p Fk(P)5 b Fj(\()p Fk(j)r Fj(\))1430 1974 y Fi(\014)1430 1999 y(\014)1430 2024 y(\014)1444 2023 y Fl(\025)1471 2030 y Fk(i;j)1520 2023 y Fm(\000)11 b Fl(\025)1593 2006 y Fy(0)1593 2035 y Fk(i;j)1633 1974 y Fi(\014)1633 1999 y(\014)1633 2024 y(\014)1646 1987 y Fj(2)1666 2023 y Fm(j)p Fl(\013)1708 2030 y Fk(i)1722 2023 y Fm(j)1734 2002 y Fj(2)751 2091 y Fm(\024)42 b Fs(2)15 b(card\(\006\))g(card\()p Fl(Q)p Fs(\))o Fl(\017)1211 2075 y Fj(2)1239 2059 y Fi(P)1282 2103 y Fk(j)1308 2059 y Fi(P)1352 2103 y Fk(i)p Fy(2)p Fk(P)5 b Fj(\()p Fk(j)r Fj(\))1468 2091 y Fm(j)p Fl(\013)1510 2098 y Fk(i)1524 2091 y Fm(j)1537 2070 y Fj(2)-4 2202 y Fs(Finally)14 b(since)h(for)d(an)o(y)h(con\014guration)h Fl(c)703 2209 y Fk(j)721 2202 y Fs(,)f(there)g(are)g(at)g(most)f(2)h (card\(\006\))f(card\()p Fl(Q)p Fs(\))h(con\014gurations)g(that)g(can)g (lead)-75 2259 y(to)i Fl(c)1 2266 y Fk(j)34 2259 y Fs(in)h(a)f(single)h (step,)f(w)o(e)g(ha)o(v)o(e)249 2380 y(2)g(card\(\006\))f(card\()p Fl(Q)p Fs(\))o Fl(\017)631 2363 y Fj(2)659 2347 y Fi(P)703 2391 y Fk(j)729 2347 y Fi(P)772 2391 y Fk(i)p Fy(2)p Fk(P)5 b Fj(\()p Fk(j)r Fj(\))888 2380 y Fm(j)p Fl(\013)930 2387 y Fk(i)944 2380 y Fm(j)957 2359 y Fj(2)1018 2380 y Fm(\024)42 b Fs(4)15 b(card\(\006\))1287 2359 y Fj(2)1322 2380 y Fs(card\()p Fl(Q)p Fs(\))1479 2359 y Fj(2)1499 2380 y Fl(\017)1517 2363 y Fj(2)1545 2347 y Fi(P)1588 2391 y Fk(i)1610 2380 y Fm(j)p Fl(\013)1652 2387 y Fk(i)1666 2380 y Fm(j)1679 2359 y Fj(2)1018 2442 y Fs(=)42 b(4)15 b(card\(\006\))1287 2421 y Fj(2)1322 2442 y Fs(card\()p Fl(Q)p Fs(\))1479 2421 y Fj(2)1499 2442 y Fl(\017)1517 2425 y Fj(2)-4 2537 y Fs(Therefore,)f(for)h(an)o(y)g(unit)h(length)g (sup)q(erp)q(osition)g Fm(j)p Fl(\036)p Fm(i)598 2629 y(k)p Fs(\()p Fl(U)e Fm(\000)d Fl(U)766 2610 y Fy(0)777 2629 y Fs(\))p Fm(j)p Fl(\036)p Fm(ik)27 b(\024)h Fs(2)15 b(card\(\006\))f(card\()p Fl(Q)p Fs(\))p Fl(\017)951 2779 y Fs(13)p eop %%Page: 14 14 14 13 bop 1988 -26 a Fd(2)-4 105 y Fs(The)14 b(follo)o(wing)h (corollary)f(sho)o(ws)g(that)f Fl(O)q Fs(\(log)8 b Fl(T)e Fs(\))13 b(bits)i(of)f(precision)h(are)f(su\016cien)o(t)h(in)g(the)f (transition)h(amplitudes)-75 161 y(to)g(sim)o(ulate)g Fl(T)21 b Fs(steps)16 b(of)e(a)h(QTM)g(to)g(within)h(accuracy)f Fl(\017)h Fs(for)f(an)o(y)f(constan)o(t)h Fl(\017)p Fs(.)-75 285 y Fn(Corollary)j(3.4.4)k Fr(L)n(et)11 b Fl(M)17 b Fs(=)c(\(\006)p Fl(;)8 b(Q;)g(\016)r Fs(\))h Fr(b)n(e)i(a)h(wel)r (l-forme)n(d)g(QTM.)f(and)g(let)h Fl(M)1301 269 y Fy(0)1324 285 y Fr(b)n(e)f(a)h(QTM)f(which)h(is)1877 267 y Fk(\017)p 1702 274 366 2 v 1702 308 a Fj(24)j Fr(c)n(ar)n(d)p Fj(\(\006\))i Fr(c)n(ar)n(d)o Fj(\()p Fk(Q)p Fj(\))p Fk(T)2072 285 y Fr(-)-75 355 y(close)d(to)h Fl(M)5 b Fr(,)15 b(wher)n(e)g Fl(\017)e(>)g Fs(0)p Fr(.)20 b(Then)14 b Fl(M)595 339 y Fy(0)621 355 y Fr(simulates)g Fl(M)20 b Fr(for)15 b(time)g Fl(T)21 b Fr(with)15 b(ac)n(cur)n(acy)g Fl(\017)p Fr(.)20 b(Mor)n(e)n(over,)15 b(this)g(statment)f(holds)-75 412 y(even)i(if)g Fl(M)123 395 y Fy(0)151 412 y Fr(is)f(not)i(wel)r (l-forme)n(d.)-75 536 y Fn(Pro)q(of.)j Fs(Let)15 b Fl(b)d Fs(=)430 518 y Fj(1)p 253 525 372 2 v 253 559 a(24)j Fs(card)p Fj(\(\006\))g Fs(card)p Fj(\()p Fk(Q)p Fj(\))o Fk(T)629 536 y Fs(.)20 b(Without)15 b(loss)h(of)e(generalit)o(y)l(,)i (w)o(e)f(further)g(assume)g Fl(\017)e(<)g Fs(1)p Fl(=)p Fs(2.)-4 624 y(Consider)18 b(running)h Fl(M)k Fs(and)18 b Fl(M)567 607 y Fy(0)597 624 y Fs(with)g(the)g(same)g(initial)i(sup)q (erp)q(osition.)30 b(Since)19 b Fl(M)k Fs(is)c(w)o(ell-formed,)g(b)o(y) f(Theo-)-75 680 y(rem)h(A.0.12,)e(its)i(time)g(ev)o(olution)g(op)q (erator)f Fl(U)24 b Fs(is)19 b(unitary)l(.)31 b(By)19 b(Theorem)f(3.4.3)f(on)i(page)f(13)g(the)h(time)g(ev)o(olution)-75 737 y(op)q(erator)14 b(of)h Fl(M)209 720 y Fy(0)221 737 y Fs(,)g Fl(U)285 720 y 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y(alignmen)o(t)i(of)e(the)h(tap)q(e)h(con)o(ten)o(ts.)36 b(In)22 b(this)f(subsection,)i(w)o(e)e(in)o(tro)q(duce)h(sev)o(eral)f (input/output)h(con)o(v)o(en)o(tions)f(on)-75 1618 y(QTMs)f(and)h (deterministic)i(TMs)c(whic)o(h)j(will)g(help)g(us)f(main)o(tain)g (these)f(relationships)j(while)f(manipulating)g(and)-75 1675 y(com)o(bining)16 b(mac)o(hines.)-4 1749 y(W)l(e)i(w)o(ould)h(lik) o(e)g(to)f(think)h(of)f(our)g(QTMs)g(as)g(\014nishing)i(their)f (computation)f(when)h(they)f(reac)o(h)g(the)h(\014nal)g(state)-75 1805 y Fl(q)-55 1812 y Fk(f)-32 1805 y Fs(.)k(Ho)o(w)o(ev)o(er,)16 b(it)g(is)h(unclear)h(ho)o(w)d(w)o(e)i(should)g(regard)f(a)g(mac)o (hine)h(whic)o(h)g(reac)o(hes)g(a)f(sup)q(erp)q(osition)i(in)f(whic)o (h)g(some)-75 1862 y(con\014gurations)f(are)g(in)g(state)f Fl(q)481 1869 y Fk(f)520 1862 y Fs(but)h(others)f(are)h(not.)21 b(W)l(e)16 b(try)g(to)f(a)o(v)o(oid)h(suc)o(h)g(di\016culties)i(b)o(y)e (sa)o(ying)f(that)h(a)f(QTM)-75 1918 y(halts)g(on)g(a)g(particular)h 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b(from)h(the)g(leftmost)g(non-blank)h(sym)o(b) q(ol)g(to)e(the)h(righ)o(tmost)f(non-blank)j(sym)o(b)q(ol.)j(This)c (means)f(that)f(giving)951 2779 y(14)p eop %%Page: 15 15 15 14 bop -75 -26 a Fs(an)17 b(output)g(string)g(lea)o(v)o(es)g(unsp)q (eci\014ed)i(the)e(alignmen)o(t)h(of)f(this)g(string)g(on)g(the)g(tap)q (e)g(and)g(the)g(lo)q(cation)h(of)f(the)g(tap)q(e)-75 31 y(head)g(to)e(b)q(e)i(iden)o(ti\014ed.)25 b(When)17 b(describing)h(the)e(input/output)h(b)q(eha)o(vior)g(of)e(a)h(QTM)g(w)o (e)g(will)i(sometimes)e(describ)q(e)-75 87 y(this)h(additional)h (information.)24 b(When)17 b(w)o(e)f(do)g(not,)g(the)h(additional)h (information)e(will)j(b)q(e)e(clear)g(from)e(con)o(text.)24 b(F)l(or)-75 144 y(example,)15 b(w)o(e)e(will)j(often)d(build)j(mac)o (hines)f(in)f(whic)o(h)h(all)g(\014nal)f(con\014gurations)g(ha)o(v)o(e) g(the)g(output)f(string)h(b)q(eginning)i(in)-75 200 y(the)f(start)f (cell)j(with)f(the)f(tap)q(e)g(head)h(scanning)g(its)f(\014rst)g(sym)o (b)q(ol.)-75 314 y 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y Fs(=)13 b Fl(\017)k Fr(is)f(abbr)n(eviate)n(d)g(as)g Fl(x)1113 2625 y Fj(1)1133 2618 y Fs(;)8 b Fl(x)1180 2625 y Fj(2)1199 2618 y Fs(;)g Fl(:)g(:)g(:)t Fs(;)g Fl(x)1326 2625 y Fk(l)1339 2618 y Fr(.)951 2779 y Fs(15)p eop %%Page: 16 16 16 15 bop -75 -26 a Ft(4)69 b(Programming)23 b(a)g(QTM)-75 93 y Fs(In)f(this)g(section)g(w)o(e)f(explore)h(the)g(fundamen)o(tals)g (of)f(building)j(up)e(a)f(QTM)g(from)g(sev)o(eral)g(simpler)i(QTMs.)39 b(Im-)-75 150 y(plemen)o(ting)22 b(basic)g(programming)f(primitiv)o (es,)i(suc)o(h)e(as)g(lo)q(oping,)i(branc)o(hing,)g(and)e(rev)o(ersing) h(a)e(computation)h(is)-75 206 y(straigh)o(tforw)o(ard)13 b(for)h(deterministic)j(TMs.)i(Ho)o(w)o(ev)o(er,)14 b(these)h (constructions)g(are)f(more)h(di\016cult)h(for)e(QTMs)h(b)q(ecause)-75 263 y(one)k(m)o(ust)e(b)q(e)j(v)o(ery)e(careful)h(to)f(main)o(tain)h (rev)o(ersibilit)o(y)l(.)31 b(In)19 b(fact,)g(the)f(same)g (di\016culties)j(arise)e(when)g(building)i(re-)-75 319 y(v)o(ersible)16 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(osition)j(of)d(con\014gurations)675 2353 y Fi(P)719 2397 y Fk(i)741 2385 y Fl(\013)770 2392 y Fk(i)784 2385 y Fm(j)p Fl(c)817 2369 y Fy(0)817 2397 y Fk(i)830 2385 y Fm(i)p Fs(.)24 b(So,)17 b(the)f(time)h(ev)o(olution)h(preserv)o(es)e (length,)i(and)e(the)h(QTM)f(is)-75 2442 y(w)o(ell-formed.)1821 b Fd(2)-4 2572 y Fs(Previous)18 b(w)o(ork)e(of)h(Bennett)h(sho)o(ws)f (that)g(rev)o(ersible)i(mac)o(hines)g(can)e(e\016cien)o(tly)i(sim)o (ulate)f(deterministic)i(TMs.)-75 2629 y(Of)15 b(course,)g(if)g(a)g (deterministic)i(TM)d(computes)h(a)g(function)h(whic)o(h)g(is)f(not)g (one-to-one,)g(then)g(no)g(rev)o(ersible)h(mac)o(hine)951 2779 y(16)p eop %%Page: 17 17 17 16 bop -75 -26 a Fs(can)16 b(sim)o(ulate)h(it)g(exactly)l(.)23 b(Bennett)17 b([7)o(])f(sho)o(w)o(ed)g(that)f(a)h(generalized)i(rev)o (ersible)g(TM)d(can)i(do)f(the)g(next)g(b)q(est)h(thing,)-75 31 y(whic)o(h)h(is)f(to)g(tak)o(e)f(an)o(y)g(input)i Fl(x)f Fs(and)g(compute)g Fl(x)p Fs(;)8 b Fl(M)d Fs(\()p Fl(x)p Fs(\))16 b(where)h Fl(M)5 b Fs(\()p Fl(x)p Fs(\))16 b(is)i(the)f(output)g(of)f Fl(M)22 b Fs(on)17 b(input)h Fl(x)p Fs(.)25 b(He)17 b(also)-75 87 y(sho)o(w)o(ed)h(that)f(if)i(a)f (deterministic)i(TM)d(computes)h(a)g(function)h(whic)o(h)g(is)g (one-to-one,)f(then)h(there)f(is)g(a)g(generalized)-75 144 y(rev)o(ersible)h(TM)f(that)f(computes)h(the)g(same)g(function.)29 b(F)l(or)18 b(b)q(oth)g(constructions,)g(he)h(used)g(a)e(m)o(ulti-tap)q (e)i(TM)f(and)-75 200 y(also)c(suggested)f(ho)o(w)g(the)h(sim)o (ulation)g(could)h(b)q(e)f(carried)g(out)f(using)i(only)f(a)f (single-tap)q(e)i(mac)o(hine.)21 b(Morita)12 b(et.al.)h([33)o(])-75 257 y(use)h(Bennett's)h(ideas,)f(and)g(some)g(further)g(tec)o(hniques,) h(to)f(sho)o(w)f(that)h(an)o(y)g(deterministic)i(TM)d(can)h(b)q(e)h (sim)o(ulated)g(b)o(y)-75 313 y(a)g(generalized)i(rev)o(ersible)f(TM)f (with)h(a)e(t)o(w)o(o)g(sym)o(b)q(ol)i(alphab)q(et.)-4 387 y(W)l(e)22 b(will)j(giv)o(e)e(a)f(sligh)o(tly)i(di\013eren)o(t)f (sim)o(ulation)h(of)e(a)h(deterministic)h(TM)e(with)i(a)e(rev)o (ersible)i(mac)o(hine)g(that)-75 444 y(preserv)o(es)15 b(an)g(imp)q(ortan)o(t)g(timing)h(prop)q(ert)o(y)l(.)-4 518 y(First,)j(w)o(e)f(describ)q(e)j(wh)o(y)e(timing)g(is)h(of)e (critical)j(imp)q(ortance.)31 b(In)20 b(later)f(sections,)h(w)o(e)e (will)j(build)g(QTMs)e(with)-75 574 y(in)o(teresting)13 b(and)g(complex)g(in)o(terference)g(patterns.)19 b(Ho)o(w)o(ev)o(er,)12 b(t)o(w)o(o)f(computational)h(paths)h(can)f(only)h(in)o(terfere)g(if)g (they)-75 631 y(reac)o(h)j(the)g(same)g(con\014guration)h(at)e(the)h (same)g(time.)23 b(W)l(e)17 b(will)h(often)e(w)o(an)o(t)f(paths)h(to)f (in)o(terfere)i(whic)o(h)g(run)f(m)o(uc)o(h)g(of)-75 687 y(the)e(same)f(computation,)h(but)g(with)g(di\013eren)o(t)g (inputs.)20 b(W)l(e)14 b(can)g(only)g(b)q(e)h(sure)f(they)g(in)o (terfere)g(if)g(w)o(e)f(kno)o(w)h(that)f(these)-75 744 y(computations)j(can)h(b)q(e)g(carried)h(out)e(in)h(exactly)g(the)g (same)f(running)i(time.)24 b(W)l(e)17 b(therefore)f(w)o(an)o(t)f(to)h (sho)o(w)g(that)g(an)o(y)-75 800 y(function)i(computable)g(in)g (deterministic)i(p)q(olynomial)f(time)e(can)h(b)q(e)g(computed)g(b)o(y) f(a)g(p)q(olynomial)i(time)f(rev)o(ersible)-75 857 y(TM)f(in)i(suc)o(h) g(a)e(w)o(a)o(y)g(that)h(the)g(running)h(time)f(of)g(the)g(latter)g(is) g(determined)i(en)o(tirely)f(b)o(y)f(the)g(length)h(of)e(its)h(input.) -75 913 y(Then,)f(pro)o(vided)h(that)d(all)j(computation)f(paths)f (carry)g(out)g(the)h(same)f(deterministic)j(algorithms)e(on)f(the)h (inputs)g(of)-75 969 y(the)e(same)g(length,)h(they)f(will)i(all)f(tak)o (e)e(exactly)i(the)f(same)g(n)o(um)o(b)q(er)h(of)e(steps.)-4 1044 y(W)l(e)d(pro)o(v)o(e)g(the)g(follo)o(wing)h(theorem)f(in)h(App)q (endix)h(B)f(on)f(page)g(55)g(using)h(ideas)f(from)g(the)g (constructions)h(of)f(Bennett)-75 1100 y(and)k(Morita)g(et.al.)