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84
The Complexity of the Local Hamiltonian Problem
 In Proc. of 24th FSTTCS
, 2004
"... The kLOCAL HAMILTONIAN problem is a natural complete problem for the complexity class QMA, the quantum analog of NP. It is similar in spirit to MAXkSAT, which is NPcomplete for k ≥ 2. It was known that the problem is QMAcomplete for any k ≥ 3. On the other hand 1LOCAL HAMILTONIAN is in P, and h ..."
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Cited by 80 (6 self)
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The kLOCAL HAMILTONIAN problem is a natural complete problem for the complexity class QMA, the quantum analog of NP. It is similar in spirit to MAXkSAT, which is NPcomplete for k ≥ 2. It was known that the problem is QMAcomplete for any k ≥ 3. On the other hand 1LOCAL HAMILTONIAN is in P, and hence not believed to be QMAcomplete. The complexity of the 2LOCAL HAMILTONIAN problem has long been outstanding. Here we settle the question and prove that it is QMAcomplete. One component in our proof is a powerful technique for analyzing the sum of two Hamiltonians; this technique is based on perturbation theory and we believe that it might prove useful elsewhere. Our proof also implies that adiabatic computation with twolocal interactions on qubits is equivalent to standard quantum computation. 1
Quantum ArthurMerlin games
 Computational Complexity
"... Abstract This paper studies quantum ArthurMerlin games, whichare a restricted form of quantum interactive proof system in which the verifier's messages are given by unbiased coinflips. The following results are proved. ffl For onemessage quantum ArthurMerlin games, whichcorrespond to the co ..."
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Cited by 74 (3 self)
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Abstract This paper studies quantum ArthurMerlin games, whichare a restricted form of quantum interactive proof system in which the verifier's messages are given by unbiased coinflips. The following results are proved. ffl For onemessage quantum ArthurMerlin games, whichcorrespond to the complexity class QMA, completeness and soundness errors can be reduced exponentially without increasing the length of Merlin's message. Previous constructions for reducing error required a polynomial increase in the length of Merlin's message. Applications of this fact include a proof that logarithmic length quantum certificates yield no increase in powerover BQP and a simple proof that QMA ` PP. ffl In the case of three or more messages, quantum ArthurMerlin games are equivalent in power to ordinary quantum interactive proof systems. In fact, for any languagehaving a quantum interactive proof system there exists a threemessage quantum ArthurMerlin game in whichArthur's only message consists of just a single coinflip that achieves perfect completeness and soundness errorexponentially close to 1/2. ffl Any language having a twomessage quantum ArthurMerlin game is contained in BP \Delta PP. This gives somesuggestion that three messages are stronger than two in
Quantum Lower Bound for the Collision Problem
, 2002
"... The collision problem is to decide whether a function X : . . . , n} is onetoone or twotoone, given that one of these is the case. We show a lower bound of on the number of queries needed by a quantum computer to solve this problem with bounded error probability. The best known upper bou ..."
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Cited by 70 (17 self)
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The collision problem is to decide whether a function X : . . . , n} is onetoone or twotoone, given that one of these is the case. We show a lower bound of on the number of queries needed by a quantum computer to solve this problem with bounded error probability. The best known upper bound is O , but obtaining any lower bound better than# (1) was an open problem since 1997. Our proof uses the polynomial method augmented by some new ideas. We also give a lower bound for the problem of deciding whether two sets are equal or disjoint on a constant fraction of elements. Finally we give implications of these results for quantum complexity theory.
Quantum computing, postselection, and probabilistic polynomialtime
, 2004
"... I study the class of problems efficiently solvable by a quantum computer, given the ability to “postselect” on the outcomes of measurements. I prove that this class coincides with a classical complexity class called PP, or Probabilistic PolynomialTime. Using this result, I show that several simple ..."
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Cited by 70 (14 self)
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I study the class of problems efficiently solvable by a quantum computer, given the ability to “postselect” on the outcomes of measurements. I prove that this class coincides with a classical complexity class called PP, or Probabilistic PolynomialTime. Using this result, I show that several simple changes to the axioms of quantum mechanics would let us solve PPcomplete problems efficiently. The result also implies, as an easy corollary, a celebrated theorem of Beigel, Reingold, and Spielman that PP is closed under intersection, as well as a generalization of that theorem due to Fortnow and Reingold. This illustrates that quantum computing can yield new and simpler proofs of major results about classical computation.
Limitations of Quantum Advice and OneWay Communication
 Theory of Computing
, 2004
"... Although a quantum state requires exponentially many classical bits to describe, the laws of quantum mechanics impose severe restrictions on how that state can be accessed. This paper shows in three settings that quantum messages have only limited advantages over classical ones. ..."
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Cited by 59 (16 self)
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Although a quantum state requires exponentially many classical bits to describe, the laws of quantum mechanics impose severe restrictions on how that state can be accessed. This paper shows in three settings that quantum messages have only limited advantages over classical ones.
