The class QMA(k), introduced by , 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? 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. First, we give a protocol by which a verifier can be convinced that a 3SAT formula of size n is satisfiable, with constant soundness, given O(sqrt(n)) unentangled quantum witnesses with O(log n) qubits each. Our protocol relies on the existence of very short PCPs. Second, we show that assuming a weak version of the 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. Third, we prove the nonexistence of "perfect disentanglers" for simulating multiple Merlins with one.
The class @math , or Quantum Merlin-Arthur, consists of all languages that admit a proof protocol in which Merlin sends Arthur a polynomial-size quantum state @math , and then Arthur decides whether to accept or reject in quantum polynomial time. This class was introduced by Knill @cite_9 , Kitaev @cite_24 , and Watrous @cite_23 as a quantum analogue of @math . By now we know a reasonable amount about @math : for example, it allows amplification of success probabilities, is contained in @math , and has natural complete promise problems. (See Aharonov and Naveh @cite_19 for a survey.)
@cite_24 @cite_19 @cite_9 @cite_23
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This paper introduces quantum multiple-Merlin''-Arthur proof systems in which Arthur receives multiple quantum proofs that are unentangled with each other. Although classical multi-proof systems are obviously equivalent to classical single-proof systems (i.e., usual Merlin-Arthur proof systems), it is unclear whether or not quantum multi-proof systems collapse to quantum single-proof systems (i.e., usual quantum Merlin-Arthur 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 does not increase the power of quantum Merlin-Arthur proof systems. It is well known that probabilistic boolean decision trees cannot be much more powerful than deterministic ones (N. Nisan, SIAM J. Comput.20, No. 6 (1991), 999?1007). Motivated by a question if randomization can significantly speed up a nondeterministic computation via a boolean decision tree, we address structural properties of Arthur?Merlin games in this model and prove some lower bounds. We consider two cases of interest, the first when the length of communication between the players is limited and the second, if it is not. While in the first case we can carry over the relations between the corresponding Turing complexity classes, in the second case we observe in contrast with Turing complexity that a one-round Merlin?Arthur protocol is as powerful as a general interactive proof system and, in particular, can simulate a one-round Arthur?Merlin protocol. Moreover, we show that sometimes a Merlin?Arthur protocol can be more efficient than an Arthur?Merlin protocol and than a Merlin?Arthur protocol with limited communication. This is the case for a boolean function whose set of zeroes is a code with high minimum distance and a natural uniformity condition. Such functions provide an example when the Merlin?Arthur complexity is 1 with one-sided error ??(23, 1), but at the same time the nondeterministic decision tree complexity is ?(n). The latter should be contrasted with another fact we prove. Namely, if a function has Merlin?Arthur complexity 1 with one-sided error probability ??(0, 23, then its nondeterministic complexity is bounded by a constant. Other results of the paper include connections with the block sensitivity and related combinatorial properties of a boolean function. We show that any language in nondeterministic time @math , where the number of iterated exponentials is an arbitrary function @math , can be decided by a multiprover interactive proof system with a classical polynomial-time verifier and a constant number of quantum entangled provers, with completeness @math and soundness @math , where the number of iterated exponentials is @math and @math is a universal constant. The result was previously known for @math and @math ; we obtain it for any time-constructible function @math . The result is based on a compression technique for interactive proof systems with entangled provers that significantly simplifies and strengthens a protocol compression result of Ji (STOC'17). As a separate consequence of this technique we obtain a different proof of Slofstra's recent result (unpublished) on the uncomputability of the entangled value of multiprover games. Finally, we show that even minor improvements to our compression result would yield remarkable consequences in computational complexity theory and the foundations of quantum mechanics: first, it would imply that the class MIP* contains all computable languages; second, it would provide a negative resolution to a multipartite version of Tsirelson's problem on the relation between the commuting operator and tensor product models for quantum correlations. Many interesting computational problems can be reformulated in terms of decision trees. A natural classical algorithm is to then run a random walk on the tree, starting at the root, to see if the tree contains a node @math level from the root. We devise a quantum-mechanical algorithm that evolves a state, initially localized at the root, through the tree. We prove that if the classical strategy succeeds in reaching level @math in time polynomial in @math then so does the quantum algorithm. Moreover, we find examples of trees for which the classical algorithm requires time exponential in @math but for which the quantum algorithm succeeds in polynomial time. The examples we have so far, however, could also be solved in polynomial time by different classical algorithms. We study the problem of approximating the commuting-operator value of a two-player non-local game. It is well-known that it is @math -complete to decide whether the classical value of a non-local game is 1 or @math . Furthermore, as long as @math is small enough, this result does not depend on the gap @math . In contrast, a recent result of Fitzsimons, Ji, Vidick, and Yuen shows that the complexity of computing the quantum value grows without bound as the gap @math decreases. In this paper, we show that this also holds for the commuting-operator value of a game. Specifically, in the language of multi-prover interactive proofs, we show that the power of @math (proofs with two provers, one round, completeness probability @math , soundness probability @math , and commuting-operator strategies) can increase without bound as the gap @math gets arbitrarily small. Our results also extend naturally in two ways, to perfect zero-knowledge protocols, and to lower bounds on the complexity of computing the approximately-commuting value of a game. Thus we get lower bounds on the complexity class @math - @math of perfect zero-knowledge multi-prover proofs with approximately-commuting operator strategies, as the gap @math gets arbitrarily small. While we do not know any computable time upper bound on the class @math , a result of the first author and Vidick shows that for @math and @math , the class @math , with constant communication from the provers, is contained in @math . We give a lower bound of @math (ignoring constants inside the function) for this class, which is tight up to polynomial factors assuming the exponential time hypothesis. Recently a great deal of attention has been focused on quantum computation following a sequence of results [Bernstein and Vazirani, in Proc. 25th Annual ACM Symposium Theory Comput., 1993, pp. 11--20, SIAM J. Comput., 26 (1997), pp. 1277--1339], [Simon, in Proc. 35th Annual IEEE Symposium Foundations Comput. Sci., 1994, pp. 116--123, SIAM J. Comput., 26 (1997), pp. 1340--1349], [Shor, in Proc. 35th Annual IEEE Symposium Foundations Comput. Sci., 1994, pp. 124--134] suggesting that quantum computers are more powerful than classical probabilistic computers. Following Shor's result that factoring and the extraction of discrete logarithms are both solvable in quantum polynomial time, it is natural to ask whether all of @math can be efficiently solved in quantum polynomial time. In this paper, we address this question by proving that relative to an oracle chosen uniformly at random with probability 1 the class @math cannot be solved on a quantum Turing machine (QTM) in time @math . We also show that relative to a permutation oracle chosen uniformly at random with probability 1 the class @math cannot be solved on a QTM in time @math . The former bound is tight since recent work of Grover [in Proc. @math th Annual ACM Symposium Theory Comput. , 1996] shows how to accept the class @math relative to any oracle on a quantum computer in time @math . We investigate the minimization of Newton's functional for the problem of the body of minimal resistance with maximal height @math cite butt in the class of convex developable functions defined in a disc. This class is a natural candidate to find a (non-radial) minimizer in accordance with the results of cite lrp2. We prove that the minimizer in this class has a minimal set in the form of a regular polygon with @math sides centered in the disc, and numerical experiments indicate that the natural number @math is a non-decreasing function of @math . The corresponding functions all achieve a lower value of the functional than the optimal radially symmetric function with the same height @math . We give two new quantum algorithms for solving semidefinite programs (SDPs) providing quantum speed-ups. We consider SDP instances with @math constraint matrices, each of dimension @math , rank @math , and sparsity @math . The first algorithm assumes an input model where one is given access to entries of the matrices at unit cost. We show that it has run time @math , where @math is the error. This gives an optimal dependence in terms of @math and quadratic improvement over previous quantum algorithms when @math . The second algorithm assumes a fully quantum input model in which the matrices are given as quantum states. We show that its run time is @math , with @math an upper bound on the trace-norm of all input matrices. In particular the complexity depends only poly-logarithmically in @math and polynomially in @math . We apply the second SDP solver to the problem of learning a good description of a quantum state with respect to a set of measurements: Given @math measurements and copies of an unknown state @math , we show we can find in time @math a description of the state as a quantum circuit preparing a density matrix which has the same expectation values as @math on the @math measurements, up to error @math . The density matrix obtained is an approximation to the maximum entropy state consistent with the