-75 1224 y Fn(Theorem)i(4.1.3)g(\(Sync)o(hronization)i(Theorem\))k Fr(If)16 b Fl(f)23 b Fr(is)17 b(a)g(function)g(mapping)h(strings)e(to)i (strings)e(which)i(c)n(an)-75 1280 y(b)n(e)e(c)n(ompute)n(d)h(in)e (deterministic)h(p)n(olynomial)g(time)g(and)h(such)f(that)h(the)g (length)e(of)i Fl(f)5 b Fs(\()p Fl(x)p Fs(\))16 b Fr(dep)n(ends)f(only) h(on)g(the)h(length)-75 1337 y(of)d Fl(x)p Fr(,)h(then)f(ther)n(e)g(is) f(a)i(p)n(olynomial)e(time,)i(stationary,)g(normal)f(form)g(r)n (eversible)f(TM)h(which)g(given)g(input)g Fl(x)p Fr(,)g(pr)n(o)n(duc)n (es)-75 1393 y(output)k Fl(x)p Fs(;)8 b Fl(f)d Fs(\()p Fl(x)p Fs(\))p Fr(,)15 b(and)h(whose)g(running)g(time)g(dep)n(ends)g (only)f(on)h(the)h(length)e(of)i Fl(x)p Fr(.)-4 1468 y(If)c Fl(f)19 b Fr(is)13 b(a)h(function)f(fr)n(om)h(strings)e(to)i (strings)f(that)h(such)g(that)h(b)n(oth)e Fl(f)19 b Fr(and)14 b Fl(f)1318 1451 y Fy(\000)p Fj(1)1379 1468 y Fr(c)n(an)f(b)n(e)g(c)n (ompute)n(d)h(in)g(deterministic)-75 1524 y(p)n(olynomial)e(time,)i (and)f(such)g(that)g(the)g(length)f(of)h Fl(f)5 b Fs(\()p Fl(x)p Fs(\))12 b Fr(dep)n(ends)g(only)h(on)f(the)h(length)f(of)h Fl(x)p Fr(,)h(then)e(ther)n(e)h(is)f(a)h(p)n(olynomial)-75 1580 y(time,)h(stationary,)g(normal)e(form)i(r)n(eversible)e(TM)g (which)i(given)e(input)h Fl(x)p Fr(,)h(pr)n(o)n(duc)n(es)e(output)j Fl(f)5 b Fs(\()p Fl(x)p Fs(\))p Fr(,)13 b(and)g(whose)g(running)-75 1637 y(time)j(dep)n(ends)g(only)g(on)g(the)g(length)g(of)g Fl(x)p Fr(.)-75 1776 y Fh(4.2)56 b(Programming)16 b(primitiv)n(es)-75 1880 y Fs(W)l(e)g(no)o(w)f(sho)o(w)g(ho)o(w)g(to)g(carry)g(out)g(sev)o (eral)h(programming)f(primitiv)o(es)i(rev)o(ersibly)l(.)23 b(The)16 b(Branc)o(hing,)g(Rev)o(ersal,)g(and)-75 1936 y(Lo)q(oping)g(Lemmas)f(will)i(b)q(e)f(used)g(frequen)o(tly)g(in)g (subsequen)o(t)f(sections.)-4 2011 y(The)j(pro)q(ofs)f(of)h(the)g (follo)o(wing)g(t)o(w)o(o)f(lemmas)h(are)f(straigh)o(tforw)o(ard)f(and) i(are)g(omitted.)28 b(Ho)o(w)o(ev)o(er,)17 b(they)h(will)h(b)q(e)-75 2067 y(quite)f(useful)h(as)e(w)o(e)g(build)i(complicated)g(mac)o (hines,)g(since)f(they)g(allo)o(w)f(us)h(to)f(build)i(a)e(series)h(of)f (simpler)i(mac)o(hines)-75 2123 y(while)e(ignoring)f(the)f(con)o(ten)o (ts)f(of)h(trac)o(ks)f(not)h(curren)o(tly)h(b)q(eing)g(used.)-75 2247 y Fn(Lemma)h(4.2.1)23 b Fr(Given)13 b(any)h(QTM)f(\(r)n(eversible) f(TM\))h Fl(M)k Fs(=)c(\(\006)p Fl(;)8 b(Q;)g(\016)r Fs(\))j Fr(and)j(any)g(set)g Fs(\006)1463 2231 y Fy(0)1474 2247 y Fr(,)g(ther)n(e)g(is)g(a)g(QTM)f(\(r)n(eversible)-75 2304 y(TM\))i Fl(M)82 2287 y Fy(0)106 2304 y Fs(=)e(\(\006)d Fm(\002)g Fs(\006)293 2287 y Fy(0)305 2304 y Fl(;)e(Q;)g(\016)405 2287 y Fy(0)415 2304 y Fs(\))16 b Fr(such)g(that)h Fl(M)694 2287 y Fy(0)722 2304 y Fr(b)n(ehaves)f(exactly)g(as)g Fl(M)21 b Fr(while)16 b(le)n(aving)f(its)h(se)n(c)n(ond)f(tr)n(ack)i (unchange)n(d.)-75 2428 y Fn(Lemma)g(4.2.2)23 b Fr(Given)16 b(any)h(QTM)g(\(r)n(eversible)e(TM\))h Fl(M)k Fs(=)14 b(\(\006)1081 2435 y Fj(1)1111 2428 y Fm(\002)e(\001)c(\001)g(\001)g (\002)k Fs(\006)1301 2435 y Fk(k)1322 2428 y Fl(;)c(Q;)g(\016)r Fs(\))15 b Fr(and)i(p)n(ermutation)h Fl(\031)e Fs(:)e([1)p Fl(;)8 b(k)q Fs(])13 b Fm(!)-75 2484 y Fs([1)p Fl(;)8 b(k)q Fs(])p Fr(,)16 b(ther)n(e)h(is)g(a)h(QTM)e(\(r)n(eversible)g (TM\))g Fl(M)759 2468 y Fy(0)786 2484 y Fs(=)f(\(\006)887 2494 y Fk(\031)q Fj(\(1\))966 2484 y Fm(\002)c(\001)d(\001)g(\001)h (\002)i Fs(\006)1155 2494 y Fk(\031)q Fj(\()p Fk(k)q Fj(\))1225 2484 y Fl(;)d(Q;)g(\016)1325 2468 y Fy(0)1335 2484 y Fs(\))17 b Fr(such)h(that)g(the)f Fl(M)1694 2468 y Fy(0)1723 2484 y Fr(b)n(ehaves)g(exactly)-75 2541 y(as)f Fl(M)21 b Fr(exc)n(ept)16 b(that)h(its)f(tr)n(acks)g(ar)n(e)g(p)n (ermute)n(d)h(ac)n(c)n(or)n(ding)e(to)i Fl(\031)r Fr(.)951 2779 y Fs(17)p eop %%Page: 18 18 18 17 bop -4 -26 a Fs(The)17 b(follo)o(wing)i(t)o(w)o(o)d(lemmas)i(are) f(also)g(straigh)o(tforw)o(ard,)f(but)i(stating)f(them)g(separately)h (mak)o(es)f(Lemma)h(4.2.5)-75 31 y(b)q(elo)o(w)j(easy)e(to)h(pro)o(v)o (e.)34 b(The)20 b(\014rst)g(deals)g(with)h(sw)o(apping)f(transitions)g (of)g(states)f(in)i(a)f(QTM.)f(W)l(e)h(can)g(sw)o(ap)g(the)-75 87 y(outgoing)13 b(transitions)g(of)g(states)f Fl(p)533 94 y Fj(1)565 87 y Fs(and)h Fl(p)674 94 y Fj(2)707 87 y Fs(for)f(transition)i(function)g Fl(\016)h Fs(b)o(y)e(de\014ning)h Fl(\016)1443 71 y Fy(0)1455 87 y Fs(\()p Fl(p)1496 94 y Fj(1)1515 87 y Fl(;)8 b(\033)r Fs(\))j(=)i Fl(\016)r Fs(\()p Fl(p)1704 94 y Fj(2)1723 87 y Fl(;)8 b(\033)r Fs(\),)k Fl(\016)1837 71 y Fy(0)1848 87 y Fs(\()p Fl(p)1889 94 y Fj(2)1909 87 y Fl(;)c(\033)r Fs(\))j(=)-75 144 y Fl(\016)r Fs(\()p Fl(p)-12 151 y Fj(1)7 144 y Fl(;)d(\033)r Fs(\))15 b(and)i Fl(\016)201 127 y Fy(0)212 144 y Fs(\()p Fl(p;)8 b(\033)r Fs(\))13 b(=)i Fl(\016)r Fs(\()p Fl(p;)8 b(\033)r Fs(\))14 b(for)i Fl(p)e Fm(6)p Fs(=)h Fl(p)708 151 y Fj(1)728 144 y Fl(;)8 b(p)772 151 y Fj(2)791 144 y Fs(.)23 b(Similarly)l(,)c(w)o(e)d(can)g(sw)o(ap)g(the)g(incoming)i (transitions)f(of)f(states)f Fl(q)2002 151 y Fj(1)-75 200 y Fs(and)i Fl(q)35 207 y Fj(2)72 200 y Fs(b)o(y)g(de\014ning)h Fl(\016)332 184 y Fy(0)344 200 y Fs(\()p Fl(p;)8 b(\033)o(;)g(\034)s(;) g(q)516 207 y Fj(1)533 200 y Fl(;)g(d)p Fs(\))14 b(=)i Fl(\016)r Fs(\()p Fl(p;)8 b(\033)o(;)g(\034)s(;)f(q)854 207 y Fj(2)871 200 y Fl(;)h(d)p Fs(\),)16 b Fl(\016)985 184 y Fy(0)996 200 y Fs(\()p Fl(p;)8 b(\033)o(;)g(\034)s(;)g(q)1168 207 y Fj(2)1185 200 y Fl(;)g(d)p Fs(\))14 b(=)i Fl(\016)r Fs(\()p Fl(p;)8 b(\033)o(;)g(\034)s(;)g(q)1507 207 y Fj(1)1524 200 y Fl(;)g(d)p Fs(\),)15 b(and)i Fl(\016)1727 184 y Fy(0)1739 200 y Fs(\()p Fl(p;)8 b(\033)o(;)g(\034)s(;)g(q)r(;)f (d)o Fs(\))13 b(=)-75 257 y Fl(\016)r Fs(\()p Fl(p;)8 b(\033)o(;)g(\034)s(;)g(q)r(;)f(d)o Fs(\))12 b(for)j Fl(q)f Fm(6)p Fs(=)f Fl(q)366 264 y Fj(1)386 257 y Fl(;)8 b(q)427 264 y Fj(2)447 257 y Fs(.)-75 381 y Fn(Lemma)17 b(4.2.3)23 b Fr(If)e Fl(M)27 b Fr(is)22 b(a)g(wel)r(l-forme)n(d)h(QTM)e (\(r)n(eversible)g(TM\),)g(then)h(swapping)g(the)g(inc)n(oming)f(or)i (outgoing)-75 437 y(tr)n(ansitions)15 b(b)n(etwe)n(en)g(a)i(p)n(air)f (of)g(states)g(in)g Fl(M)21 b Fr(gives)16 b(another)h(wel)r(l-forme)n (d)f(QTM)f(\(r)n(eversible)g(TM\).)-75 561 y Fn(Lemma)i(4.2.4)23 b Fr(L)n(et)12 b Fl(M)360 568 y Fj(1)392 561 y Fs(=)h(\(\006)p Fl(;)8 b(Q)548 568 y Fj(1)567 561 y Fl(;)g(\016)608 568 y Fj(1)627 561 y Fs(\))k Fr(and)i Fl(M)787 568 y Fj(2)819 561 y Fs(=)f(\(\006)p Fl(;)8 b(Q)975 568 y Fj(2)993 561 y Fl(;)g(\016)1034 568 y Fj(2)1053 561 y Fs(\))13 b Fr(b)n(e)g(two)g (wel)r(l-forme)n(d)h(QTMs)e(\(r)n(eversible)f(TMs\))h(with)-75 617 y(the)g(same)g(alphab)n(et)g(and)g(disjoint)f(state)h(sets.)18 b(Then)11 b(then)h(the)g(union)g(of)g(the)g(two)g(machines,)g Fl(M)18 b Fs(=)13 b(\(\006)p Fl(;)8 b(Q)1768 624 y Fj(1)1787 617 y Fm([)p Fl(Q)1853 624 y Fj(2)1873 617 y Fl(;)g(\016)1914 624 y Fj(1)1934 617 y Fm([)p Fl(\016)1984 624 y Fj(2)2004 617 y Fs(\))-75 674 y Fr(and)16 b(with)h(arbitr)n(arily)f(chosen)g (start)h(state)f Fl(q)713 681 y Fj(0)745 674 y Fm(2)d Fl(Q)824 681 y Fj(1)854 674 y Fm([)e Fl(Q)931 681 y Fj(2)966 674 y Fr(is)16 b(also)g(a)h(wel)r(l-forme)n(d)f(QTM)g(\(r)n(eversible)e 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b(sw)o(ap)e(the)h(incoming)g(transitions) g(of)f Fl(q)j Fs(and)e Fl(q)1629 1748 y Fj(2)p Fk(;)p Fj(0)1689 1741 y Fs(and)g(the)f(outgoing)-75 1797 y(transitions)j(of)g Fl(q)i Fs(and)f Fl(q)348 1804 y Fj(2)p Fk(;f)413 1797 y Fs(according)g(to)e(Lemma)i(4.2.3)d(on)i(page)g(18)g(to)g(get)f(the)i (w)o(ell-formed)g(mac)o(hine)g Fl(M)5 b Fs(.)-4 1872 y(Since)17 b Fl(M)159 1879 y Fj(1)194 1872 y Fs(is)g(in)f(normal)g (form,)f(the)h(\014nal)h(state)e(of)g Fl(M)21 b Fs(leads)16 b(bac)o(k)g(to)f(its)h(initial)i(state)d(no)h(matter)f(whether)h Fl(q)h Fs(is)-75 1928 y(the)e(initial)j(state)c(of)h Fl(M)342 1935 y Fj(1)362 1928 y Fs(,)f(the)i(\014nal)g(state)e(of)h Fl(M)776 1935 y Fj(1)796 1928 y Fs(,)f(or)h(neither.)958 b Fd(2)-4 2059 y Fs(Next,)17 b(w)o(e)g(sho)o(w)g(ho)o(w)g(to)f(tak)o(e) h(t)o(w)o(o)f(mac)o(hines)i(and)f(form)g(a)g(third)h(b)o(y)f(\\do)o(v)o (etailing")h(one)f(on)o(to)g(the)g(end)h(of)f(the)-75 2115 y(other.)h(Notice,)13 b(that)e(when)h(w)o(e)f(do)o(v)o(etail)h 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y(ev)o(olv)o(es)i(just)f(as)g(the)h(\014rst)f(under)i(the)e(c)o (hange)h(of)f(basis.)22 b(Note,)15 b(that)g(in)i(this)f(construction,)f (the)h(new)g(QTM)f(has)h(the)-75 2312 y(same)e(time)h(ev)o(olution)g (op)q(erator)f(as)g(the)h(original)h(mac)o(hine.)k(Ho)o(w)o(ev)o(er,)13 b(the)i(states)f(of)g(the)h(new)f(mac)o(hine)i(di\013er)f(from)-75 2368 y(those)g(of)g(the)g(old.)20 b(This)c(c)o(hange)f(of)g(basis)h (will)h(allo)o(w)e(us)g(to)g(pro)o(v)o(e)g(Lemmas)g(5.3.2)e(and)j (5.3.4)d(b)q(elo)o(w.)-75 2492 y Fn(Lemma)k(5.1.1)23 b Fr(Given)14 b(a)h(QTM)f Fl(M)k Fs(=)13 b(\(\006)p Fl(;)8 b(Q;)g(\016)r Fs(\))k Fr(and)j(a)g(set)g(of)g(ve)n(ctors)f Fl(B)k Fr(fr)n(om)1391 2481 y Fs(~)1380 2492 y Fl(C)1416 2476 y Fk(Q)1461 2492 y Fr(which)d(forms)g(an)g(orthonormal)-75 2549 y(b)n(asis)g(for)h Fn(C)144 2532 y Fk(Q)174 2549 y Fr(,)g(ther)n(e)f(is)h(a)g(QTM)f Fl(M)577 2532 y Fy(0)601 2549 y Fs(=)e(\(\006)p Fl(;)8 b(B)r(;)g(\016)800 2532 y Fy(0)811 2549 y Fs(\))15 b Fr(which)i(evolves)e(exactly)g(as)h Fl(M)21 b Fr(under)16 b(a)g(change)g(of)g(b)n(asis)f(fr)n(om)h Fl(Q)-75 2605 y Fr(to)h Fl(B)r Fr(.)951 2779 y Fs(22)p eop %%Page: 23 23 23 22 bop -75 -26 a Fn(Pro)q(of.)20 b Fs(Let)15 b Fl(M)j Fs(=)13 b(\(\006)p Fl(;)8 b(Q;)g(\016)r Fs(\))k(b)q(e)k(a)f(QTM)g(and)h Fl(B)h Fs(an)f(orthonormal)e(basis)i(for)e Fn(C)1367 -42 y Fk(Q)1397 -26 y Fs(.)-4 48 y(Since)j Fl(B)h Fs(is)f(an)e (orthonormal)g(basis,)h(this)h(establishes)g(a)e(unitary)h (transformation)f(from)g(the)g(space)i(of)e(sup)q(erp)q(o-)-75 105 y(sitions)g(of)e(states)h(in)g Fl(Q)g Fs(to)g(the)g(space)g(of)g (sup)q(erp)q(ositions)h(of)f(states)f(in)i Fl(B)r Fs(.)20 b(Sp)q(eci\014call)q(y)l(,)d(for)c(eac)o(h)h Fl(p)f Fm(2)g Fl(Q)g Fs(w)o(e)h(ha)o(v)o(e)g(the)-75 161 y(mapping)785 218 y Fm(j)p Fl(p)p Fm(i)d(!)914 177 y Fi(X)908 269 y Fk(v)q Fy(2)p Fk(B)986 218 y Fm(h)p Fl(p)p Fm(j)o(j)p Fl(v)r Fm(i)j(j)p Fl(v)r Fm(i)-75 337 y Fs(Similarly)l(,)j(w)o(e)d(ha)o (v)o(e)h(a)f(unitary)h(transformation)f(from)g(the)h(space)g(of)f(sup)q (erp)q(ositions)i(of)f(con\014gurations)g(with)g(states)-75 394 y(in)i Fl(Q)f Fs(to)g(the)g(space)h(of)f(con\014gurations)g(with)h (states)e(in)i Fl(B)r Fs(.)24 b(In)17 b(this)g(second)f (transformation,)f(a)h(con\014guration)h(with)-75 450 y(state)d Fl(p)g Fs(is)h(mapp)q(ed)h(to)e(the)h(sup)q(erp)q(osition)h (of)e(con\014gurations)h(where)g(the)f(corresp)q(onding)i (con\014guration)f(with)g(state)-75 506 y Fl(v)i Fs(app)q(ears)e(with)h (amplitude)g Fm(h)p Fl(p)p Fm(j)o(j)p Fl(v)r Fm(i)o Fs(.)