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71
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 78 (7 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
Zeroknowledge against quantum attacks
 STOC
, 2006
"... This paper proves that several interactive proof systems are zeroknowledge against general quantum attacks. This includes the wellknown GoldreichMicaliWigderson classical zeroknowledge protocols for Graph Isomorphism and Graph 3Coloring (assuming the existence of quantum computationally conce ..."
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Cited by 54 (0 self)
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This paper proves that several interactive proof systems are zeroknowledge against general quantum attacks. This includes the wellknown GoldreichMicaliWigderson classical zeroknowledge protocols for Graph Isomorphism and Graph 3Coloring (assuming the existence of quantum computationally concealing commitment schemes in the second case). Also included is a quantum interactive protocol for a complete problem for the complexity class of problems having "honest verifier" quantum statistical zeroknowledge proofs, which therefore establishes that honest verifier and general quantum statistical zeroknowledge are equal: QSZK = QSZKHV. Previously no nontrivial proof systems were known to be zeroknowledge against quantum attacks, except in restricted settings such as the honestverifier and common reference string models. This paper therefore establishes for the first time that true zeroknowledge is indeed possible in the presence of quantum information and computation.
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 41 (8 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
The Jones polynomial: quantum algorithms and applications in quantum complexity theory
"... We analyze relationships between the Jones polynomial and quantum computation. First, we present two polynomialtime quantum algorithms which give additive approximations of the Jones polynomial, in the sense of Bordewich, Freedman, Lovász and Welsh, of any link obtained from a certain general famil ..."
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Cited by 37 (6 self)
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We analyze relationships between the Jones polynomial and quantum computation. First, we present two polynomialtime quantum algorithms which give additive approximations of the Jones polynomial, in the sense of Bordewich, Freedman, Lovász and Welsh, of any link obtained from a certain general family of closures of braids, evaluated at any primitive root of unity. This family encompasses the wellknown plat and trace closures, generalizing results recently obtained by Aharonov, Jones and Landau. We base our algorithms on a local qubit implementation of the unitary JonesWenzl representations of the braid group which makes the underlying representation theory apparent, allowing us to provide an algorithm for approximating the HOMFLYPT twovariable polynomial of the trace closure of a braid at certain pairs of values as well. Next, we provide a selfcontained proof that any quantum computation can be replaced by an additive approximation of the Jones polynomial, evaluated at almost any primitive root of unity. This theorem was originally proved by Freedman, Larsen and Wang in the context of topological quantum computation, and the necessary notion of approximation was later provided by Bordewich et al. Our proof is simpler as it uses a more natural encoding of twoqubit unitaries into the rectangular representation of the eightstrand braid group. We then give QCMAcomplete and PSPACEcomplete problems which are based on braids. Finally, we conclude with direct proofs that evaluating the Jones polynomial of the plat closure at most primitive roots of unity is a #Phard problem, while learning its most significant bit is PPhard, without taking the usual route through the Tutte polynomial and graph coloring. 1
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 29 (15 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
The power of unentanglement
, 2008
"... The class QMA(k), introduced by Kobayashi et al., consists of all languages that can be verified using k unentangled quantum proofs. Many of the simplest questions about this class have remained embarrassingly open: for example, can we give any evidence that k quantum proofs are more powerful than o ..."
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Cited by 27 (3 self)
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The class QMA(k), introduced by Kobayashi et al., consists of all languages that can be verified using k unentangled quantum proofs. Many of the simplest questions about this class have remained embarrassingly open: for example, can we give any evidence that k quantum proofs are more powerful than one? Can we show any upper bound on QMA(k), besides the trivial NEXP? Does QMA(k) = QMA(2) for k ≥ 2? Can QMA(k) protocols be amplified to exponentially small error? In this paper, we make progress on all of the above questions. • We give a protocol by which a verifier can be convinced that a 3Sat formula of size n is satisfiable, with constant soundness, given Õ ( √ n) unentangled quantum witnesses with O (log n) qubits each. Our protocol relies on Dinur’s version of the PCP Theorem and is inherently nonrelativizing. • We show that assuming the famous Additivity Conjecture from quantum information theory, any QMA(2) protocol can be amplified to exponentially small error, and QMA(k) = QMA(2) for all k ≥ 2. • We give evidence that QMA(2) ⊆ PSPACE, by showing that this would follow from “strong amplification ” of QMA(2) protocols. • We prove the nonexistence of “perfect disentanglers” for simulating multiple Merlins with one.
All languages in NP have very short quantum proofs
"... In this note we show that all languages in NP have very short (logarithmic size) quantum proofs which can be verified provided that two unentangled copies are given. We thus introduce a new complexity class QMAlog(2) and show that NP ⊆ QMAlog(2). This gives a new perspective when compared to the pre ..."
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Cited by 26 (0 self)
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In this note we show that all languages in NP have very short (logarithmic size) quantum proofs which can be verified provided that two unentangled copies are given. We thus introduce a new complexity class QMAlog(2) and show that NP ⊆ QMAlog(2). This gives a new perspective when compared to the previously known result QMAlog = BQP. 1
Fast amplification of QMA
 Quantum Inf. Comput
"... Abstract Given a verifier circuit for a problem in QMA, we show how to exponentially amplify the gap between its acceptance probabilities in the 'yes' and 'no' cases, with a method that is quadratically faster than the procedure given by Marriott and Watrous [1]. Our constructio ..."
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Cited by 18 (2 self)
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Abstract Given a verifier circuit for a problem in QMA, we show how to exponentially amplify the gap between its acceptance probabilities in the 'yes' and 'no' cases, with a method that is quadratically faster than the procedure given by Marriott and Watrous [1]. Our construction is natively quantum, based on the analogy of a product of two reflections and a quantum walk. Second, in some special cases we show how to amplify the acceptance probability for good witnesses to 1, making a step towards the proof that QMA with onesided error (QMA 1 ) is equal to QMA. Finally, we simplify the filterstate method to search for QMA witnesses by Poulin and Wocjan
Entanglement in interactive proof systems with binary answers
 In Proceedings of STACS 2006
, 2006
"... If two classical provers share an entangled state, the resulting interactive proof system is significantly weakened [6]. We show that for the case where the verifier computes the XOR of two binary answers, the resulting proof system is in fact no more powerful than a system based on a single quantum ..."
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Cited by 17 (1 self)
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If two classical provers share an entangled state, the resulting interactive proof system is significantly weakened [6]. We show that for the case where the verifier computes the XOR of two binary answers, the resulting proof system is in fact no more powerful than a system based on a single quantum prover: ⊕MIP ∗ [2] ⊆ QIP(2). This also implies that ⊕MIP ∗ [2] ⊆ EXP which was previously shown using a different method [7]. This contrasts with an interactive proof system where the two provers do not share entanglement. In that case, ⊕MIP[2] = NEXP for certain soundness and completeness parameters [6]. 1