Note on quantum counting classes

Transcription

1 Note on quantum countng classes Yaoyun Sh Shengyu Zhang Abstract We defne countng classes #BPP and #BQP as natural extensons of the classcal well-studed one #P to the randomzed and quantum cases. It s then shown that P #P = P #BQP. 1 Introducton One mportant type of computatonal tasks are countng problems, and a partcularly nterestng class of such problems are for countng the number of the certfcates. These countng problems naturally arse n a plethora of felds rangng from enumeratve combnatorcs, to statstcal physcs and economcs; see [AB09] (Chapter 17) for a number of specfc examples. Countng certfcates s n general harder than the correspondng NP problems, whch merely requres to decde for an nput nstance whether at least one certfcate exsts. In [Val79b] Valant ntalzed the study of countng problems from a complexty perspectve by defnng a countng class called #P. Precsely, the class contans the functons f : {0, 1} N whch have a polynomal-tme nondetermnstc Turng machne M wth the number of acceptng paths on an nput x equal to f(x). In another paper [Val79a] of Valant, t was showed that computng Perm, the permanent of {0,1}- matrces, s #P-complete. That s to say, Perm s n #P and any other functon n #P can be computed n polynomal tme gven Perm as an oracle. Ths mples the computatonal hardness of computng Perm, n a strong contrast to the exstence of effcent algorthms for computng the determnant, a quantty closely related to permanent, of a matrx. More propertes of the class #P were later dscovered, ncludng Toda s dscovery that the polynomal herarchy PH s n P #P. In ths paper, we consder countng classes n the randomzed and quantum computatonal modes. When the probablty s nvolved, one ssue needs to be handled properly. For a predcate P (x, y) decded by a probablstc or quantum Turng machne (PTM or QTM), a (canddate) certfcate y s usually not perfectly good or perfectly bad as n the determnstc Turng machne case, but has an acceptng probablty, and the probabltes for dfferent y s can be qute densely dstrbuted n [0, 1]. As n bounded-error classes, we assume a gap between the good and bad certfcates by requrng that each certfcate has acceptance probablty at least 2/3 or at most 1/3. Then f(x) s equal Ths work s supported by Hong Kong grants RGC and RGC Yaoyun: 1. Please add your grant nformaton. 2. I m not sure ths s the most sutable ttle; please feel free to suggest somethng else. Yaoyun: Please add your malng address. Department of Computer Scence and Engneerng, The Chnese Unversty of Hong Kong. Emal: syzhang@cse.cuhk.edu.hk 1

2 to the number of the certfcates wth large acceptance probablty; see the next secton for precse defnton of #BPP. When t comes to the quantum countng, more ssues arse. Snce the certfcate space s a contnuous Hlbert space, there are nfntely many good certfcates n general, and thus countng may not be meanngful. The contnuty also makes the gap between acceptance probabltes mpossble: If certfcates states y 1 and y 2 have acceptance probabltes, say, ɛ and 1 ɛ, respectvely, then the acceptance probablty for a superposton of y 1 and y 2 can be any real number between ɛ and 1 ɛ. A natural measure to use s the dmenson of subspace of good certfcates. However, how can we guarantee that the set of good certfcates form a subspace? All these ssues shall be settled by a spectral result n [MW05]. Bascally, what was shown there s a useful fact that f the verfer s a unform famly of quantum crcut, then there s a natural spectral decomposton of the wtness space V = p V p, such that n each egenspace V p, the correspondng egenvalue p s the acceptance probablty of a wtness ψ p V p. In addton, a general wtness ψ = p α p ψ p, where ψ p V p, has acceptance probablty p α p 2 p. Based on ths, we defne the countng class #BQP as those functons f wth a unform famly of quantum crcut of polynomal sze such that all these egenvalues are ether at least 2/3 or at most 1/3, and f(x) s equal to the number of egenvalues at least 2/3. One mportant queston n quantum computng s to understand the ultmate power of quantum computers n varous computatonal modes. Snce countng problems form an mportant class of computatonal tasks, t s desrable to understand the power of quantum countng classes. In ths paper, we shall show the followng relaton of #BQP and #P. Theorem 1.1 P #BQP = P #P. Another way to look at the result s from a central queston of where BQP sts n classcal complexty classes. Whle t has been known for more than a decade that BQP PP P #P, pursung a better upper bound turns out to be much harder. One way to mprove the contanment s to see how much we can boost the power of BQP so that t s stll upper bounded by a classcal complexty class such as PP. Observng that the proof that BQP PP does not use the bounded-error assumpton of BQP, Watrous [Wat09] mproved the relaton to PQP = PP, where PQP s the same as BQP but only requrng the acceptance probablty to be strctly larger than 1/2 for Yes nstances and at most 1/2 for No nstances. The result of ths note can be vewed n the same sprt as an mprovement of BQP P #P by pushng up the BQP to P #BQP. Related work One classcal result about the number of wtness and ts mplcaton on hardness s the UnqueNP problem studed by Valant and Vazran [VV86], who showed that an NP problem wth the promse of at most one certfcate exsts s no easer than the problem wthout the promse. Recently the result was extended to the randomzed [ABOBS08] and quantum [JKK + 10] cases, where a gap s also assumed between the good and bad certfcates. 2