-4 581 y(Let)10 b(us)h(see)g(what)f(the)h(time)g(ev)o(olution)g(of)f Fl(M)16 b Fs(should)11 b(lo)q(ok)g(lik)o(e)h(under)f(this)g(c)o(hange)g (of)f(basis.)19 b(In)11 b Fl(M)16 b Fs(a)10 b(con\014guration)-75 637 y(in)15 b(state)f Fl(p)g Fs(reading)i(a)e Fl(\033)i Fs(ev)o(olv)o(es)f(in)g(a)f(single)i(time)f(step)f(to)g(the)h(sup)q (erp)q(osition)h(of)e(con\014gurations)h(corresp)q(onding)h(to)-75 694 y(the)f(sup)q(erp)q(osition)i Fl(\016)r Fs(\()p Fl(p;)8 b(\033)r Fs(\).)624 750 y Fl(\016)r Fs(\()p Fl(p;)g(\033)r Fs(\))j(=)819 710 y Fi(X)812 802 y Fk(\034)r(;q)q(;d)892 750 y Fl(\016)r Fs(\()p Fl(p;)d(\033)o(;)g(\034)s(;)g(q)r(;)f(d)p Fs(\))12 b Fm(j)p Fl(\034)5 b Fm(i)o(j)p Fl(q)r Fm(i)o(j)p Fl(d)p Fm(i)-75 874 y Fs(With)15 b(the)h(c)o(hange)f(of)g(basis,)g(the) g(sup)q(erp)q(osition)i(on)e(the)h(righ)o(t)f(hand)g(side)h(will)h (instead)f(b)q(e)606 964 y Fi(X)599 1056 y Fk(\034)r(;v)q(;d)681 932 y Fi( )714 964 y(X)735 1051 y Fk(q)781 1004 y Fm(h)p Fl(q)r Fm(j)p Fl(v)r Fm(i)e Fl(\016)r Fs(\()p Fl(p;)8 b(\033)o(;)g(\034)s(;)f(q)r(;)h(d)o Fs(\))1147 932 y Fi(!)1184 1004 y Fm(j)p Fl(\034)d Fm(ij)p Fl(v)r Fm(i)o(j)p Fl(d)p Fm(i)-75 1149 y Fs(No)o(w,)14 b(since)j(the)e(state)f(sym)o(b)q (ol)i(pair)f Fm(j)p Fl(v)r Fm(i)o(j)p Fl(\033)r Fm(i)f Fs(in)i Fl(M)820 1133 y Fy(0)847 1149 y Fs(corresp)q(onds)g(to)e(the)i (sup)q(erp)q(osition)829 1209 y Fi(X)850 1296 y Fk(p)897 1249 y Fm(h)p Fl(v)r Fm(j)p Fl(p)p Fm(i)e(j)p Fl(p)p Fm(i)o(j)p Fl(\033)r Fm(i)-75 1389 y Fs(in)i Fl(M)5 b Fs(,)15 b(w)o(e)g(should)h(ha)o(v)o(e)f(in)h Fl(M)473 1373 y Fy(0)361 1532 y Fl(\016)383 1513 y Fy(0)394 1532 y Fs(\()p Fl(v)r(;)8 b(\033)r Fs(\))j(=)562 1491 y Fi(X)583 1578 y Fk(p)629 1532 y Fm(h)p Fl(v)r Fm(j)p Fl(p)p Fm(i)731 1447 y Fi(0)731 1522 y(@)780 1491 y(X)768 1584 y Fk(\034)r(;v)814 1574 y Fq(0)824 1584 y Fk(;d)860 1460 y Fi( )893 1491 y(X)914 1578 y Fk(q)960 1532 y Fm(h)p Fl(q)r Fm(j)p Fl(v)1037 1513 y Fy(0)1048 1532 y Fm(i)k Fl(\016)r Fs(\()p Fl(p;)8 b(\033)o(;)g(\034)s(;)f(q)r(;)g(d)p Fs(\))1338 1460 y Fi(!)1368 1447 y(1)1368 1522 y(A)1412 1532 y Fm(j)p Fl(\034)e Fm(i)o(j)p Fl(v)1504 1513 y Fy(0)1515 1532 y Fm(i)o(j)p Fl(d)p Fm(i)-4 1695 y Fs(Therefore,)18 b Fl(M)267 1679 y Fy(0)296 1695 y Fs(will)i(b)q(eha)o(v)o(e)e(exactly)h(as)e Fl(M)23 b Fs(under)c(the)f(c)o(hange)g(of)f(basis)i(if)f(w)o(e)g (de\014ne)h Fl(\016)1629 1679 y Fy(0)1659 1695 y Fs(b)o(y)e(sa)o(ying)h (that)g(for)-75 1752 y(eac)o(h)d Fl(v)r(;)8 b(\033)13 b Fm(2)g Fl(B)g Fm(\002)e Fs(\006)427 1838 y Fl(\016)449 1820 y Fy(0)460 1838 y Fs(\()p Fl(v)r(;)d(\033)r Fs(\))j(=)640 1798 y Fi(X)628 1891 y Fk(\034)r(;v)674 1881 y Fq(0)685 1891 y Fk(;d)720 1767 y Fi( )753 1798 y(X)761 1885 y Fk(p;q)821 1838 y Fm(h)p Fl(v)r Fm(j)p Fl(p)p Fm(i)i(h)p Fl(q)r Fm(j)p Fl(v)1007 1820 y Fy(0)1018 1838 y Fm(i)i Fl(\016)r Fs(\()p Fl(p;)8 b(\033)o(;)g(\034)s(;)f(q)r(;)g(d)p Fs(\))1308 1767 y Fi(!)1345 1838 y Fm(j)p Fl(\034)e Fm(ij)p Fl(v)1438 1820 y Fy(0)1449 1838 y Fm(i)o(j)p Fl(d)p Fm(i)-75 1973 y Fs(Since)17 b(the)e(v)o(ectors)f(in)i Fl(B)i Fs(are)d(con)o (tained)h(in)726 1962 y(~)716 1973 y Fl(C)752 1957 y Fk(Q)782 1973 y Fs(,)e(eac)o(h)i(amplitude)g(of)f Fl(\016)1200 1957 y Fy(0)1227 1973 y Fs(is)h(con)o(tained)f(in)1542 1962 y(~)1531 1973 y Fl(C)s Fs(.)-4 2047 y(Finally)l(,)h(note)e(that)g (the)h(time)g(ev)o(olution)h(of)e Fl(M)835 2031 y Fy(0)861 2047 y Fs(m)o(ust)h(preserv)o(e)f(Euclidean)j(length)f(since)f(it)g(is) 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(condition)g(sp)q(eci\014es)h(the)f(orthogonalit)o(y)e(of)h(pairs)g(of) g(columns)h(for)e(con\014gurations)h(whic)o(h)h(are)f(iden)o(tical)-75 1956 y(except)f(that)e(the)h(second)h(has)e(its)i(tap)q(e)f(head)g(t)o (w)o(o)f(cells)j(to)d(the)h(left,)g(is)h(in)g(state)e Fl(p)1388 1963 y Fj(2)1422 1956 y Fs(instead)i(of)f Fl(p)1654 1963 y Fj(1)1673 1956 y Fs(,)g(has)g(a)g Fl(\033)1846 1963 y Fj(2)1880 1956 y Fs(instead)-75 2013 y(of)g(a)g Fl(\034)35 2020 y Fj(2)69 2013 y Fs(in)h(the)g(cell)h(under)e(its)h (tap)q(e)f(head,)g(and)h(has)f(a)g Fl(\034)925 2020 y Fj(1)959 2013 y Fs(instead)h(of)f(a)g Fl(\033)1232 2020 y Fj(1)1267 2013 y Fs(t)o(w)o(o)f(cells)i(to)f(the)g(left.)326 b Fd(2)-4 2143 y Fs(No)o(w)17 b(consider)i(again)f(unidirectional)j (QTMs,)d(those)g(in)h(whic)o(h)g(eac)o(h)f(state)f(can)h(b)q(e)h(en)o (tered)g(while)g(mo)o(ving)f(in)-75 2200 y(only)e(one)f(direction.)22 b(When)16 b(w)o(e)f(considered)i(this)f(prop)q(ert)o(y)f(for)f 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Fy(\003)1162 565 y Fs(=)13 b(0)-75 689 y(Therefore,)i(for)f(an)o (y)h Fl(v)r(;)8 b(\033)o(;)g(\034)s(;)g(v)458 672 y Fy(0)479 689 y Fm(2)13 b Fl(B)g Fm(\002)d Fs(\006)g Fm(\002)h Fs(\006)e Fm(\002)i Fl(B)p 825 676 V 15 x Fk(d)543 799 y Fl(\016)565 780 y Fy(0)576 799 y Fs(\()p Fl(v)r(;)d(\033)o(;)g(\034)s (;)g(v)752 780 y Fy(0)761 799 y Fl(;)g(d)p Fs(\))j(=)883 758 y Fi(X)891 845 y Fk(p;q)951 799 y Fm(h)p Fl(v)r Fm(j)p Fl(p)p Fm(i)n(h)p Fl(q)r Fm(j)p Fl(v)1122 780 y Fy(0)1133 799 y Fm(i)o Fl(\016)r Fs(\()p Fl(p;)d(\033)o(;)g(\034)s(;)g(q)r(;)f(d) p Fs(\))610 950 y(=)658 910 y Fi(X)679 997 y Fk(p)725 950 y Fm(h)p Fl(v)r Fm(j)p Fl(p)p Fm(i)827 910 y Fi(X)849 997 y Fk(q)895 950 y Fm(h)p Fl(q)r Fm(j)p Fl(v)972 932 y Fy(0)982 950 y Fm(i)p Fl(\016)r Fs(\()p Fl(p;)h(\033)o(;)g(\034)s(;)g (q)r(;)f(d)o Fs(\))j(=)j(0)-75 1071 y(Therefore)i(an)o(y)g(state)f(in)i Fl(B)i Fs(can)e(b)q(e)f(en)o(tered)h(while)h(tra)o(v)o(eling)e(in)h (only)g(one)f(direction.)-4 1145 y(Unfortunately)l(,)k(this)g(new)g 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Fj(0)879 1626 y Fs(\()p Fl(p;)8 b(\033)r Fs(\))i(=)j Fm(j)p Fl(\033)r Fm(i)o(j)p Fl(p)p Fm(ij)p Fl(R)p Fm(i)-19 1728 y Fs(2.)22 b(Change)15 b(basis)g(from)g Fl(Q)g Fs(to)f Fl(B)k Fs(while)f(stepping)f(left.)760 1860 y Fl(\016)780 1867 y Fj(1)800 1860 y Fs(\()p Fl(p;)8 b(\033)r Fs(\))j(=)970 1819 y Fi(X)967 1912 y Fk(b)p Fy(2)p Fk(B)1041 1860 y Fm(h)p Fl(p)p Fm(j)p Fl(b)p Fm(i)o(j)p Fl(\033)r Fm(i)n(j)p Fl(b)p Fm(i)o(j)p Fl(L)p Fm(i)-19 2003 y Fs(3.)22 b Fl(M)88 1986 y Fy(0)118 2003 y Fs(carries)c(out)g(a)g (step)h(of)e(the)i(computation)f(of)g Fl(M)5 b Fs(.)29 b(So,)19 b Fl(\016)1140 2010 y Fj(2)1178 2003 y Fs(is)g(just)f(the)h (quan)o(tum)f(transition)g(function)h Fl(\016)2010 1986 y Fy(0)39 2059 y Fs(from)14 b(QTM)h Fl(M)320 2043 y Fy(0)347 2059 y Fs(constructed)g(ab)q(o)o(v)o(e.)-19 2153 y(4.)22 b(Change)15 b(basis)g(bac)o(k)g(from)g Fl(B)j Fs(to)c Fl(Q)h Fs(while)i(stepping)f(left.)759 2284 y Fl(\016)779 2291 y Fj(3)799 2284 y Fs(\()p Fl(b;)8 b(\033)r Fs(\))i(=)967 2244 y Fi(X)962 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b(assume)g(that)f(an)h(appro) o(ximation)g(to)f(the)h(unitary)g(transformation)e(is)j(sp)q(eci\014ed) h(b)o(y)d(a)h Fl(d)s Fm(\002)s Fl(d)g Fs(complex)-75 2629 y(matrix)j(whose)h(en)o(tries)g(are)f(appro)o(ximations)h(to)f (the)h(en)o(tries)g(of)f(the)h(actual)g(unitary)g(matrix)f(corresp)q (onding)i(to)e(the)951 2779 y(28)p eop %%Page: 29 29 29 28 bop -75 -26 a Fs(desired)16 b(transformation.)j(W)l(e)c(sho)o(w)f (in)i(Theorem)f(6.4.1)e(on)i(page)g(37)f(that)h(there)g(is)g(a)g(QTM)g (whic)o(h)g(on)g(input)h Fl(\017)g Fs(and)-75 31 y(a)e Fl(d)g Fs(dimensional)i(transformation)d(whic)o(h)i(is)g(within)h (distance)1120 13 y Fk(\017)p 1054 20 147 2 v 1054 53 a Fj(2\(10)1122 25 y Fy(p)p 1150 25 19 2 v 1150 53 a Fk(d)p Fj(\))1182 44 y Fb(d)1219 31 y Fs(of)e(a)g(unitary)h (transformation)e(carries)h(out)g(a)-75 97 y(transformation)f(whic)o(h) i(is)g(an)g Fl(\017)f Fs(appro)o(ximation)h(to)f(the)g(desired)i (unitary)e(transformation.)19 b(Moreo)o(v)o(er,)12 b(the)j(running)-75 154 y(time)h(of)e(the)i(QTM)f(is)g(b)q(ounded)i(b)o(y)e(a)g(p)q (olynomial)i(in)f Fl(d)f Fs(and)g(1)p Fl(=\017)p Fs(.)-4 228 y(In)i(a)f(single)h(step,)g(a)f(QTM)g(can)g(map)g(a)g(single)i (con\014guration)f(in)o(to)f(a)g(sup)q(erp)q(osition)i(of)e(a)g(b)q (ounded)i(n)o(um)o(b)q(er)e(of)-75 284 y(con\014gurations.)j (Therefore,)13 b(in)h(order)e(to)h(carry)f(out)h(an)g(\(appro)o (ximation)f(to)g(an\))h(arbitrary)f(unitary)h(transformation)-75 341 y(on)18 b(a)f(QTM,)g(w)o(e)h(sho)o(w)f(ho)o(w)h(to)f(appro)o (ximate)g(it)h(b)o(y)g(a)f(pro)q(duct)i(of)e(simple)i(unitary)f (transformations)f(-)h(eac)o(h)g(suc)o(h)-75 397 y(simple)12 b(transformation)e(acts)g(as)h(the)g(iden)o(tit)o(y)h(in)g(all)f(but)h (t)o(w)o(o)d(dimensions.)20 b(W)l(e)11 b(then)h(sho)o(w)e(that)g(there) h(is)h(a)e(particular)-75 454 y(simple)k(unitary)f(transformation,)f (suc)o(h)h(that)f(an)o(y)h(giv)o(en)g(simple)i(transformation)c(can)i (b)q(e)h(expressed)f(as)g(a)f(pro)q(duct)h(of)-75 510 y(p)q(erm)o(utation)g(matrices)g(and)g(p)q(o)o(w)o(ers)f(of)h(this)g (\014xed)h(simple)g(matrix.)19 b(Finally)l(,)14 b(w)o(e)f(put)g(it)g (all)h(together,)e(and)h(sho)o(w)f(ho)o(w)-75 567 y(to)j(design)j(a)d (single)j(QTM)e(that)f(carries)h(out)g(an)g(arbitrary)g(unitary)g (transformation)f(-)h(this)h(QTM)e(is)i(deterministic)-75 623 y(except)f(for)e(a)h(single)i(kind)f(of)f(quan)o(tum)g (coin-\015ip.)-4 697 y(The)e(decomp)q(osition)h(of)e(an)h(arbitrary)f (unitary)h(transformation)e(in)o(to)i(a)g(pro)q(duct)g(of)f(simple)i (unitary)f(transforma-)-75 754 y(tions)f(is)h(similar)h(to)d(w)o(ork)h (carried)g(out)g(b)o(y)g(Deutsc)o(h)h([21)o(].)18 b(Deutsc)o(h's)12 b(w)o(ork,)f(although)i(phrased)g(in)g(terms)e(of)h(quan)o(tum)-75 810 y(computation)j(net)o(w)o(orks,)f(can)i(b)q(e)g(view)o(ed)g(as)f (sho)o(wing)h(that)e(a)h Fl(d)g Fs(dimensional)j(unitary)e (transformation)e(can)h(b)q(e)h(de-)-75 867 y(comp)q(osed)h(in)o(to)h 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y(computation)g(net)o(w)o(ork)f(see)i([5)o(])f(and)g(the) h(references)f(therein.)-75 1345 y Fh(6.1)56 b(Measuring)18 b(errors)g(in)h(appro)n(ximated)e(transformations)-75 1448 y Fs(In)g(this)g(section,)h(w)o(e)e(will)i(deal)g(with)f(op)q (erators)f(\(linear)h(transformations\))e(on)i(\014nite)g(dimensional)i (Hilb)q(ert)f(spaces.)