Hidden translation and orbit coset in quantum computing
 IN PROC. 35TH ACM STOC
, 2003
"... We give efficient quantum algorithms for the problems of Hidden Translation and Hidden Subgroup in a large class of nonabelian solvable groups including solvable groups of constant exponent and of constant length derived series. Our algorithms are recursive. For the base case, we solve efficiently ..."
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Cited by 48 (8 self)
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We give efficient quantum algorithms for the problems of Hidden Translation and Hidden Subgroup in a large class of nonabelian solvable groups including solvable groups of constant exponent and of constant length derived series. Our algorithms are recursive. For the base case, we solve efficiently Hidden Translation in Z n p, whenever p is a fixed prime. For the induction step, we introduce the problem Orbit Coset generalizing both Hidden Translation and Hidden Subgroup, and prove a powerful selfreducibility result: Orbit Coset in a finite group G is reducible to Orbit Coset in G/N and subgroups of N, for any solvable normal subgroup N of G. Our selfreducibility framework combined with Kuperberg’s subexponential quantum algorithm for solving Hidden Translation in any abelian group, leads to subexponential quantum algorithms for Hidden Translation and Hidden Subgroup in any solvable group.
Quantum algorithms for solvable groups
 In Proceedings of the 33rd ACM Symposium on Theory of Computing
, 2001
"... ABSTRACT In this paper we give a polynomialtime quantum algorithm for computing orders of solvable groups. Several other problems, such as testing membership in solvable groups, testing equality of subgroups in a given solvable group, and testing normality of a subgroup in a given solvable group, r ..."
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Cited by 45 (1 self)
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ABSTRACT In this paper we give a polynomialtime quantum algorithm for computing orders of solvable groups. Several other problems, such as testing membership in solvable groups, testing equality of subgroups in a given solvable group, and testing normality of a subgroup in a given solvable group, reduce to computing orders of solvable groups and therefore admit polynomialtime quantum algorithms as well. Our algorithm works in the setting of blackbox groups, wherein none of these problems have polynomialtime classical algorithms. As an important byproduct, our algorithm is able to produce a pure quantum state that is uniform over the elements in any chosen subgroup of a solvable group, which yields a natural way to apply existing quantum algorithms to factor groups of solvable groups. 1.
Quantum MerlinArthur Proof Systems: Are Multiple Merlins More Helpful to Arthur?
, 2008
"... This paper introduces quantum “multipleMerlin”Arthur proof systems in which Arthur receives multiple quantum proofs that are unentangled with each other. Although classical multiproof systems are obviously equivalent to classical singleproof systems (i.e., usual MerlinArthur proof systems), it ..."
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Cited by 40 (7 self)
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This paper introduces quantum “multipleMerlin”Arthur proof systems in which Arthur receives multiple quantum proofs that are unentangled with each other. Although classical multiproof systems are obviously equivalent to classical singleproof systems (i.e., usual MerlinArthur proof systems), it is unclear whether or not quantum multiproof systems collapse to quantum singleproof systems (i.e., usual quantum MerlinArthur proof systems). This paper presents a necessary and sufficient condition under which the number of quantum proofs is reducible to two. It is also proved that, in the case of perfect soundness, using multiple quantum proofs
BQP and the polynomial hierarchy
 in Proceedings of the 42nd ACM symposium on Theory of computing, STOC ’10
, 2010
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Quantum versus classical proofs and advice
 In preparation
, 2006
"... Abstract: This paper studies whether quantum proofs are more powerful than classical proofs, or in complexity terms, whether QMA = QCMA. We prove three results about this question. First, we give a “quantum oracle separation ” between QMA and QCMA. More concretely, we show that any quantum algorithm ..."
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Cited by 30 (16 self)
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Abstract: This paper studies whether quantum proofs are more powerful than classical proofs, or in complexity terms, whether QMA = QCMA. We prove three results about this question. First, we give a “quantum oracle separation ” between QMA and QCMA. More concretely, we show that any quantum algorithm needs Ω queries to find an n� � 2 n m+1 qubit “marked state ” ψ〉, even if given an mbit classical description of ψ 〉 together with a quantum black box that recognizes ψ〉. Second, we give an explicit QCMA protocol that nearly achieves this lower bound. Third, we show that, in the one previouslyknown case where quantum proofs seemed to provide an exponential advantage, classical proofs are basically just as powerful. In particular, Watrous gave a QMA protocol for verifying nonmembership in finite groups. Under plausible grouptheoretic assumptions, we give a QCMA protocol for the same problem. Even with no assumptions, our protocol makes only polynomially many queries to the group oracle. We end with some conjectures about quantum versus classical oracles, and about the possibility of a classical oracle separation between QMA and QCMA. ACM Classification: F.1.2, F.1.3