3 2 Prelmnares and notaton In the model for #BQP, there s a unform famly of quantum crcuts of polynomal sze. We can equvalently thnk of the crcuts as dependng on nput, so the verfer for nput x s V x. Suppose that V x has an nput space W S, where W s the space for an m-qubt potental wtness ψ W, and S s a k-qubt workng space, ntalzed as 0 k. On a partcular wtness ψ, the crcut operates on ψ 0 k and, at the end of the output, measures the frst qubt n the { 0, 1 } bass and output the result. Overloadng the notaton, we also use V x to denote the untary operaton that the crcut apples. Defne two projectons 1. Π acc = V x Π acc V x : projecton onto the subspace correspondng to the frst qubt beng 1 f the computaton V x f performed. 2. Π nt = I m 0 k S 0 k S : projecton onto the subspace of S contanng 0 k. A smple but mportant property for analyss of QMA feasble s to consder the egensystem of Π nt Π acc Π nt. Snce t s a postve operator, t enjoys a spectral decomposton. There are 2 m egenvectors φ of Π nt Π acc Π nt, all n the form of ψ 0 k. The egenvalue for φ turns out to be the accept probablty p of the wtness ψ : λ (Π nt Π acc Π nt ) = ψ 0 k Π nt Π acc Π nt ψ 0 k = Π acc Π nt ψ 0 k 2 = Π acc ψ 0 k 2 (1) Now the acceptance probablty for a general wtness state α φ can also be computed easly: Π acc α φ 2 = α φ Π acc α φ = α α j φ Π acc φ j = α 2 p j (2) where the last equalty uses the fact that the egenvectors are orthogonal. Wth all these setup, we can defne the quantum countng class. Frst recall that #P s defned as follows. Defnton 2.1 f #P f a polynomal-tme Turng machne V s.t. f(x) = {w : V (x, w) = 1}. (3) The countng classes #BPP and #BQP are defned as follows. Defnton 2.2 f #BPP f a polynomal-tme probablstc Turng machne V s.t. 1. w, V (x, w) = 1 wth probablty ether at least 2/3 or at most 1/3, 2. f(x) = {w : V (x, w) = 1 wth probablty at least 2/3}. Defnton 2.3 f #BQP f a unform famly of quantum crcuts {V x } of sze polynomal n x s.t. 1. The egenvalues of Π nt Π acc Π nt are all ether at least 2/3 or at most 1/3, 2. f(x) = the number of egenvalues of Π nt Π acc Π nt that are at least 2/3. 3

4 3 On the lmtaton of the quantum countng class In ths secton we shall show that #BQP and #P, when workng as oracles for P, gve the same class. The dea s to frst amplfy the correct probablty, and then use the double-countng on trace. The same dea was used n [MW05] to prove that QMA PP. Let us frst recall the followng amplfcaton result for QMA. Lemma 3.1 (Strong Amplfcaton of QMA, [MW05]) For any problem n PromseQMA, and any polynomals m(n) and r(n), there s a unversal famly of polynomal-sze verfers {V x : x = n} wth m(n)-qubt wtness space and O(r(n))-qubt workng space s.t. the egenvalues of Π nt Π acc Π nt are ether more than 1 2 r(n) or less than 2 r(n). Proof (of Theorem 1.1) We shall smulate each query to a #BQP oracle by a #P oracle wth some further polynomal-tme post-processng. For a query to a functon f #BQP, there exsts a quantum verfer V x wth m-qubt wtness space. The strong amplfcaton n Lemma 3.1 gves both the soundness error and completeness error smaller than 2 r. Actually, denote by Π nt and Π acc the correspondng projectons for the new verfer, then the egenvectors of Q = Π nt Π accπ nt are the same as those of Q = Π nt Π acc Π nt, but the correspondng egenvalues change to ether more than 1 2 r or less than 2 r. By the defnton of #BQP, f(x) s equal to the number of egenvalues of Q that are at least 2/3, whch s n turn equal to the the number of egenvalues of Q that are at least 1 2 r. Now consder the trace of Q. On the one hand, we have tr(q ) = λ (Q ) f(x)(1 2 r ). (4) On the other hand, snce all the small egenvalues are less than 2 r(n), we have tr(q ) = λ (Q ) (2 m f(x)) 2 r + f(x) 1 = f(x)(1 2 r ) + 2 m r. (5) Note that one desrable property of strong amplfcaton s the flexblty of choce of r to be any polynomal of n, ndependent of m. Let r = m + 2, then f(x) 1/4 f(x)(1 2 (m+2) ) tr(q ) f(x)(1 2 (m+2) )+1/4 f(x)+1/4. (6) where the frst nequalty s because f(x) 2 m due to the assumpton that the verfer for f has m-qubt wtness space. Wthout loss of generalty, we can assume that the quantum crcut s made of Toffol, Hadamard and -shft gates. Each entry (, j) of the whole matrx Q equals to h(, j)/2 g where h s an #P functon and g = poly(n) s the number of Hadamard gates n the crcut. Snce the trace also equals to the summaton of the dagonal entres, and #P s closed under exponental sum, we get tr(q ) = h/2 g (7) for some GapP functon h. Suppose h = h 1 h 2 for some #P functons h 1 and h 2. Now we use our #P oracle to get h 1 and h 2, and then compute [(h 1 h 2 )/2 g ] n determnstc polynomal tme, where the bracket s to round the number to the closest nteger. By Eq. (6), ths s equal to f(x). Ths completes the smulaton of the query to the #BQP oracle. 4

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