-75 1505 y(It)h(is)h(often)e(con)o(v)o(enien)o(t) i(to)e(\014x)i(an)f(orthonormal)f(basis)h(for)g(the)g(Hilb)q(ert)i (space)e(and)g(describ)q(e)i(the)e(op)q(erator)f(b)o(y)h(a)-75 1561 y(\014nite)g(matrix)f(with)h(resp)q(ect)g(to)e(the)i(c)o(hosen)f (basis.)30 b(Let)18 b Fl(e)993 1568 y Fj(1)1013 1561 y Fl(;)8 b(:)g(:)g(:)d(;)j(e)1136 1568 y Fk(d)1174 1561 y Fs(b)q(e)19 b(an)f(orthonormal)g(basis)g(for)g(Hilb)q(ert)i(space)-75 1618 y Fm(H)d Fs(=)g Fn(C)70 1601 y Fk(d)90 1618 y Fs(.)27 b(Then)19 b(w)o(e)e(can)h(represen)o(t)f(an)h(op)q(erator)f Fl(U)22 b Fs(on)c Fm(H)g Fs(b)o(y)g(a)f Fl(d)11 b Fm(\002)h Fl(d)17 b Fs(complex)i(matrix)e Fl(M)5 b Fs(,)18 b(whose)f Fl(i;)8 b(j)1867 1601 y Fk(th)1918 1618 y Fs(en)o(try)-75 1674 y Fl(m)-35 1681 y Fk(i;j)20 1674 y Fs(is)16 b(\()p Fl(U)5 b(e)141 1681 y Fk(j)159 1674 y Fl(;)j(e)201 1681 y Fk(i)215 1674 y Fs(\).)20 b(The)15 b Fl(i)375 1658 y Fk(th)425 1674 y Fs(ro)o(w)g(of)g(the)g(matrix)g Fl(M)21 b Fs(is)16 b(giv)o(en)f(b)o(y)h Fl(e)1105 1681 y Fk(i)1119 1658 y(T)1146 1674 y Fl(M)5 b Fs(,)16 b(and)f(w)o(e)g(will)i(denote)f (it)f(b)o(y)h Fl(M)1765 1681 y Fk(i)1779 1674 y Fs(.)k(W)l(e)c(denote) -75 1731 y(b)o(y)h Fl(M)34 1738 y Fk(i)49 1713 y Fy(\003)86 1731 y Fs(the)g(conjugate)h(transp)q(ose)f(of)g Fl(M)679 1738 y Fk(i)693 1731 y Fs(.)27 b(The)18 b Fl(j)851 1714 y Fk(th)903 1731 y Fs(column)g(of)f Fl(M)23 b Fs(is)18 b(giv)o(en)g(b)o(y)f Fl(M)5 b(e)1489 1738 y Fk(j)1508 1731 y Fs(.)26 b Fl(U)1583 1714 y Fy(\003)1603 1731 y Fs(,)18 b(the)f(adjoin)o(t)h(of)f Fl(U)5 b Fs(,)17 b(is)-75 1787 y(represen)o(ted)f(b)o(y)g(the)g Fl(d)10 b Fm(\002)g Fl(d)16 b Fs(matrix)f Fl(M)626 1771 y Fy(\003)646 1787 y Fs(.)21 b Fl(M)g Fs(is)16 b(unitary)g(i\013)g Fl(M)5 b(M)1105 1771 y Fy(\003)1138 1787 y Fs(=)14 b Fl(M)1236 1771 y Fy(\003)1256 1787 y Fl(M)k Fs(=)c Fl(I)t Fs(.)21 b(It)16 b(follo)o(ws)g(that)f(if)h Fl(M)k Fs(is)d(unitary)-75 1843 y(then)f(the)f(ro)o(ws)f(\(and)h(columns\))h(of)f Fl(M)20 b Fs(are)15 b(orthonormal.)-4 1918 y(Recall)i(that)d(for)h(a)f (b)q(ounded)j(linear)f(op)q(erator)f Fl(U)20 b Fs(on)15 b(a)g(Hilb)q(ert)i(space)e Fm(H)p Fs(,)h(the)f(norm)g(of)g Fl(U)20 b Fs(is)15 b(de\014ned)i(as)800 2020 y Fm(j)p Fl(U)5 b Fm(j)27 b Fs(=)44 b(sup)952 2060 y Fy(k)p Fk(x)p Fy(k)o Fj(=1)1060 2020 y Fm(j)o Fl(U)5 b(x)p Fm(j)-75 2154 y Fs(If)14 b(w)o(e)g(represen)o(t)h Fl(U)k Fs(b)o(y)14 b(the)g(matrix)g Fl(M)5 b Fs(,)14 b(then)h(w)o(e)f(can)g(de\014ne)h (the)g(norm)f(of)f(the)i(matrix)f Fl(M)19 b Fs(to)13 b(b)q(e)i(same)f(as)g(the)g(norm)-75 2211 y(of)h Fl(U)5 b Fs(.)20 b(Th)o(us,)14 b(since)j(w)o(e're)d(w)o(orking)h(in)h(a)f (\014nite)h(dimensional)h(space,)769 2313 y Fm(k)p Fl(M)5 b Fm(k)27 b Fs(=)35 b(max)954 2345 y Fy(k)p Fk(v)q Fy(k)p Fj(=1)1060 2313 y Fm(k)p Fl(M)5 b(v)r Fm(k)-75 2415 y Fs(.)-75 2539 y Fn(F)l(act)18 b(6.1.1)k Fr(If)16 b Fl(U)21 b Fr(is)16 b(unitary)h(then)f Fm(k)p Fl(U)5 b Fm(k)12 b Fs(=)h Fm(k)p Fl(U)781 2522 y Fy(\003)800 2539 y Fm(k)f Fs(=)h(1)p Fr(.)951 2779 y Fs(29)p eop %%Page: 30 30 30 29 bop -75 -26 a Fn(Pro)q(of.)36 b Fm(8)p Fl(x)21 b Fm(2)h(H)p Fs(,)g Fm(k)p Fl(U)5 b(x)p Fm(k)408 -47 y Fj(2)449 -26 y Fs(=)21 b(\()p Fl(U)5 b(x;)j(U)d(x)p Fs(\))20 b(=)i(\()p Fl(x;)8 b(U)864 -42 y Fy(\003)883 -26 y Fl(U)d(x)p Fs(\))21 b(=)h(\()p Fl(x;)8 b(x)p Fs(\))19 b(=)j Fm(k)p Fl(x)p Fm(k)1298 -47 y Fj(2)1317 -26 y Fs(.)36 b(Therefore)20 b Fm(k)p Fl(U)5 b Fm(k)21 b Fs(=)h(1,)f(and)g(similar) -75 31 y(reasoning)15 b(sho)o(ws)g Fm(k)p Fl(U)317 14 y Fy(\003)336 31 y Fm(k)e Fs(=)g(1.)1532 b Fd(2)-4 161 y Fs(W)l(e)15 b(will)h(\014nd)g(it)f(useful)h(to)e(k)o(eep)h(trac)o(k)f (of)h(ho)o(w)f(far)g(our)h(appro)o(ximations)g(are)f(from)h(b)q(eing)h 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Fj(\))p 104 2078 V 122 2105 a Fy(k)p Fk(v)157 2110 y Fb(i)170 2105 y Fy(k)228 2089 y Fs(dep)q(ending)j(whether)d Fl(I)t(m)p Fs(\()p Fl(v)731 2096 y Fk(i)744 2089 y Fs(\))g(is)h(p)q (ositiv)o(e)g(or)f(negativ)o(e.)29 b(Then)19 b Fl(P)1416 2096 y Fj(1)1443 2089 y Fm(\001)8 b(\001)g(\001)e Fl(P)1533 2096 y Fk(d)1553 2089 y Fl(v)20 b Fs(is)f(the)f(v)o(ector)g(with)g Fl(i)1987 2072 y Fk(th)-75 2151 y Fs(co)q(ordinate)e Fm(k)p Fl(v)194 2158 y Fk(i)207 2151 y Fm(k)p Fs(.)-4 2225 y(Next,)e(w)o(e)g(use)i Fl(d)8 b Fm(\000)i Fs(1)15 b(rotations)e(to)i(mo)o(v)o(e)f(all)h(of)g(the)f(w)o(eigh)o(t)h(of)f (the)h(v)o(ector)f(in)o(to)h(dimension)h(1.)k(So,)14 b(w)o(e)h(let)g Fl(R)1947 2232 y Fk(i)1975 2225 y Fs(b)q(e)-75 2282 y(the)g(near-trivial)i(matrix)e(whic)o(h)h(applies)g(to)f (dimensions)i Fl(i)e Fs(and)g Fl(i)10 b Fs(+)g(1)15 b(the)g(rotation)g (b)o(y)g(angle)g Fl(\022)1641 2289 y Fk(i)1671 2282 y Fs(where)737 2395 y Fl(\022)758 2402 y Fk(i)785 2395 y Fs(=)e Fl(cos)896 2376 y Fy(\000)p Fj(1)1036 2364 y Fm(k)p Fl(v)1081 2371 y Fk(i)1095 2364 y Fm(k)p 949 2384 257 2 v 949 2393 a Fi(q)p 990 2393 215 2 v 990 2410 a(P)1034 2424 y Fk(d)1034 2454 y(j)r Fj(=)p Fk(i)1099 2443 y Fm(k)p Fl(v)1144 2450 y Fk(j)1162 2443 y Fm(k)1185 2422 y Fj(2)-75 2537 y Fs(if)j(the)f(sum)g(in)h(the)f(denominator)h(is)f(not)g(0)g(and) h Fl(\022)812 2544 y Fk(i)839 2537 y Fs(=)d(0)h(otherwise.)21 b(Then)682 2629 y Fl(R)717 2636 y Fj(1)744 2629 y Fm(\001)8 b(\001)g(\001)d Fl(R)839 2636 y Fk(d)p Fy(\000)p Fj(1)904 2629 y Fl(P)933 2636 y Fj(1)961 2629 y Fm(\001)j(\001)g(\001)d Fl(P)1050 2636 y Fk(d)1071 2629 y Fl(v)14 b Fs(=)f Fm(k)p Fl(v)r Fm(k)o Fl(e)1245 2636 y Fj(1)951 2779 y Fs(31)p eop %%Page: 32 32 32 31 bop -4 -26 a Fs(No)o(w,)14 b(instead)i(of)f(pro)q(ducing)i(these) f(v)m(alues)h Fl(\036)815 -19 y Fk(i)829 -26 y Fl(;)8 b(\022)871 -19 y Fk(i)900 -26 y Fs(exactly)l(,)16 b(w)o(e)f(can)h(in)g (time)g(p)q(olynomial)h(in)g Fl(d)e Fs(and)g(log)9 b(1)p Fl(=\017)p Fs(,)15 b(and)-75 31 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y Fj(1)p Fk(;j)910 207 y Fs(=)13 b(0)i(for)g Fl(j)g Fm(6)p Fs(=)e(1;)22 b Fl(x)1232 214 y Fk(j;l)1281 207 y Fs(=)13 b Fl(w)1362 214 y Fk(j;l)1414 207 y Fs(for)i Fl(j;)8 b(l)k(>)h(l)d Fs(+)h(1)-75 309 y(It)k(follo)o(ws)h(from)f(our)g(b)q(ounds)h(on)f(the) h(norm)f(of)g(the)g(\014rst)h(ro)o(w)e(of)h Fl(W)22 b Fs(and)15 b(on)h(the)f(en)o(tries)h(of)f(the)g(\014rst)g(column)i(of)e Fl(W)-75 366 y Fs(that)d Fm(k)p Fl(W)k Fm(\000)11 b Fl(X)t Fm(k)g(\024)i Fs(2)p Fl(\016)7 b Fs(+)f(6)387 327 y Fm(p)p 425 327 24 2 v 39 x Fl(d\016)r Fs(.)19 b(Since)c Fl(W)k Fs(is)13 b Fl(\016)r Fs(-close)h(to)e(unitary)l(,)i(w)o(e)f(can)g(then) g(conclude)i(that)d Fl(X)k Fs(is)e(3)p Fl(\016)7 b Fs(+)f(6)1830 327 y Fm(p)p 1868 327 V 39 x Fl(d)o(\016)r Fs(-close)-75 422 y(to)15 b(unitary)l(.)-4 496 y(Unfortunately)l(,)e(w)o(e)f(cannot)g (compute)h(the)g(en)o(tries)g(of)f Fl(W)19 b Fs(=)13 b Fl(U)5 b(V)1140 480 y Fy(\003)1172 496 y Fs(exactly)l(.)19 b(Instead,)14 b(app)q(ealing)g(to)e(Lemma)h(6.1.4)-75 553 y(on)18 b(page)g(30,)g(w)o(e)g(compute)g(them)g(to)f(within)j Fl(\016)r(=d)d Fs(to)h(obtain)g(a)g(matrix)1250 541 y(^)1236 553 y Fl(W)25 b Fs(suc)o(h)18 b(that)g Fm(k)1546 541 y Fs(^)1534 553 y Fl(W)e Fm(\000)10 b Fl(W)c Fm(k)17 b(\024)h Fl(\016)r Fs(.)29 b(Let's)18 b(use)-75 609 y(the)j(en)o(tries) h(of)232 598 y(^)219 609 y Fl(W)27 b Fs(to)21 b(de\014ne)h(matrix)655 598 y(^)642 609 y Fl(X)j Fs(analogous)c(to)f Fl(X)t Fs(.)37 b(Using)22 b(the)g(triangle)f(inequalit)o(y)l(,)k(it)c(is)h(easy)f(to)g (see)-75 666 y(that)c Fm(k)p Fl(W)f Fm(\000)166 654 y Fs(^)153 666 y Fl(X)t Fm(k)g(\024)h Fs(3)p Fl(\016)c Fs(+)f(6)412 627 y Fm(p)p 450 627 V 39 x Fl(d)o(\016)20 b Fs(and)617 654 y(^)603 666 y Fl(X)h Fs(is)d(4)p Fl(\016)c Fs(+)e(6)837 627 y Fm(p)p 874 627 V 874 666 a Fl(d\016)r Fs(-close)18 b(to)f(unitary)l(.)27 b(If)18 b(w)o(e)f(are)h(willing)i (to)c(incur)j(an)f(error)e(of)-75 722 y Fm(k)p Fl(W)g Fm(\000)66 711 y Fs(^)53 722 y Fl(X)s Fm(k)c(\024)h Fs(3)p Fl(\016)5 b Fs(+)s(6)286 683 y Fm(p)p 324 683 V 39 x Fl(d\016)r Fs(,)11 b(then)h(w)o(e)g(are)f(left)h(with)g(the)g(problem)g (of)g(appro)o(ximating)f(the)h Fl(k)5 b Fs(+)s(1-simple)1733 711 y(^)1720 722 y Fl(X)15 b Fs(b)o(y)c(a)h(pro)q(duct)-75 778 y(of)18 b(near-trivial)h(matrices.)29 b(Therefore,)19 b(w)o(e)f(ha)o(v)o(e)f(reduced)j(the)e(problem)h(of)f(appro)o(ximating) g(the)g Fl(k)q Fs(-simple)i(matrix)-75 835 y Fl(U)25 b Fs(b)o(y)20 b(near-trivial)h(matrices)e(to)h(the)g(problem)g(of)g (appro)o(ximating)f(the)h Fl(k)15 b Fs(+)e(1-simple)21 b(matrix)1685 823 y(^)1672 835 y Fl(X)i Fs(b)o(y)d(near-trivial)-75 891 y(matrices)15 b(while)i(incurring)g(t)o(w)o(o)c(sources)j(of)e (error:)-19 1003 y(1.)22 b(An)15 b(error)g(of)f Fm(k)p Fl(W)i Fm(\000)417 991 y Fs(^)404 1003 y Fl(X)s Fm(k)c(\024)h Fs(3)p Fl(\016)f Fs(+)f(6)652 964 y Fm(p)p 689 964 V 689 1003 a Fl(d\016)r Fs(,)k(since)h(w)o(e)f(are)g(appro)o(ximating) 1333 991 y(^)1320 1003 y Fl(X)j Fs(instead)e(of)f Fl(W)6 b Fs(.)-19 1097 y(2.)22 b(The)15 b(new)h(matrix)386 1085 y(^)373 1097 y Fl(X)j Fs(is)c(only)h(4)p Fl(\016)c Fs(+)e(6)698 1058 y Fm(p)p 736 1058 V 39 x Fl(d\016)r Fs(-close)16 b(to)e(unitary)l(.)-4 1208 y(Let)k Fl(\016)102 1192 y Fy(0)132 1208 y Fs(=)g(10)231 1170 y Fm(p)p 269 1170 V 38 x Fl(d)o(\016)r Fs(.)30 b(Clearly)19 b Fl(\016)540 1192 y Fy(0)570 1208 y Fs(is)g(an)f(upp)q(er-b)q(ound)j(on)d(b)q(oth)h (the)g(sources)f(of)g(error)g(cited)h(ab)q(o)o(v)o(e.)29 b(Therefore,)-75 1265 y(the)15 b(total)g(error)g(in)h(the)f(appro)o (ximation)h(is)f(just)794 1233 y Fi(P)838 1246 y Fk(d)838 1276 y(j)r Fj(=1)902 1265 y Fs(\(10)966 1226 y Fm(p)p 1003 1226 V 1003 1265 a Fl(d)o Fs(\))1044 1248 y Fk(j)1062 1265 y Fl(\016)g Fm(\024)e Fs(2\(10)1232 1226 y Fm(p)p 1269 1226 V 1269 1265 a Fl(d)p Fs(\))1311 1248 y Fk(d)1330 1265 y Fl(\016)r Fs(.)21 b(The)15 b(last)g(inequalit)o(y)i(follo)o(ws)f (since)-75 1329 y(10)-29 1290 y Fm(p)p 8 1290 V 8 1329 a Fl(d)f Fm(\025)f Fs(2)i(and)h(therefore)f(the)g(sum)h(can)f(b)q(e)h (b)q(ounded)h(b)o(y)e(a)g(geometric)h(series.)24 b(Therefore,)16 b(the)g(total)g(error)g(in)h(the)-75 1386 y(appro)o(ximation)e(is)h(b)q (ounded)g(b)o(y)g Fl(\017)p Fs(,)f(since)h(b)o(y)f(assumption)h Fl(U)k Fs(is)c Fl(\016)r Fs(-close)f(to)g(unitary)h(for)e Fl(\016)h Fs(=)1665 1368 y Fk(\017)p 1600 1375 147 2 v 1600 1408 a Fj(2\(10)1668 1380 y Fy(p)p 1696 1380 19 2 v 1696 1408 a Fk(d)p Fj(\))1728 1399 y Fb(d)1751 1386 y Fs(.)-4 1472 y(It)e(is)h(easy)f(to)g(see)h(that)e(this)i(algorithm)g (runs)f(in)h(time)g(p)q(olynomial)h(in)g Fl(d)e Fs(and)g(log)8 b(1)p Fl(=\017)p Fs(.)20 b(Our)13 b(algorithm)h(consists)f(of)-75 1529 y Fl(d)h Fs(iterations)g(of)g(\014rst)f(calling)j(the)e(algorithm) g(from)g(Lemma)g(6.2.2)e(on)i(page)g(31)g(to)f(compute)h Fl(V)24 b Fs(and)14 b(then)g(computing)-75 1595 y(the)f(matrix)160 1584 y(^)147 1595 y Fl(X)s Fs(.)19 b(Since)14 b(the)f(eac)o(h)g (iteration)h(tak)o(es)e(time)h(p)q(olynomial)i(in)f Fl(d)e Fs(and)h(log)1388 1573 y Fj(\(10)1438 1545 y Fy(p)p 1466 1545 V 1466 1573 a Fk(d)p Fj(\))1498 1561 y Fb(d)p 1388 1585 129 2 v 1445 1611 a Fk(\017)1521 1595 y Fs(,)g(these)g Fl(d)g Fs(calls)h(tak)o(e)e(a)h(total)-75 1652 y(time)j(p)q(olynomial)g (in)g Fl(d)f Fs(and)h(log)8 b(1)p Fl(=\017)p Fs(.)-4 1726 y(Finally)l(,)20 b(w)o(e)f(sho)o(w)f(as)g(required)i(that)e(the)h Fl(k)14 b Fs(+)e(1)883 1709 y Fk(st)933 1726 y Fs(ro)o(w)18 b(of)g Fl(W)25 b Fs(satis\014es)19 b Fm(k)p Fl(W)1384 1733 y Fk(k)q Fj(+1)1460 1726 y Fm(\000)10 b Fl(e)1526 1709 y Fk(T)1526 1739 y(k)q Fj(+1)1593 1726 y Fm(k)18 b(\024)h Fs(2)p Fl(\016)r Fs(,)g(and)g(that)f(the)-75 1782 y(en)o(tries)j(of)f(the)h Fl(k)15 b Fs(+)f(1)328 1766 y Fk(st)379 1782 y Fs(column)22 b(of)e Fl(W)26 b Fs(satisfy)21 b Fm(k)p Fl(w)874 1789 y Fk(j;k)q Fj(+1)963 1782 y Fm(k)h(\024)g Fs(6)p Fl(\016)17 b Fs(for)d Fl(j)24 b Fm(6)p Fs(=)e Fl(k)15 b Fs(+)f(1.)36 b(T)l(o)20 b(see)h(this,)h (\014rst)e(recall)i(that)-75 1839 y(the)c(lo)o(w)o(er)g(dimension)i Fl(V)28 b Fs(satis\014es)19 b Fm(k)p Fl(V)9 b(Z)665 1822 y Fk(T)662 1850 y Fj(1)703 1839 y Fm(\000)i(k)p Fl(Z)803 1846 y Fj(1)822 1839 y Fm(k)p Fl(e)866 1846 y Fj(1)886 1839 y Fm(k)17 b(\024)h Fl(\016)j Fs(where)d Fl(Z)1185 1846 y Fj(1)1223 1839 y Fs(is)h(the)f(\014rst)g(ro)o(w)g(of)g(the)g(lo) o(w)o(er)g(righ)o(t)g Fl(k)13 b Fm(\002)g Fl(k)-75 1895 y Fs(submatrix)20 b(of)f Fl(U)5 b Fs(.)33 b(Therefore,)20 b(the)f(higher)i(dimension)g Fl(V)29 b Fs(satis\014es)20 b Fm(k)p Fl(V)9 b(U)1276 1879 y Fk(T)1271 1909 y(k)q Fj(+1)1348 1895 y Fm(\000)h(k)p Fl(U)1447 1902 y Fk(k)q Fj(+1)1513 1895 y Fm(k)p Fl(e)1557 1902 y Fk(k)q Fj(+1)1624 1895 y Fm(k)19 b(\024)h Fl(\016)r Fs(.)33 b(Then,)21 b(since)-75 1952 y(1)11 b Fm(\000)h Fl(\016)17 b Fm(\024)f(k)p Fl(U)148 1959 y Fk(k)q Fj(+1)214 1952 y Fm(k)g(\024)f Fs(1)c(+)h Fl(\016)r Fs(,)17 b(it)g(follo)o(ws)g(that)f Fm(k)p Fl(V)10 b(U)830 1935 y Fk(T)825 1965 y(k)q Fj(+1)901 1952 y Fm(\000)h Fl(e)968 1959 y Fk(k)q Fj(+1)1034 1952 y Fm(k)16 b(\024)g Fs(2)p Fl(\016)r Fs(.)24 b(Therefore,)17 b(the)g Fl(k)c Fs(+)e(1)1613 1935 y Fk(st)1661 1952 y Fs(ro)o(w)16 b(of)h Fl(W)23 b Fs(satis\014es)-75 2008 y Fm(k)p Fl(W)-9 2015 y Fk(k)q Fj(+1)67 2008 y Fm(\000)11 b Fl(e)134 1992 y Fk(T)134 2021 y(k)q Fj(+1)200 2008 y Fm(k)i(\024)g Fs(2)p Fl(\016)r Fs(.)-4 2082 y(Next,)18 b(w)o(e)g(will)i(sho)o(w)d(that)h(this)g(implies)j(that)c(the)h(en)o (tries)h(of)f(the)g Fl(k)13 b Fs(+)g(1)1323 2066 y Fk(st)1372 2082 y Fs(column)19 b(of)e Fl(W)25 b Fs(satisfy)18 b Fm(k)p Fl(w)1857 2089 y Fk(j;k)q Fj(+1)1946 2082 y Fm(k)g(\024)-75 2139 y Fs(6)p Fl(\016)f Fs(for)d Fl(j)h Fm(6)p Fs(=)e Fl(k)e Fs(+)f(1.)19 b(T)l(o)c(see)g(this,)g(\014rst)g(notice)g(that)g (since)h Fl(V)24 b Fs(is)16 b(unitary)f(and)g Fl(U)20 b Fs(is)15 b(delta)h(close)f(to)f(unitary)l(,)h Fl(W)22 b Fs(is)15 b(also)-75 2201 y Fl(\016)r Fs(-close)e(to)e(unitary)l(.)20 b(This)12 b(means)g(that,)g(b)o(y)g(Statemen)o(t)f(5)h(of)g(Lemma)g (6.1.3)f(on)h(page)g(30)1483 2152 y Fi(\014)1483 2177 y(\014)1483 2202 y(\014)1497 2201 y Fl(W)1540 2208 y Fk(k)q Fj(+1)1606 2201 y Fl(W)1655 2184 y Fy(\003)1649 2213 y Fk(j)1675 2152 y Fi(\014)1675 2177 y(\014)1675 2202 y(\014)1702 2201 y Fm(\024)h Fs(2)p Fl(\016)6 b Fs(+)t(3)p Fl(\016)1883 2184 y Fj(2)1902 2201 y Fs(.)19 b(No)o(w)-75 2270 y(let)14 b(us)g(use)g(the)g(condition)h Fm(k)p Fl(W)464 2277 y Fk(k)q Fj(+1)540 2270 y Fm(\000)c Fl(e)607 2253 y Fk(T)607 2283 y(k)q Fj(+1)673 2270 y Fm(k)i(\024)g Fs(2)p Fl(\016)r Fs(.)18 b(This)d(implies)h(that)d Fm(j)o Fl(w)1231 2277 y Fk(k)q Fj(+1)p Fk(;k)q Fj(+1)1371 2270 y Fm(j)g(\025)g Fs(1)7 b Fm(\000)g Fs(2)p Fl(\016)r Fs(.)19 b(Also,)14 b(let)g(us)g(denote)g(b)o(y)-56 2322 y(^)-75 2334 y Fl(W)-32 2341 y Fk(j)-1 2334 y Fs(the)e Fl(d)t Fm(\000)t Fs(1)h(dimensional)h(ro)o(w)e(v)o(ector)f(arriv)o(ed)i (at)f(b)o(y)g(dropping)h Fl(w)1129 2341 y Fk(j;k)q Fj(+1)1231 2334 y Fs(from)f Fl(W)1379 2341 y Fk(j)1397 2334 y Fs(.)19 b(Then)13 b(the)f(condition)i(that)e Fl(W)1956 2341 y Fk(k)q Fj(+1)-75 2395 y Fs(is)18 b(close)g(to)f Fl(e)163 2378 y Fk(T)163 2408 y(k)q Fj(+1)247 2395 y Fs(also)h(implies)i(that)c Fm(k)664 2383 y Fs(^)621 2395 y Fl(W)664 2402 y Fk(k)q Fj(+1)731 2395 y Fm(k)g(\024)h Fs(2)p Fl(\016)r Fs(.)26 b(Also,)19 b(the)e(fact)g(that)g Fl(W)24 b Fs(is)18 b Fl(\016)r Fs(-close)g(to)f(unitary)h(implies)i(that)-75 2464 y Fm(k)-33 2453 y Fs(^)-52 2464 y Fl(W)-9 2471 y Fk(j)9 2464 y Fm(k)13 b(\024)g Fs(1)d(+)g Fl(\016)r Fs(.)21 b(Putting)15 b(all)i(this)e(together,)g(w)o(e)g(ha)o(v)o(e)g(2)p Fl(\016)d Fs(+)e(3)p Fl(\016)1056 2448 y Fj(2)1089 2464 y Fm(\025)1137 2416 y Fi(\014)1137 2441 y(\014)1137 2465 y(\014)1151 2464 y Fl(W)1194 2471 y Fk(k)q Fj(+1)1260 2464 y Fl(W)1309 2448 y Fy(\003)1303 2476 y Fk(j)1329 2416 y Fi(\014)1329 2441 y(\014)1329 2465 y(\014)1356 2464 y Fs(=)1404 2416 y Fi(\014)1404 2441 y(\014)1404 2465 y(\014)1418 2464 y Fl(w)1451 2471 y Fk(k)q Fj(+1)p Fk(;k)q Fj(+1)1591 2464 y Fl(w)1625 2448 y Fy(\003)1624 2478 y Fk(j;k)q Fj(+1)1724 2464 y Fs(+)1813 2453 y(^)1770 2464 y Fl(W)1813 2471 y Fk(k)q Fj(+1)1898 2453 y Fs(^)1879 2464 y Fl(W)1922 2471 y Fk(j)1940 2434 y Fy(\003)1960 2416 y Fi(\014)1960 2441 y(\014)1960 2465 y(\014)1987 2464 y Fm(\025)-75 2495 y Fi(\014)-75 2519 y(\014)-75 2544 y(\014)-61 2543 y Fl(w)-28 2550 y Fk(k)q Fj(+1)p Fk(;k)q Fj(+1)112 2543 y Fl(w)146 2527 y Fy(\003)145 2556 y Fk(j;k)q Fj(+1)235 2495 y Fi(\014)235 2519 y(\014)235 2544 y(\014)253 2543 y Fm(\000)293 2495 y Fi(\014)293 2519 y(\014)293 2544 y(\014)350 2532 y Fs(^)307 2543 y Fl(W)350 2550 y Fk(k)q Fj(+1)436 2532 y Fs(^)416 2543 y Fl(W)459 2550 y Fk(j)478 2513 y Fy(\003)497 2495 y Fi(\014)497 2519 y(\014)497 2544 y(\014)511 2543 y Fs(.)19 b(Therefore)746 2495 y Fi(\014)746 2519 y(\014)746 2544 y(\014)760 2543 y Fl(w)793 2550 y Fk(k)q Fj(+1)p Fk(;k)q Fj(+1)933 2543 y Fl(w)967 2527 y Fy(\003)966 2556 y Fk(j;k)q Fj(+1)1056 2495 y Fi(\014)1056 2519 y(\014)1056 2544 y(\014)1083 2543 y Fm(\024)13 b Fs(2)p Fl(\016)6 b Fs(+)f(3)p Fl(\016)1265 2527 y Fj(2)1289 2543 y Fs(+)1329 2495 y Fi(\014)1329 2519 y(\014)1329 2544 y(\014)1386 2532 y Fs(^)1342 2543 y Fl(W)1385 2550 y Fk(k)q Fj(+1)1471 2532 y Fs(^)1452 2543 y Fl(W)1495 2550 y Fk(j)1513 2513 y Fy(\003)1533 2495 y Fi(\014)1533 2519 y(\014)1533 2544 y(\014)1559 2543 y Fm(\024)13 b Fs(2)p Fl(\016)6 b Fs(+)f(3)p Fl(\016)1741 2527 y Fj(2)1765 2543 y Fs(+)g(2)p Fl(\016)r Fs(\(1)g(+)g Fl(\016)r Fs(\))11 b Fm(\024)-75 2619 y Fs(4)p Fl(\016)g Fs(+)e(5)p Fl(\016)68 2603 y Fj(2)88 2619 y Fs(.)19 b(Therefore)c Fm(j)p Fl(w)372 2626 y Fk(j;k)q Fj(+1)462 2619 y Fm(j)d(\024)540 2601 y Fj(4)p Fk(\016)q Fj(+5)p Fk(\016)637 2589 y Fg(2)p 540 2608 114 2 v 557 2635 a Fj(1)p Fy(\000)p Fj(2)p Fk(\016)658 2619 y Fs(.)20 b(Finally)l(,)c(since)g(w)o(e)f(ma)o(y)f(assume)g(that)g Fl(\016)h Fm(\024)e Fs(1)p Fl(=)p Fs(10,)g(w)o(e)i(ha)o(v)o(e)f Fm(j)p Fl(w)1802 2626 y Fk(j;k)q Fj(+1)1891 2619 y Fm(j)f(\024)g Fs(6)p Fl(\016)r Fs(.)951 2779 y(33)p eop %%Page: 34 34 34 33 bop -75 -26 a Fd(2)-75 170 y Fh(6.3)56 b(Carrying)18 b(out)h(near-trivial)d(transformations)-75 274 y Fs(In)i(this)f (section,)h(w)o(e)f(sho)o(w)f(ho)o(w)h(to)f(construct)h(a)f(single)j (QTM)e(that)f(can)h(carry)g(out,)f(at)h(least)g(appro)o(ximately)l(,)g (an)o(y)-75 330 y(sp)q(eci\014ed)h(near-trivial)f(transformation.)k (Since)d(a)d(near-trivial)i(transformation)e(can)h(apply)h(an)f (arbitrary)f(rotation,)-75 387 y(either)22 b(b)q(et)o(w)o(een)f(t)o(w)o (o)e(dimensions)k(or)d(in)i(the)f(phase)g(of)f(a)h(single)h(dimension,) i(w)o(e)c(m)o(ust)h(\014rst)f(sho)o(w)g(ho)o(w)h(a)f(\014xed)-75 443 y(rotation)14 b(can)g(b)q(e)h(used)g(to)f(e\016cien)o(tly)i(appro)o (ximate)e(an)h(arbitrary)e(rotation.)19 b(Note)14 b(that)g(a)g(single)i (cop)o(y)e(of)g(this)h(\014xed)-75 499 y(rotation)f(giv)o(es)i(the)f (only)h("non-classical")g(amplitudes)h(\(those)e(other)g(than)g(0)p Fl(;)8 b Fs(1\))13 b(in)j(the)g(transition)f(function)h(of)f(the)-75 556 y(univ)o(ersal)g(QTM)f(constructed)g(b)q(elo)o(w.)20 b(See)14 b(Adleman,)h(et.)e(al.)h([1])f(and)h(Solo)o(v)m(a)o(y)g(and)g 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(called)g(a)f Fr(query)i(at)g(no)n(de)f Fl(x)p Fs(.)23 b(A)16 b(query)g(whic)o(h)h(is)-75 31 y(lo)q(cated)h(in)h(the)f(same)f (recursiv)o(e)i(tree)f(as)f(no)q(de)h Fl(x)p Fs(,)h(but)e(not)h(in)h (the)e(subtree)h(ro)q(oted)g(at)f Fl(x)p Fs(,)h(is)g(called)i Fr(outside)f(of)27 b Fl(x)p Fs(.)-75 87 y(Notice)14 b(that)e(for)h(an)o (y)g(candidate)h Fl(x)p Fs(,)f(the)h(v)m(alues)g(of)f Fl(V)858 94 y Fk(O)900 87 y Fs(at)g(no)q(des)h(in)g(the)f(tree)g(ro)q (oted)g(at)g Fl(x)g Fs(and)g(the)h(decision)h(whether)-75 144 y Fl(x)g Fs(is)h(in)g Fm(R)104 151 y Fk(O)148 144 y Fs(all)h(dep)q(end)f(only)g(on)f(the)g(answ)o(ers)g(of)g(queries)h (lo)q(cated)g(at)e(no)q(des)i(in)g(the)f(tree)g(ro)q(oted)g(at)g Fl(x)p Fs(.)-75 268 y Fn(Theorem)i(8.4.3)22 b Fr(Ther)n(e)17 b(is)g(an)g(or)n(acle)h(QTM)e Fl(M)23 b Fr(such)17 b(that)h(for)g (every)g(le)n(gal)e(or)n(acle)h Fl(O)q Fr(,)h Fl(M)1600 251 y Fk(O)1647 268 y Fr(runs)f(in)g(p)n(olynomial)-75 324 y(time)f(and)h(ac)n(c)n(epts)e(the)i(language)f Fm(R)573 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Fs(\))-75 748 y(can)f(b)q(e)h(found)g(b)o(y)f(querying)h(the)f(string)g Fl(x)p Fs($.)20 b(If)15 b Fl(x)g Fs(is)h(a)f(no)q(de)h(at)e(lev)o(el)j Fl(l)c Fm(\025)g Fs(0,)i(then)g(calculate)h Fl(V)1621 755 y Fk(O)1666 748 y Fs(as)f(follo)o(ws.)-19 872 y(1.)22 b(Split)16 b(in)o(to)f(an)h(equal)f(sup)q(erp)q(osition)i(of)e(the)g(2) 856 856 y Fj(2)874 844 y Fb(l)903 872 y Fs(c)o(hildren)i(of)e Fl(x)p Fs(.)-19 966 y(2.)22 b(Recursiv)o(ely)17 b(compute)e Fl(V)492 973 y Fk(O)537 966 y Fs(for)f(these)i(c)o(hildren)h(in)f(sup)q (erp)q(osition.)-19 1060 y(3.)22 b(Apply)16 b(phase)g(to)e(eac)o(h)h(c) o(hild)i(giv)o(en)f(b)o(y)f(that)f(c)o(hild's)j(v)m(alue)f Fl(V)1134 1067 y Fk(O)1164 1060 y Fs(.)-19 1153 y(4.)22 b(Rev)o(erse)15 b(the)h(computation)f(of)g(Step)g(2)g(to)f(erase)i(the) f(v)m(alue)h Fl(V)1135 1160 y Fk(O)1165 1153 y Fs(.)-19 1247 y(5.)22 b(Apply)17 b(the)f(F)l(ourier)g(transform)e(con)o(v)o (erting)i(the)g(sup)q(erp)q(osition)i(of)d(c)o(hildren)j(of)e Fl(x)g Fs(in)o(to)f(a)h(sup)q(erp)q(osition)i(con-)39 1304 y(sisting)e(en)o(tirely)g(of)f(the)g(single)h(string)g Fl(k)755 1311 y Fk(x)776 1304 y Fs(.)-19 1398 y(6.)22 b(Query)15 b Fl(x)p Fs($)p Fl(k)249 1405 y Fk(x)286 1398 y Fs(to)f(\014nd)i(the)g(v)m(alue)g Fl(V)655 1405 y Fk(O)700 1398 y Fs(for)e Fl(x)p Fs(.)-19 1491 y(7.)22 b(Rev)o(erse)15 b(Steps)h(1-5)f(to)f(erase)h Fl(k)597 1498 y Fk(x)634 1491 y Fs(\(lea)o(ving)h(only)f Fl(x)g Fs(and)h Fl(V)1062 1498 y Fk(O)1091 1491 y Fs(\()p Fl(x)p Fs(\)\).)-4 1615 y(Notice)c(that)f(the)g(n)o(um)o(b)q(er)h(of)f(steps)h(required)h(in)f (the)g(iteration)g(ob)q(eys)f(the)h(recursion)g(discussed)i(ab)q(o)o(v) o(e,)d(and)h(hence)-75 1672 y(is)17 b(p)q(olynomial)g(in)g(the)g (length)f(of)g Fl(x)p Fs(.)23 b(So,)15 b(w)o(e)h(can)h(use)f(the)g (Sync)o(hronization)i(Theorem)d(on)i(page)f(17)f(to)g(construct)h(a)-75 1728 y(p)q(olynomial)j(time)f(QTM)g(whic)o(h,)g(for)f(an)o(y)h (particular)g Fl(x)p Fs(,)g(writes)f(a)h(list)g(of)f(the)h(steps)g 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b(w)o(e)f(ha)o(v)o(e)g(stationary)l(,)g(normal)g(form)f (QTMs)h(to)g(handle)h(eac)o(h)f(step)h(at)e(lev)o(el)i Fl(l)g Fs(and)f(its)h(rev)o(erse)f(in)h(time)-75 2554 y(b)q(ounded)d(b)o(y)f(a)f(p)q(olynomial)i(in)f(2)524 2538 y Fk(l)537 2554 y Fs(,)f(w)o(e)h(can)f(use)h(the)g(Branc)o(hing)g (Lemma)g(to)f(construct)g(a)h(stationary)l(,)e(normal)i(form)-75 2611 y(QTM)h(to)f(carry)h(out)g(an)o(y)g(sp)q(eci\014ed)i(step)e(of)g (the)g(computation.)25 b(Do)o(v)o(etailing)18 b(with)f(a)g(mac)o(hine)h (whic)o(h)g(rotates)e(the)951 2779 y(49)p eop %%Page: 50 50 50 49 bop -75 -26 a Fs(\014rst)15 b(step)f(in)i(the)f(list)h(to)e(the)h (end,)g(and)g(inserting)h(the)f(resulting)h(mac)o(hine)g(in)o(to)f(the) g(rev)o(ersible)h(TM)f(of)f(the)h(Lo)q(oping)-75 31 y(Lemma)g(giv)o(es) h(the)f(desired)h(QTM.)1434 b Fd(2)-4 161 y Fs(Computing)16 b(the)g(function)h Fl(V)519 168 y Fk(O)564 161 y Fs(tak)o(es)f(time)g (\012\()p Fl(n)864 145 y Fj(log)5 b Fk(n)939 161 y Fs(\))15 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y Fk(O)1961 387 y Fs(\()p Fl(x)p Fs(\))-75 444 y(is)h Fr(\014xe)n(d)g(by)g(c)n(onstr)n(aint)p Fs(.)25 b(An)18 b(example)g(of)f(this)h(is)g(that)f(if)h(w)o(e)f(ask)o(ed)g(all)h(of)f (the)h(queries)g(in)g(the)g(subtrees)g(ro)q(oted)f(at)-75 500 y(all)g(of)f(the)g(siblings)i(of)d Fl(x)p Fs(,)h(then)h(w)o(e)e(ha) o(v)o(e)h(\014xed)h Fl(V)807 507 y Fk(O)852 500 y Fs(for)e(all)i(of)f (the)g(siblings,)i(thereb)o(y)e(\014xing)h(the)f(string)g Fl(k)h Fs(suc)o(h)g(that)-75 557 y Fl(V)-48 564 y Fk(O)-19 557 y Fs(\()p Fl(x)25 564 y Fj(1)45 557 y Fs($)8 b Fl(:)g(:)g(:)d Fs($)p Fl(x)185 564 y Fk(m)218 557 y Fs($)p Fl(y)r Fs(\))14 b(equals)i(\()p Fm(\000)p Fs(1\))529 540 y Fk(y)q Fy(\001)p Fk(k)578 557 y Fs(.)-4 631 y(The)f(only)h(other)g(w)o(a)o(y)e(that)h (the)g(v)m(alue)i(of)e(the)h(no)q(de)g Fl(x)f Fs(migh)o(t)h(b)q(e)g (\014xed)g(is)g(the)g(follo)o(wing.)21 b(If)16 b(the)g(queries)g(\014x) g(the)-75 687 y(v)m(alue)i(of)e Fl(V)124 694 y Fk(O)170 687 y Fs(for)g(some)h(of)f(the)h(c)o(hildren)h(of)f Fl(x)f Fs(then)h(this)h(will)g(restrict)f(the)f(p)q(ossible)j(v)m(alues)f(for) e(the)h(string)f Fl(k)1896 694 y Fk(x)1935 687 y Fs(suc)o(h)-75 744 y(that)d Fl(V)49 751 y Fk(O)78 744 y Fs(\()p Fl(x)p Fs($)p Fl(y)r Fs(\))f(alw)o(a)o(ys)h(equals)h(\()p Fm(\000)p Fs(1\))573 727 y Fk(y)q Fy(\010)p Fk(k)636 731 y Fb(x)659 744 y Fs(,)f(and)h(suc)o(h)f(that)g Fl(V)996 751 y Fk(O)1026 744 y Fs(\()p Fl(x)p Fs(\))e(=)i Fl(O)q Fs(\()p Fl(x)p Fs($)p Fl(k)1274 751 y Fk(x)1295 744 y Fs(\).)19 b(If)14 b(the)g(query)f Fl(x)p Fs($)p Fl(k)1663 751 y Fk(x)1698 744 y Fs(for)g(eac)o(h)g(p)q(ossible)-75 800 y Fl(k)-51 807 y Fk(x)-14 800 y Fs(has)i(b)q(een)i(ask)o(ed)e(and)g(all)i(ha)o(v)o (e)e(the)g(same)g(answ)o(ers,)f(then)i(this)g(\014xes)g(the)f(v)m(alue) i Fl(V)1448 807 y Fk(O)1492 800 y Fs(at)e Fl(x)p Fs(.)20 b(If)c(the)f(query)h Fl(x)p Fs($)p Fl(k)1931 807 y Fk(x)1968 800 y Fs(for)-75 857 y(the)f(correct)g Fl(k)179 864 y Fk(x)216 857 y Fs(has)g(b)q(een)h(ask)o(ed,)f(then)g(w)o(e)g(call)i Fl(x)e Fs(a)g 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(deterministic)i(TM.)d(So,)h(w)o(e)f(will)j(argue)d(not)h(just)g(that)f (it)h(tak)o(es)f Fl(n)1416 1158 y Fj(\012\(log)6 b Fk(n)q Fj(\))1561 1174 y Fs(queries)18 b(to)f(\014x)g(the)g(v)m(alue)-75 1231 y(of)e(a)g(candidate)h(of)f(length)h Fl(n)p Fs(,)f(but)g(that)g (if)g(few)o(er)g(than)g Fl(n)943 1214 y Fj(\012\(log)6 b Fk(n)q Fj(\))1087 1231 y Fs(queries)16 b(are)f(\014xed,)g(then)h(c)o (ho)q(osing)f(a)g(random)g(legal)-75 1287 y(oracle)f(consisten)o(t)f (with)h(those)f(queries)h(giv)o(es)g(to)f(a)g(candidate)h(of)f(length)h Fl(n)g Fs(the)f(v)m(alue)i(1)e(with)h(probabilit)o(y)g(extremely)-75 1344 y(close)k(to)e(1)p Fl(=)p Fs(2.)24 b(This)18 b(will)h(giv)o(e)e (the)g(desired)h(result.)26 b(T)l(o)17 b(see)g(this,)h(call)g(the)f (set)g(of)f(queries)i(actually)g(ask)o(ed)f(and)g(the)-75 1400 y(answ)o(ers)d(giv)o(en)i(to)e(those)h(queries)h(a)f Fr(run)k Fs(of)14 b(the)h(probabilistic)j(TM.)c(W)l(e)h(will)i(ha)o(v)o (e)d(sho)o(wn)h(that)f(if)i(w)o(e)f(tak)o(e)f(an)o(y)h(run)-75 1456 y(on)d(a)h(candidate)g(of)f(length)h Fl(n)g Fs(with)g(few)o(er)f (than)g Fl(n)797 1440 y Fj(\012\(log)6 b Fk(n)p Fj(\))938 1456 y Fs(queries,)13 b(then)g(the)g(probabilit)o(y)g(that)f(a)g (random)g(legal)i(oracle)-75 1513 y(agreeing)j(with)f(the)h(run)f (assigns)h(to)e Fl(x)i Fs(the)f(v)m(alue)i(1)e(is)h(extremely)g(close)g (to)e(1)p Fl(=)p Fs(2.)23 b(This)17 b(means)f(that)g(a)g(probabilistic) -75 1569 y(TM)e(whose)g(running)h(time)f(is)h Fl(n)489 1553 y Fk(o)p Fj(\(log)6 b Fk(n=)p Fj(2\))658 1569 y Fs(will)16 b(fail)f(with)f(probabilit)o(y)i(1)e(to)f(accept)i Fm(R)1424 1576 y Fk(O)1467 1569 y Fs(for)e(a)h(random)g(legal)h(oracle) f Fl(O)q Fs(.)-75 1682 y Fn(De\014nition)19 b(8.4.4)j Fr(A)15 b Fs(run)e(of)g(size)i Fl(k)h Fr(is)e(de\014ne)n(d)g(as)g(a)h (p)n(air)f Fl(S;)8 b(f)19 b Fr(wher)n(e)c Fl(S)i Fr(is)d(a)h(set)f(of)g Fl(k)i Fr(query)f(strings)e(and)i Fl(f)k Fr(is)14 b(map)-75 1739 y(fr)n(om)i Fl(S)j Fr(to)e Fm(f)p Fs(0)p Fl(;)8 b Fs(1)p Fm(g)14 b Fr(such)j(that)f(ther)n(e)h(is)f(at)g(le)n(ast)g (one)f(le)n(gal)h(or)n(acle)g(agr)n(e)n(eing)f(with)i Fl(f)5 b Fr(.)-4 1813 y(L)n(et)16 b Fl(r)j Fr(b)n(e)e(a)h(run,)g(let)f Fl(y)i Fr(b)n(e)e(a)h(no)n(de)f(at)h(level)f Fl(l)e Fm(\025)h Fs(2)p Fr(,)h(and)h(let)f Fl(O)i Fr(b)n(e)e(a)h(le)n(gal)e(or)n(acle)i (at)g(and)f(b)n(elow)g Fl(y)j Fr(which)e(agr)n(e)n(es)-75 1869 y(with)e Fl(r)q Fr(.)21 b(Then)15 b Fl(O)i Fr(determines)e(the)h (string)f Fl(k)709 1876 y Fk(y)746 1869 y Fr(for)h(which)g Fl(V)972 1876 y Fk(O)1002 1869 y Fs(\()p Fl(y)r Fs(\))11 b(=)i Fl(O)q Fs(\()p Fl(y)r Fs($)p Fl(k)1246 1876 y Fk(y)1266 1869 y Fs(\))p Fr(.)20 b(If)c Fl(y)r Fs($)p Fl(k)1437 1876 y Fk(y)1473 1869 y Fr(is)f(a)h(query)g(in)g Fl(r)q Fr(,)f(then)h(we)g(say)-75 1926 y Fl(O)h Fs(mak)o(es)d Fl(y)j Fs(a)d(hit)h(for)h Fl(r)q Fr(.)k(Suppr)n(essing)15 b(the)h(dep)n(endency)f(on)h Fl(r)g Fr(in)g(the)g(notation,)g(we)g (de\014ne)f Fl(P)6 b Fs(\()p Fl(y)r Fs(\))15 b Fr(as)h(the)g(pr)n(ob)n (ability)-75 1982 y(that)h Fl(y)i Fr(is)d(a)h(hit)f(for)h Fl(r)h Fr(when)e(we)h(cho)n(ose)f(a)h(le)n(gal)e(or)n(acle)i(at)f(and)h (b)n(elow)f Fl(y)j Fr(uniformly)d(at)h(r)n(andom)g(fr)n(om)g(the)f(set) g(of)h(al)r(l)-75 2039 y(such)h(or)n(acle)f(at)i(and)e(b)n(elow)h Fl(y)i Fr(which)e(agr)n(e)n(e)f(with)h Fl(r)q Fr(.)26 b(Similarly,)17 b(for)h Fl(x)g Fr(an)g(anc)n(estor)f(of)h Fl(y)r Fr(,)g(we)g(de\014ne)f Fl(P)1804 2046 y Fk(x)1826 2039 y Fs(\()p Fl(y)r Fs(\))g Fr(as)h(the)-75 2095 y(pr)n(ob)n(ability) d(that)h Fl(y)h Fr(is)d(a)i(hit)f(when)g(a)h(le)n(gal)e(or)n(acle)h(is) f(chosen)h(at)g(and)g(b)n(elow)g Fl(x)g Fr(uniformly)g(at)h(r)n(andom)f (fr)n(om)h(the)f(set)g(of)-75 2152 y(al)r(l)h(or)n(acle)g(at)h(and)f(b) n(elow)g Fl(x)g Fr(which)h(agr)n(e)n(e)e(with)i Fl(r)q Fr(.)-75 2264 y Fn(Lemma)g(8.4.5)23 b Fl(P)270 2271 y Fk(x)292 2264 y Fs(\()p Fl(y)r Fs(\))12 b Fm(\024)h Fs(2)p Fl(P)6 b Fs(\()p Fl(y)r Fs(\))-75 2377 y Fn(Pro)q(of.)19 b Fs(Let)14 b Fl(S)i Fs(b)q(e)f(the)e(set)h(of)f(legal)h(oracles)g(at)f (and)h(b)q(elo)o(w)g Fl(y)i Fs(whic)o(h)e(agree)g(with)g Fl(r)q Fs(.)19 b(W)l(e)14 b(can)f(write)h Fl(S)i Fs(as)e(the)f(disjoin) o(t)-75 2434 y(union)20 b(of)e Fl(S)138 2441 y Fk(h)179 2434 y Fs(and)h Fl(S)299 2441 y Fk(n)342 2434 y Fs(where)g(the)g (former)f(is)h(the)g(set)f(of)h(those)f(oracles)h(in)h Fl(S)h Fs(that)e(mak)o(e)f Fl(y)j Fs(a)d(hit)h(for)g Fl(r)q Fs(.)30 b(F)l(urther)-75 2490 y(splitting)13 b(according)g(to)f (the)g(v)m(alue)h Fl(V)572 2497 y Fk(O)602 2490 y Fs(\()p Fl(y)r Fs(\),)e(w)o(e)h(can)g(write)h Fl(S)h Fs(as)e(the)g(disjoin)o(t) h(union)g(of)f(four)g(sets,)g Fl(S)1664 2497 y Fk(h)p Fj(+)1713 2490 y Fl(;)c(S)1762 2497 y Fk(h)p Fy(\000)1811 2490 y Fl(;)g(S)1860 2497 y Fk(n)p Fj(+)1910 2490 y Fl(;)g(S)1959 2497 y Fk(n)p Fy(\000)2009 2490 y Fs(.)-75 2559 y(Using)17 b(this)h(notation,)e(w)o(e)g(ha)o(v)o(e)h Fl(P)6 b Fs(\()p Fl(y)r Fs(\))15 b(=)773 2537 y(card)p Fj(\()p Fk(S)894 2543 y Fb(h)915 2537 y Fj(\))p 681 2549 339 2 v 681 2582 a Fs(card)p Fj(\()p Fk(S)802 2588 y Fb(h)823 2582 y Fj(\)+)p Fs(card)p Fj(\()p Fk(S)985 2586 y Fb(n)1006 2582 y Fj(\))1025 2559 y Fs(.)25 b(It)16 b(is)i(easy)e(to)g(see)h(that,)g(since)h(the)e (oracles)h(in)h Fl(S)1934 2566 y Fk(n)1974 2559 y Fs(do)-75 2629 y(not)d(mak)o(e)f Fl(y)k Fs(a)c(hit)i(for)f Fl(r)q Fs(,)f(card\()p Fl(S)523 2636 y Fk(n)p Fj(+)574 2629 y Fs(\))e(=)h(card\()p Fl(S)784 2636 y Fk(n)p Fy(\000)834 2629 y Fs(\))g(=)g(card\()p Fl(S)1045 2636 y Fk(n)1068 2629 y Fs(\))o Fl(=)p Fs(2.)951 2779 y(50)p eop %%Page: 51 51 51 50 bop -4 -26 a Fs(Next)18 b(consider)h(the)f(set)g Fl(T)24 b Fs(of)18 b(all)h(legal)g(oracles)f(de\014ned)i(at)e(and)g(b)q (elo)o(w)h Fl(x)p Fs(,)f(but)h(outside)f Fl(y)r Fs(,)h(whic)o(h)g (agree)f(with)-75 31 y Fl(r)q Fs(.)37 b(Eac)o(h)21 b(oracle)g Fl(O)i Fm(2)f Fl(T)27 b Fs(determines)22 b(b)o(y)f(constrain)o(t)g(the) g(v)m(alue)h Fl(V)1176 38 y Fk(O)1205 31 y Fs(\()p Fl(y)r Fs(\),)f(but)g(lea)o(v)o(es)g(the)g(string)g Fl(k)1767 38 y Fk(y)1809 31 y Fs(completely)-75 87 y(undetermined.)h(If)16 b(w)o(e)f(again)h(write)f Fl(T)22 b Fs(as)15 b(the)g(disjoin)o(t)h (union)h(of)e Fl(T)1133 94 y Fj(+)1177 87 y Fs(and)h Fl(T)1293 94 y Fy(\000)1337 87 y Fs(according)g(to)f(the)h(constrained) g(v)m(alue)-75 144 y Fl(V)-48 151 y Fk(O)-19 144 y Fs(\()p Fl(y)r Fs(\),)d(w)o(e)g(notice)g(that)g(the)g(set)g(of)g(legal)h (oracles)f(at)g(and)g(b)q(elo)o(w)h Fl(x)f Fs(is)h(exactly)f(\()p Fl(T)1355 151 y Fj(+)1394 144 y Fm(\002)d Fl(S)1467 151 y Fj(+)1497 144 y Fs(\))5 b Fm([)h Fs(\()p Fl(T)1601 151 y Fy(\000)1641 144 y Fm(\002)k Fl(S)1714 151 y Fy(\000)1744 144 y Fs(\))o(.)19 b(So,)13 b(w)o(e)g(ha)o(v)o(e)236 264 y Fl(P)265 271 y Fk(x)287 264 y Fs(\()p Fl(y)r Fs(\))41 b(=)470 233 y(card\()p Fl(T)601 240 y Fj(+)629 233 y Fs(\)card\()p Fl(S)779 240 y Fk(h)p Fj(+)828 233 y Fs(\))10 b(+)h(card\()p Fl(T)1033 240 y Fy(\000)1061 233 y Fs(\)card\()p Fl(S)1211 240 y Fk(h)p Fy(\000)1261 233 y Fs(\))p 470 253 809 2 v 490 295 a(card\()p Fl(T)621 302 y Fj(+)650 295 y Fs(\))o(card\()p Fl(S)799 302 y Fj(+)828 295 y Fs(\))f(+)h(card\()p Fl(T)1033 302 y Fy(\000)1061 295 y Fs(\)card\()p Fl(S)1211 302 y Fy(\000)1240 295 y Fs(\))388 388 y(=)684 357 y(card\()p Fl(T)815 364 y Fj(+)843 357 y Fs(\)card\()p Fl(S)993 364 y Fk(h)p Fj(+)1042 357 y Fs(\))f(+)h(card\()p Fl(T)1247 364 y Fy(\000)1275 357 y Fs(\)card\()p Fl(S)1425 364 y Fk(h)p Fy(\000)1474 357 y Fs(\))p 470 377 1237 2 v 470 419 a(card\()p Fl(T)601 426 y Fj(+)629 419 y Fs(\)card\()p Fl(S)779 426 y Fk(h)p Fj(+)828 419 y Fs(\))f(+)h(card\()p Fl(T)1033 426 y Fy(\000)1061 419 y Fs(\)card\()p Fl(S)1211 426 y Fk(h)p Fy(\000)1261 419 y Fs(\))e(+)i(card\()p Fl(T)6 b Fs(\))o(card\()p Fl(S)1620 426 y Fk(n)1643 419 y Fs(\))p Fl(=)p Fs(2)-75 509 y(Without)15 b(loss)h(of)f(generalit)o(y)l(,)h(let)g(card\()p Fl(T)663 516 y Fj(+)692 509 y Fs(\))d Fm(\025)g Fs(card\()p Fl(T)902 516 y Fy(\000)931 509 y Fs(\).)20 b(Then)c(since)h Fl(n=)p Fs(\()p Fl(n)10 b Fs(+)h Fl(c)p Fs(\))k(with)h Fl(c;)8 b(n)k(>)i Fs(0)h(increases)h(with)g Fl(n)p Fs(,)-75 565 y(w)o(e)f(ha)o(v)o(e)454 674 y Fl(P)483 681 y Fk(x)505 674 y Fs(\()p Fl(y)r Fs(\))41 b Fm(\024)902 643 y Fs(card\()p Fl(T)1033 650 y Fj(+)1062 643 y Fs(\))o(card\()p Fl(S)1211 650 y Fk(h)1233 643 y Fs(\))p 688 664 777 2 v 688 705 a(card\()p Fl(T)819 712 y Fj(+)848 705 y Fs(\))o(card\()p Fl(S)997 712 y Fk(h)1019 705 y Fs(\))10 b(+)g(card\()p Fl(T)c Fs(\)card\()p Fl(S)1379 712 y Fk(n)1402 705 y Fs(\))o Fl(=)p Fs(2)606 798 y Fm(\024)913 767 y Fs(card\()p Fl(T)1044 774 y Fj(+)1073 767 y Fs(\)card\()p Fl(S)1223 774 y Fk(h)1245 767 y Fs(\))p 688 788 801 2 v 688 829 a(card\()p Fl(T)819 836 y Fj(+)848 829 y Fs(\))o(card\()p Fl(S)997 836 y Fk(h)1019 829 y Fs(\))k(+)g(card\()p Fl(T)1223 836 y Fj(+)1252 829 y Fs(\)card\()p Fl(S)1402 836 y Fk(n)1425 829 y Fs(\))p Fl(=)p Fs(2)606 922 y(=)802 891 y(2card\()p Fl(S)957 898 y Fk(h)979 891 y Fs(\))p 688 912 423 2 v 688 953 a(2card\()p Fl(S)843 960 y Fk(h)864 953 y Fs(\))g(+)h(card\()p Fl(S)1070 960 y Fk(n)1093 953 y Fs(\))1143 922 y Fm(\024)28 b Fs(2)p Fl(P)6 b Fs(\()p Fl(y)r Fs(\))1988 1029 y Fd(2)-4 1160 y Fs(F)l(or)14 b(a)h(p)q(ositiv)o(e)h(in)o(teger)g Fl(n)p Fs(,)f(w)o(e)g(de\014ne)h Fl(\015)s Fs(\()p Fl(n)p Fs(\))11 b(=)i Fl(n)p Fs(\()p Fl(n=)p Fs(2\))8 b Fm(\001)g(\001)g(\001) d Fs(1.)19 b(Notice)d(that)e Fl(\015)s Fs(\()p Fl(n)p Fs(\))e Fl(>)h(n)1513 1143 y Fj(\(log)5 b Fk(n)p Fj(\))p Fk(=)p Fj(2)1651 1160 y Fs(.)-75 1277 y Fn(Theorem)17 b(8.4.6)22 b Fr(Supp)n(ose)16 b Fl(r)h Fr(is)f(a)h(run,)f Fl(y)i Fr(is)e(a)g(no)n(de)g(at)h(level)e Fl(l)e Fm(\025)g Fs(2)j Fr(with)h Fl(q)h Fr(queries)e(fr)n(om)h Fl(r)g Fr(at)f(or)h(b)n(elow)f Fl(y)r Fr(,)g(and)g Fl(x)-75 1333 y Fr(is)g(an)g(anc)n(estor)f(of)i Fl(y)r Fr(.)k(Then)15 b Fl(P)479 1340 y Fk(x)501 1333 y Fs(\()p Fl(y)r Fs(\))d Fm(\024)h Fl(q)r(=\015)s Fs(\()p Fl(n=)p Fs(4\))h Fr(wher)n(e)j Fl(n)12 b Fs(=)h(2)1055 1317 y Fk(l)1068 1333 y Fr(.)-75 1451 y Fn(Pro)q(of.)20 b Fs(W)l(e)15 b(pro)o(v)o(e)g(the)g(theorem)g(b) o(y)g(induction)i(on)e Fl(l)q Fs(.)-4 1525 y(So,)g(\014x)h(a)f(run)h Fl(r)g Fs(and)g(a)f(no)q(de)i Fl(y)g Fs(at)e(lev)o(el)i(2)e(with)h Fl(q)i Fs(queries)e(from)f Fl(r)i Fs(at)e(or)g(b)q(elo)o(w)h Fl(y)r Fs(.)21 b(If)16 b Fl(q)f Fs(=)f(0,)h Fl(y)i Fs(can)f(nev)o(er)g (b)q(e)g(a)-75 1581 y(hit.)k(So,)15 b(certainly)h(the)g(probabilit)o(y) g(that)f Fl(y)i Fs(is)e(a)g(hit)h(is)g(at)e(most)h Fl(q)i Fs(as)d(desired.)-4 1656 y(Next,)i(w)o(e)g(p)q(erform)g(the)g(inductiv) o(e)i(step.)24 b(So,)16 b(assume)g(the)g(theorem)g(holds)h(true)g(for)e (an)o(y)h Fl(r)h Fs(and)g Fl(y)h Fs(at)e(lev)o(el)h(less)-75 1712 y(than)h Fl(l)h Fs(with)g Fl(l)g Fm(\025)f Fs(2.)30 b(Then,)19 b(\014x)g(a)f(run)h Fl(r)g Fs(and)g(a)f(no)q(de)h Fl(y)i Fs(at)d(lev)o(el)h Fl(l)g Fs(with)g Fl(q)i Fs(queries)e(from)f Fl(r)h Fs(at)f(or)g(b)q(elo)o(w)h Fl(y)r Fs(.)30 b(Let)-75 1769 y Fl(n)13 b Fs(=)g(2)36 1752 y Fk(l)49 1769 y Fs(.)19 b(W)l(e)14 b(will)i(sho)o(w)d(that)g Fl(P)6 b Fs(\()p Fl(y)r Fs(\))13 b Fm(\024)664 1748 y Fk(q)p 612 1758 122 2 v 612 1785 a Fj(2)p Fk(\015)r Fj(\()p Fk(n=)p Fj(4\))739 1769 y Fs(,)h(and)g(then)g(the)g(theorem)g(will)h(follo)o(w)g(from)e (Lemma)h(8.4.5.)k(So,)c(for)f(the)-75 1825 y(remainder)h(of)e(the)h (pro)q(of,)f(all)i(probabilities)h(are)e(tak)o(en)f(o)o(v)o(er)g(the)h (c)o(hoice)h(of)e(a)h(legal)h(oracle)f(at)f(and)h(b)q(elo)o(w)g Fl(y)i Fs(uniformly)-75 1881 y(at)g(random)f(from)h(the)g(set)g(of)g (all)h(those)f(legal)h(oracles)f(at)g(and)g(b)q(elo)o(w)h Fl(y)h Fs(whic)o(h)f(agree)f(with)h Fl(r)q Fs(.)-4 1956 y(No)o(w,)j(supp)q(ose)h(that)e Fl(q)416 1939 y Fy(0)447 1956 y Fs(of)h(the)g Fl(q)i Fs(queries)f(are)f(actually)h(at)f Fl(y)r Fs(.)31 b(Clearly)l(,)21 b(if)f(w)o(e)f(condition)h(on)f(there)h (b)q(eing)g(no)-75 2012 y(hits)g(among)f(the)h(c)o(hildren)i(of)d Fl(y)r Fs(,)h(then)g Fl(k)675 2019 y Fk(y)715 2012 y Fs(will)i(b)q(e)e(c)o(hosen)g(uniformly)h(among)e(all)h Fl(n)p Fs(-bit)h(strings,)f(and)g(hence)h(the)-75 2069 y(probabilit)o(y)c Fl(y)h Fs(is)f(a)f(hit)g(w)o(ould)h(b)q(e)g Fl(q)574 2052 y Fy(0)585 2069 y Fl(=)p Fs(2)631 2052 y Fk(n)654 2069 y Fs(.)23 b(If)16 b(w)o(e)g(instead)h(condition)g(on)f (there)g(b)q(eing)i(exactly)e Fl(c)g Fs(hits)g(among)g(the)g(2)1999 2052 y Fk(n)-75 2125 y Fs(c)o(hildren)h(of)f Fl(y)r Fs(,)f(then)h(the)f (probabilit)o(y)i(that)e Fl(y)j Fs(is)e(a)f(hit)h(m)o(ust)f(b)q(e)h(at) f(most)g Fl(q)1277 2109 y Fy(0)1288 2125 y Fl(=)p Fs(2)1334 2109 y Fk(n)p Fy(\000)p Fk(c)1400 2125 y Fs(.)21 b(Therefore)16 b(the)f(probabilit)o(y)i Fl(y)g Fs(is)-75 2181 y(a)d(hit)h(is)g(b)q (ounded)h(ab)q(o)o(v)o(e)e(b)o(y)g(the)h(sum)f(of)g Fl(q)698 2165 y Fy(0)710 2181 y Fl(=)p Fs(2)756 2165 y Fk(n=)p Fj(2)829 2181 y Fs(and)g(the)h(probabilit)o(y)h(that)d(at)h(least)h Fl(n=)p Fs(2)f(of)g(the)h(c)o(hildren)h(of)e Fl(y)i Fs(are)-75 2238 y(hits.)-4 2312 y(No)o(w)g(consider)i(an)o(y)e(c)o(hild)j Fl(z)g Fs(of)d Fl(y)r Fs(.)25 b(Applying)18 b(the)f(inductiv)o(e)i(h)o (yp)q(othesis)f(with)f Fl(y)i Fs(and)e Fl(z)i Fs(taking)e(the)f(roles)i (of)e Fl(x)-75 2369 y Fs(and)h Fl(y)r Fs(,)f(w)o(e)h(kno)o(w)f(that)g (if)h Fl(r)g Fs(has)g Fl(q)541 2376 y Fk(z)578 2369 y Fs(queries)g(at)f(and)h(b)q(elo)o(w)g Fl(z)r Fs(,)g(then)g Fl(P)1196 2376 y Fk(y)1217 2369 y Fs(\()p Fl(z)r Fs(\))e Fm(\024)g Fl(q)1361 2376 y Fk(z)1381 2369 y Fl(=\015)s Fs(\()p Fl(n=)p Fs(8\).)22 b(Therefore)17 b(the)g Fr(exp)n(e)n(cte)n(d) -75 2425 y Fs(n)o(um)o(b)q(er)f(of)g(hits)h(among)e(the)h(c)o(hildren)i (of)e Fl(y)i Fs(is)e(at)g(most)f(\()p Fl(q)d Fm(\000)f Fl(q)1060 2409 y Fy(0)1072 2425 y Fs(\))p Fl(=\015)s Fs(\()p Fl(n=)p Fs(8\).)20 b(This)d(means)f(that)f(the)h(probabilit)o (y)i(that)-75 2481 y(at)d(least)g Fl(n=)p Fs(2)g(of)g(the)g(c)o (hildren)i(of)e Fl(y)i Fs(are)e(hits)g(is)h(at)f(most)771 2572 y(\()p Fl(q)d Fm(\000)f Fl(q)889 2555 y Fy(0)900 2572 y Fs(\))p 741 2592 207 2 v 741 2634 a Fl(\015)s Fs(\()p Fl(n=)p Fs(8\))p Fl(n=)p Fs(2)981 2603 y(=)1054 2572 y(\()p Fl(q)h Fm(\000)e Fl(q)1171 2555 y Fy(0)1183 2572 y Fs(\))p 1049 2592 157 2 v 1049 2634 a(2)p Fl(\015)s Fs(\()p 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Fl(n)p Fs(\))15 b(is)743 1434 y Fi(X)764 1521 y Fk(r)810 1475 y Fl(P)6 b(r)q Fs([)p Fl(r)q Fs(])p Fl(P)g(r)q Fs([)p Fl(cor)q(r)q(ect)14 b Fm(j)h Fl(r)q Fs(])-75 1609 y(where)f Fl(P)6 b(r)q Fs([)p Fl(r)q Fs(])14 b(is)g(the)h(probabilit)o(y)g(of)f (run)g Fl(r)q Fs(,)g(where)g Fl(P)6 b(r)q Fs([)p Fl(cor)q(r)q(ect)15 b Fm(j)g Fl(r)q Fs(])e(is)i(the)f(probabilit)o(y)h(the)g(answ)o(er)e (is)i(correct)e(giv)o(en)-75 1666 y(run)20 b Fl(r)q Fs(,)h(and)f(where) g Fl(r)g Fs(ranges)g(o)o(v)o(er)f(all)i(runs)f(with)g(at)f(most)g Fl(T)6 b Fs(\()p Fl(n)p Fs(\))20 b(queries.)35 b(Theorem)19 b(8.4.6)g(tells)i(us)f(that)f(if)h(w)o(e)-75 1722 y(condition)15 b(on)f(an)o(y)f(run)h Fl(r)g Fs(with)g(few)o(er)g(than)724 1704 y Fj(1)p 715 1711 36 2 v 715 1738 a(12)756 1722 y Fl(\015)s Fs(\()p Fl(n=)p Fs(4\))d(queries,)k(then)f(the)g (probabilit)o(y)h(0)1501 1706 y Fk(n)1538 1722 y Fs(is)f(a)f(hit)h(is)h (less)f(than)f(1)p Fl(=)p Fs(12.)-75 1779 y(This)j(means)g(that)e(the)i 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y(W)l(e)i(sa)o(y)f (that)g(a)h(con\014guration)g Fl(c)f Fs(is)i(of)e(t)o(yp)q(e)h Fl(q)r(;)8 b(\033)794 1932 y Fj(1)813 1925 y Fl(;)g(\033)860 1932 y Fj(2)879 1925 y Fl(;)g(\033)926 1932 y Fj(3)960 1925 y Fs(if)15 b Fl(c)g Fs(is)g(in)h(state)e Fl(q)g Fm(2)f Fl(Q)i Fs(and)g(the)g(three)g(adjacen)o(t)f(tap)q(e)h(cells)-75 1981 y(cen)o(tered)i(ab)q(out)g(the)g(tap)q(e)g(head)h(con)o(tain)f (the)g(three)g(sym)o(b)q(ols)g Fl(\033)1095 1988 y Fj(1)1115 1981 y Fl(;)8 b(\033)1162 1988 y Fj(2)1181 1981 y Fl(;)g(\033)1228 1988 y Fj(3)1262 1981 y Fm(2)16 b Fs(\006.)25 b(The)18 b(en)o(tries)f(of)g(a)f(ro)o(w)g(indexed)j(b)o(y)-75 2038 y Fl(c)f Fs(m)o(ust)g(b)q(e)g(a)g(p)q(erm)o(utation)h(of)e(the)i (en)o(tries)f(of)g(a)g(ro)o(w)f(indexed)j(b)o(y)f(an)o(y)f (con\014guration)g Fl(c)1543 2021 y Fy(0)1572 2038 y Fs(of)g(the)h(same)f(t)o(yp)q(e)g(as)g Fl(c)p Fs(.)-75 2094 y(This)c(is)h(b)q(ecause)f(a)g(transition)g(of)f(the)h(QTM)f(M)g 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bop -75 -26 a Fm(j)p Fl(T)6 b Fm(j)12 b Fs(=)h(\()p Fl(k)8 b Fm(\000)g Fs(2\)card)o(\(\006\))332 -47 y Fk(k)q Fy(\000)p Fj(3)398 -26 y Fs(.)19 b(Eac)o(h)14 b(ro)o(w)f(of)h Fl(B)i Fs(indexed)g(b)o(y)e(a)f(con\014guration)h Fl(c)f Fm(2)g Fl(T)19 b Fs(has)14 b(the)g(prop)q(ert)o(y)g(that)f(if)h Fl(d)g Fs(is)g(an)o(y)-75 31 y(con\014guration)f(that)f(yields)i Fl(c)e Fs(with)h(non-zero)g(amplitude)h(in)g(a)e(single)i(step,)f(then) g Fl(d)f Fm(2)h Fl(S)s Fs(.)18 b(Therefore)13 b(the)g(ro)o(w)e(indexed) -75 87 y(b)o(y)17 b Fl(c)f Fs(in)h(the)g(\014nite)h(matrix)e Fl(B)j Fs(has)e(the)g(same)f(non-zero)h(en)o(tries)g(as)f(the)h(ro)o(w) f(indexed)i(b)o(y)f Fl(c)f Fs(in)h(the)g(in\014nite)i(matrix.)-75 144 y(Therefore)14 b(it)f(mak)o(es)h(sense)g(to)f(sa)o(y)g(that)g(ro)o (w)f Fl(c)i Fs(of)f Fl(B)j Fs(is)f(of)e(the)h(same)f(t)o(yp)q(e)h(as)f (ro)o(w)g Fl(c)g Fs(of)g(the)h(in\014nite)i(matrix.)j(Finally)l(,)-75 200 y(the)d(ro)o(ws)e(of)i(eac)o(h)g(t)o(yp)q(e)f(constitute)h(a)g 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(lowing)g(two)h(c)n(onditions)e(b)n(oth)i(hold)-21 1230 y(1.)23 b(Each)18 b(state)h(of)f Fl(M)24 b Fr(c)n(an)18 b(b)n(e)g(enter)n(e)n(d)f(while)h(moving)h(in)f(only)f(one)i(dir)n(e)n (ction.)26 b(In)18 b(other)h(wor)n(ds,)g(if)f Fl(\016)r Fs(\()p Fl(p)1867 1237 y Fj(1)1886 1230 y Fl(;)8 b(\033)1933 1237 y Fj(1)1952 1230 y Fs(\))17 b(=)39 1287 y(\()p Fl(\034)77 1294 y Fj(1)96 1287 y Fl(;)8 b(q)r(;)g(d)184 1294 y Fj(1)202 1287 y Fs(\))16 b Fr(and)g Fl(\016)r Fs(\()p Fl(p)387 1294 y Fj(2)406 1287 y Fl(;)8 b(\033)453 1294 y Fj(2)472 1287 y Fs(\))k(=)h(\()p Fl(\034)588 1294 y Fj(2)607 1287 y Fl(;)8 b(q)r(;)g(d)695 1294 y Fj(2)713 1287 y Fs(\))16 b Fr(then)g Fl(d)872 1294 y Fj(1)904 1287 y Fs(=)d Fl(d)976 1294 y Fj(2)996 1287 y Fr(.)-21 1380 y(2.)23 b(The)16 b(tr)n(ansition)f(function)h Fl(\016)i Fr(is)e(one-to-one)g(when)g(dir) n(e)n(ction)g(is)g(ignor)n(e)n(d.)-75 1503 y Fn(Pro)q(of.)k Fs(First)15 b(w)o(e)f(sho)o(w)h(that)g(these)g(t)o(w)o(o)f(conditions)i (imply)h(rev)o(ersibilit)o(y)l(.)-4 1577 y(Supp)q(ose)d Fl(M)k Fs(=)13 b(\(\006)p Fl(;)8 b(Q;)g(\016)r Fs(\))j(is)j(a)f(TM)g (or)g(generalized)i(TM)e(satisfying)g(these)h(t)o(w)o(o)e(conditions.) 21 b(Then,)14 b(the)f(follo)o(wing)-75 1633 y(pro)q(cedure)j(lets)g(us) f(tak)o(e)g(an)o(y)f(con\014guration)i(of)f Fl(M)20 b Fs(and)15 b(compute)h(its)f(predecessor)h(if)f(it)h(has)f(one.)20 b(First,)15 b(since)h(eac)o(h)-75 1690 y(state)d(can)g(b)q(e)h(en)o (tered)g(while)h(mo)o(ving)e(in)h(only)g(one)f(direction,)i(the)f (state)e(of)h(the)g(con\014guration)h(tells)g(us)g(in)g(whic)o(h)g (cell)-75 1746 y(the)j(tap)q(e)h(head)f(m)o(ust)g(ha)o(v)o(e)g(b)q(een) h(in)g(the)f(previous)h(con\014guration.)26 b(Lo)q(oking)18 b(at)f(this)g(cell,)i(w)o(e)e(can)h(see)f(what)g(tap)q(e)-75 1803 y(sym)o(b)q(ol)e(w)o(as)f(written)g(in)i(the)e(last)h(step.)20 b(Then,)14 b(since)i Fl(\016)h Fs(is)e(one-to-one)f(w)o(e)h(kno)o(w)f (the)g(up)q(date)h(rule,)h(if)f(an)o(y)l(,)f(that)g(w)o(as)-75 1859 y(used)i(on)f(the)g(previous)h(step,)f(allo)o(wing)h(us)f(to)g (reconstruct)g(the)g(previous)h(con\014guration.)-4 1933 y(Next,)i(w)o(e)h(sho)o(w)f(that)g(the)g(\014rst)g(prop)q(ert)o(y)h(is) g(necessary)f(for)g(rev)o(ersibilit)o(y)l(.)32 b(So,)19 b(for)f(example,)i(consider)f(a)g(TM)-75 1990 y(or)d(generalized)i(TM)e Fl(M)k Fs(=)15 b(\(\006)p Fl(;)8 b(Q;)g(\016)r Fs(\))14 b(suc)o(h)i(that)g Fl(\016)r Fs(\()p Fl(p)872 1997 y Fj(1)891 1990 y Fl(;)8 b(\033)938 1997 y Fj(1)957 1990 y Fs(\))14 b(=)i(\()p Fl(\034)1078 1997 y Fj(1)1097 1990 y Fl(;)8 b(q)r(;)g(L)p Fs(\))14 b(and)j Fl(\016)r Fs(\()p Fl(p)1377 1997 y Fj(2)1396 1990 y Fl(;)8 b(\033)1443 1997 y Fj(2)1462 1990 y Fs(\))14 b(=)h(\()p Fl(\034)1582 1997 y Fj(2)1601 1990 y Fl(;)8 b(q)r(;)g(R)p Fs(\).)22 b(Then,)17 b(w)o(e)f(can)-75 2046 y(easily)e(construct)f(t)o(w)o(o)f (con\014gurations)i(whic)o(h)g(lead)g(to)f(the)g(same)h(next)f (con\014guration:)19 b(Let)14 b Fr(c)1580 2053 y Fa(1)1618 2046 y Fs(b)q(e)g(an)o(y)f(con\014guration)-75 2103 y(where)j(the)f (mac)o(hine)h(is)g(in)g(state)f Fl(p)547 2110 y Fj(1)582 2103 y Fs(reading)h(a)f Fl(\033)808 2110 y Fj(1)843 2103 y Fs(and)g(where)h(the)f(sym)o(b)q(ol)h(t)o(w)o(o)e(cells)j(to)e(the)g (left)h(of)f(the)g(tap)q(e)h(head)-75 2159 y(is)j(a)g Fl(\034)36 2166 y Fj(2)55 2159 y Fs(,)g(and)g(let)g Fr(c)269 2166 y Fa(2)313 2159 y Fs(b)q(e)g(iden)o(tical)h(to)e Fr(c)644 2166 y Fa(1)688 2159 y Fs(except)h(that)f(the)h Fl(\033)1044 2166 y Fj(1)1082 2159 y Fs(and)g Fl(\034)1194 2166 y Fj(2)1232 2159 y Fs(are)f(c)o(hanged)h(to)f Fl(\034)1569 2166 y Fj(1)1608 2159 y Fs(and)h Fl(\033)1726 2166 y Fj(2)1745 2159 y Fs(,)g(the)g(mac)o(hine)-75 2216 y(is)h(in)f(state)g Fl(p)170 2223 y Fj(2)189 2216 y Fs(,)h(and)f(the)g(tap)q(e)g(head)g(is) h(t)o(w)o(o)e(cells)i(further)f(left.)32 b(Therefore)19 b Fl(M)24 b Fs(is)19 b(not)g(rev)o(ersible.)32 b(Since)21 b(similar)-75 2272 y(argumen)o(ts)15 b(apply)h(for)f(eac)o(h)h(pair)g (of)g(distinct)h(directions,)f(the)g(\014rst)f(condition)i(in)g(the)f (theorem)f(m)o(ust)g(b)q(e)i(necessary)-75 2329 y(for)e(rev)o (ersibilit)o(y)l(.)-4 2403 y(Finally)l(,)d(w)o(e)e(sho)o(w)g(that)g (the)h(second)g(condition)g(is)g(also)g(necessary)f(for)g(rev)o (ersibilit)o(y)l(.)21 b(Supp)q(ose)11 b(that)f Fl(M)18 b Fs(=)13 b(\(\006)p Fl(;)8 b(Q;)g(\016)r Fs(\))-75 2459 y(is)14 b(a)f(TM)g(or)g(generalized)i(TM)e(with)h Fl(\016)r Fs(\()p Fl(p)633 2466 y Fj(1)652 2459 y Fl(;)8 b(\033)699 2466 y Fj(1)718 2459 y Fs(\))k(=)h Fl(\016)r Fs(\()p Fl(p)859 2466 y Fj(2)879 2459 y Fl(;)8 b(\033)926 2466 y Fj(2)944 2459 y Fs(\).)19 b(Then,)14 b(an)o(y)f(pair)h(of)f (con\014gurations)h(whic)o(h)g(di\013er)g(only)g(in)-75 2516 y(the)h(state)f(and)h(sym)o(b)q(ol)g(under)g(the)g(tap)q(e)g (head,)g(where)g(one)g(has)g(\()p Fl(p)1122 2523 y Fj(1)1141 2516 y Fl(;)8 b(\033)1188 2523 y Fj(1)1207 2516 y Fs(\))14 b(and)h(the)g(other)f(\()p Fl(p)1564 2523 y Fj(2)1584 2516 y Fl(;)8 b(\033)1631 2523 y Fj(2)1650 2516 y Fs(\),)14 b(lead)h(to)f(the)h(same)-75 2572 y(next)g(con\014guration,)g(and)h (again)f Fl(M)20 b Fs(is)c(not)f(rev)o(ersible.)1073 b Fd(2)951 2779 y Fs(55)p eop %%Page: 56 56 56 55 bop -75 -26 a Fn(Corollary)18 b(B.0.14)k Fr(If)16 b Fl(M)21 b Fr(is)16 b(a)g(r)n(eversible)f(TM)h(then)g(every)g(c)n (on\014gur)n(ation)f(of)i Fl(M)k Fr(has)16 b(exactly)h(one)f(pr)n(e)n (de)n(c)n(essor.)-75 98 y Fn(Pro)q(of.)23 b Fs(Let)16 b Fl(M)j Fs(=)c(\(\006)p Fl(;)8 b(Q;)g(\016)r Fs(\))13 b(b)q(e)k(a)f(rev)o(ersible)h(TM.)f(By)g(the)g(de\014nition)i(of)e(rev) o(ersibilit)o(y)l(,)i(eac)o(h)e(con\014guration)g(of)g Fl(M)-75 155 y Fs(has)f(at)g(most)f(one)h(predecessor.)-4 229 y(So,)h(let)i Fl(c)e Fs(b)q(e)i(a)e(con\014guration)i(of)e Fl(M)22 b Fs(in)c(state)e Fl(q)r Fs(.)25 b(Theorem)16 b(B.0.13)g(tells)i(us)f(that)f Fl(M)22 b Fs(can)17 b(en)o(ter)g(state)f Fl(q)i Fs(while)-75 285 y(mo)o(ving)d(its)g(tap)q(e)h(head)f(in)h(only) g(one)f(direction)h Fl(d)807 292 y Fk(q)826 285 y Fs(.)k(Since)c (Theorem)f(B.0.13)f(tells)i(us)f(that,)g(ignoring)g(direction,)h Fl(\016)h Fs(is)-75 342 y(one-to-one,)e(taking)f(the)h(in)o(v)o(erse)h (of)e Fl(\016)j Fs(on)e(the)g(state)f Fl(q)i Fs(and)f(the)g(sym)o(b)q (ol)h(in)f(direction)1475 330 y(\026)1467 342 y Fl(d)1491 349 y Fk(q)1525 342 y Fs(tells)h(us)f(ho)o(w)f(to)g(transform)-75 398 y Fl(c)h Fs(in)o(to)g(its)g(predecessor.)1635 b Fd(2)-75 579 y Fn(Corollary)18 b(B.0.15)k Fr(If)15 b Fl(\016)i Fr(is)e(a)g(p)n(artial)h(function)f(fr)n(om)h Fl(Q)23 b Fm(\002)h Fs(\006)15 b Fr(to)h Fs(\006)8 b Fm(\002)g Fl(Q)g Fm(\002)h(f)p Fl(L;)f(R)p Fm(g)13 b Fr(satisfying)i(the)g(two)h (c)n(onditions)-75 635 y(of)g(The)n(or)n(em)g(B.0.13)h(then)f Fl(\016)i Fr(c)n(an)e(b)n(e)g(extende)n(d)g(to)g(a)h(total)f(function)g (that)h(stil)r(l)e(satis\014es)g(The)n(or)n(em)h(B.0.13.)-75 759 y Fn(Pro)q(of.)32 b Fs(Supp)q(ose)21 b Fl(\016)g Fs(is)f(a)f(partial)h(function)g(from)f Fl(Q)12 b Fm(\002)i Fs(\006)19 b(to)g(\006)12 b Fm(\002)i Fl(Q)e Fm(\002)i(f)p Fl(L;)8 b(R)p Fm(g)18 b Fs(that)g(satis\014es)i(the)f(prop)q(erties)h (of)-75 816 y(Theorem)c(B.0.13.)k(Then,)c(for)g(eac)o(h)g Fl(q)f Fm(2)f Fl(Q)i Fs(let)g Fl(d)805 823 y Fk(q)840 816 y Fs(b)q(e)g(the)g(one)g(direction,)h(if)g(an)o(y)l(,)e(in)i(whic)o (h)g Fl(q)g Fs(can)f(b)q(e)h(en)o(tered,)f(and)-75 872 y(let)g Fl(d)15 879 y Fk(q)49 872 y Fs(b)q(e)h(\(arbitrarily\))e Fl(L)h Fs(otherwise.)21 b(Then)16 b(w)o(e)f(can)h(\014ll)h(in)f (unde\014ned)i(v)m(alues)e(of)g Fl(\016)h Fs(with)f(as)f(y)o(et)g(un)o (used)i(triples)f(of)-75 928 y(the)i(form)e(\()p Fl(\034)s(;)8 b(q)r(;)g(d)244 935 y Fk(q)260 928 y Fs(\))17 b(so)g(as)g(to)g(main)o (tain)h(the)g(conditions)g(of)f(Theorem)h(B.0.13.)25 b(Since)19 b(there)e(are)g(card\(\006\))g(card\()p Fl(Q)p Fs(\))-75 985 y(suc)o(h)f(triples)g(there)f(will)i(b)q(e)f(exactly)f (enough)h(to)e(fully)j(de\014ne)f Fl(\016)r Fs(.)919 b Fd(2)-75 1165 y Fn(Theorem)17 b(B.0.16)22 b Fr(If)c Fl(M)23 b Fr(is)17 b(a)h(gener)n(alize)n(d)e(r)n(eversible)h(TM,)g (then)h(ther)n(e)g(is)f(a)h(r)n(eversible)f(TM)g Fl(M)1714 1149 y Fy(0)1744 1165 y Fr(that)h(simulates)-75 1222 y Fl(M)j Fr(with)c(slowdown)f(at)h(most)f(2.)-75 1346 y Fn(Pro)q(of.)k Fs(The)c(idea)g(is)g(to)f(replace)h(an)o(y)f (transition)h(that)f(has)g(the)h(tap)q(e)f(head)h(stand)f(still)i(with) f(t)o(w)o(o)e(transitions.)21 b(The)-75 1402 y(\014rst)15 b(up)q(dates)g(the)g(tap)q(e)g(and)g(mo)o(v)o(es)f(to)h(the)g(righ)o (t,)f(remem)o(b)q(ering)i(whic)o(h)g(state)e(it)h(should)h(en)o(ter.)j (The)d(second)f(steps)-75 1459 y(bac)o(k)g(to)g(the)g(left)g(and)h(en)o (ters)f(the)g(desired)h(state.)-4 1533 y(So,)g(if)i Fl(M)i Fs(=)c(\(\006)p Fl(;)8 b(Q;)g(\016)r Fs(\))14 b(is)j(a)g(generalized)h (rev)o(ersible)g(TM)f(then)g(w)o(e)f(let)i Fl(M)1324 1516 y Fy(0)1352 1533 y Fs(b)q(e)g(iden)o(tical)g(to)e Fl(M)22 b Fs(except)17 b(that)g(for)-75 1589 y(eac)o(h)e(state)f Fl(q)j Fs(with)f(a)e(transition)i(of)e(the)h(form)g Fl(\016)r Fs(\()p Fl(p;)8 b(\033)r Fs(\))i(=)j(\()p Fl(\034)s(;)8 b(q)r(;)g(N)d Fs(\))12 b(w)o(e)j(add)g(a)g(new)g(state)f Fl(q)1545 1573 y Fy(0)1572 1589 y Fs(and)h(w)o(e)g(also)g(add)g(a)g (new)-75 1646 y(transition)g(rule)h Fl(\016)r Fs(\()p Fl(q)285 1629 y Fy(0)296 1646 y Fl(;)8 b(\033)r Fs(\))j(=)i Fl(\033)o(;)8 b(q)r(;)g(L)13 b Fs(for)i(eac)o(h)g Fl(\033)f Fm(2)f Fs(\006.)19 b(Finally)l(,)e(w)o(e)d(replace)i(eac)o(h)f (transition)h Fl(\016)r Fs(\()p Fl(p;)8 b(\033)r Fs(\))i(=)j(\()p Fl(\034)s(;)8 b(q)r(;)g(N)d Fs(\))12 b(with)-75 1702 y(the)j(transition)h Fl(\016)r Fs(\()p Fl(p;)8 b(\033)r Fs(\))i(=)j(\()p Fl(\034)s(;)8 b(q)483 1686 y Fy(0)493 1702 y Fl(;)g(R)p Fs(\).)19 b(Clearly)d Fl(M)806 1686 y Fy(0)833 1702 y Fs(sim)o(ulates)f Fl(M)21 b Fs(with)15 b(slo)o(wdo)o(wn)g(b)o(y)g(a)g(factor)f(of)h(at)f(most)h(2.)-4 1776 y(T)l(o)f(complete)i(the)g(pro)q(of,)e(w)o(e)h(need)h(to)f(sho)o (w)f(that)h Fl(M)953 1760 y Fy(0)980 1776 y Fs(is)g(also)g(rev)o (ersible.)-4 1851 y(So,)h(consider)h(a)f(con\014guration)h Fr(c)h Fs(of)e Fl(M)708 1834 y Fy(0)720 1851 y Fs(.)23 b(W)l(e)17 b(need)g(to)f(sho)o(w)g(that)f Fr(c)k Fs(has)d(at)g(most)g (one)g(predecessor.)24 b(If)16 b Fr(c)j Fs(is)e(in)-75 1907 y(state)f Fl(q)h Fm(2)e Fl(Q)i Fs(and)f Fl(M)22 b Fs(en)o(ters)16 b Fl(q)j Fs(mo)o(ving)d(left)h(or)g(righ)o(t,)f(then) h(the)g(transitions)g(in)o(to)f Fl(q)j Fs(in)e Fl(M)1571 1891 y Fy(0)1599 1907 y Fs(are)g(iden)o(tical)h(to)e(those)-75 1964 y(in)g Fl(M)21 b Fs(and)15 b(therefore)g(since)i Fl(M)k Fs(is)15 b(rev)o(ersible,)i Fr(c)h Fs(has)d(at)g(most)f(one)i (predecessor)g(in)g Fl(M)1474 1947 y Fy(0)1486 1964 y Fs(.)k(Similarly)l(,)e(if)e Fr(c)i Fs(is)e(in)g(one)f(of)-75 2020 y(the)g(new)f(states)g Fl(q)246 2003 y Fy(0)272 2020 y Fs(then)h(the)g(transitions)f(in)o(to)h Fl(q)791 2003 y Fy(0)817 2020 y Fs(in)g Fl(M)918 2003 y Fy(0)945 2020 y Fs(are)f(exactly)h(the)f(same)h(as)f(those)g(in)o(to)h Fl(q)h Fs(in)f Fl(M)5 b Fs(,)15 b(except)g(that)-75 2076 y(the)i(tap)q(e)g(head)h(mo)o(v)o(es)e(righ)o(t)h(instead)g(of)g(sta)o (ying)f(still.)27 b(So,)17 b(again)g(the)g(rev)o(ersibilit)o(y)i(of)e Fl(M)22 b Fs(implies)d(that)d Fr(c)k Fs(has)d(at)-75 2133 y(most)f(one)h(predecessor.)25 b(Finally)l(,)19 b(supp)q(ose)f Fr(c)h Fs(is)e(in)h(state)e Fl(q)h Fm(2)f Fl(Q)h Fs(where)g Fl(M)22 b Fs(en)o(ters)16 b Fl(q)j Fs(while)f(standing)g(still.)26 b(Then,)-75 2189 y(Theorem)12 b(B.0.13)f(tells)i(us)g(that)e(all)i(transitions)g(in)g Fl(M)k Fs(that)12 b(en)o(ter)g Fl(q)i Fs(ha)o(v)o(e)e(direction)h Fl(N)5 b Fs(.)19 b(Therefore,)12 b(all)h(of)f(them)g(ha)o(v)o(e)-75 2246 y(b)q(een)19 b(remo)o(v)o(ed,)e(and)g(the)h(only)g(transitions)g (en)o(tering)f Fl(q)j Fs(in)e Fl(M)1053 2229 y Fy(0)1082 2246 y Fs(are)f(the)h(new)f(ones)h(of)f(the)h(form)e Fl(\016)r Fs(\()p Fl(q)1746 2229 y Fy(0)1757 2246 y Fl(;)8 b(\033)r Fs(\))15 b(=)i Fl(\033)o(;)8 b(q)r(;)g(L)p Fs(.)-75 2302 y(Again,)15 b(this)h(means)f(that)f Fr(c)k Fs(can)d(ha)o(v)o(e)g 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