measurement data considered in Jaynes' principle from statistical mechanics. As in previous work, we obtain our algorithm by "quantizing" classical SDP solvers based on the matrix multiplicative weight method. One of our main technical contributions is a quantum Gibbs state sampler for low-rank Hamiltonians with a poly-logarithmic dependence on its dimension, which could be of independent interest. It is a fundamental problem to decide how many copies of an unknown mixed quantum state are necessary and sufficient to determine the state. Previously, it was known only that estimating states to error @math in trace distance required @math copies for a @math -dimensional density matrix of rank @math . Here, we give a theoretical measurement scheme (POVM) that requires @math copies to estimate @math to error @math in infidelity, and a matching lower bound up to logarithmic factors. This implies @math copies suffice to achieve error @math in trace distance. We also prove that for independent (product) measurements, @math copies are necessary in order to achieve error @math in infidelity. For fixed @math , our measurement can be implemented on a quantum computer in time polynomial in @math . In quantum zero knowledge, the assumption was made that the verifier is only using unitary operations. Under this assumption, many nice properties have been shown about quantum zero knowledge, including the fact that Honest-Verifier Quantum Statistical Zero Knowledge ( @math ) is equal to Cheating-Verifier Quantum Statistical Zero Knowledge ( @math ) (see Wat02,Wat06 ). In this paper, we study what happens when we allow an honest verifier to flip some coins in addition to using unitary operations. Flipping a coin is a non-unitary operation but doesn 't seem at first to enhance the cheating possibilities of the verifier since a classical honest verifier can flip coins. In this setting, we show an unexpected result: any classical Interactive Proof has an Honest-Verifier Quantum Statistical Zero Knowledge proof with coins. Note that in the classical case, honest verifier @math is no more powerful than @math and hence it is not believed to contain even @math . On the other hand, in the case of cheating verifiers, we show that Quantum Statistical Zero Knowledge where the verifier applies any non-unitary operation is equal to Quantum Zero-Knowledge where the verifier uses only unitaries. One can think of our results in two complementary ways. If we would like to use the honest verifier model as a means to study the general model by taking advantage of their equivalence, then it is imperative to use the unitary definition without coins, since with the general one this equivalence is most probably not true. On the other hand, if we would like to use quantum zero knowledge protocols in a cryptographic scenario where the honest-but-curious model is sufficient, then adding the unitary constraint severely decreases the power of quantum zero knowledge protocols. In a previous paper [BS] we proved, using the elements of the theory of nilpotent groups , that some of the fundamental computational problems in matriz groups belong to NP . These problems were also shown to belong to coNP , assuming an unproven hypothesis concerning finite simple groups . The aim of this paper is to replace most of the (proven and unproven) group theory of [BS] by elementary combinatorial arguments. The result we prove is that relative to a random oracle B , the mentioned matrix group problems belong to ( NP∩coNP ) B . The problems we consider are membership in and order of a matrix group given by a list of generators. These problems can be viewed as multidimensional versions of a close relative of the discrete logarithm problem. Hence NP∩coNP might be the lowest natural complexity class they may fit in. We remark that the results remain valid for black box groups where group operations are performed by an oracle. The tools we introduce seem interesting in their own right. We define a new hierarchy of complexity classes AM ( k ) “just above NP ”, introducing Arthur vs. Merlin games , the bounded-away version of Papdimitriou's Games against Nature . We prove that in spite of their analogy with the polynomial time hierarchy, the finite levels of this hierarchy collapse to AM=AM (2). Using a combinatorial lemma on finite groups [BE], we construct a game by which the nondeterministic player (Merlin) is able to convince the random player (Arthur) about the relation [ G ]= N provided Arthur trusts conclusions based on statistical evidence (such as a Slowly-Strassen type “proof” of primality). One can prove that AM consists precisely of those languages which belong to NP B for almost every oracle B . Our hierarchy has an interesting, still unclarified relation to another hierarchy, obtained by removing the central ingredient from the User vs. Expert games of Goldwasser, Micali and Rackoff. In this paper, we propose and study a master-equation based approach to drive a quantum network with @math qubits to a consensus (symmetric) state introduced by The state evolution of the quantum network is described by a Lindblad master equation with the Lindblad terms generated by continuous-time swapping operators, which also introduce an underlying interaction graph. We establish a graphical method that bridges the proposed quantum consensus scheme and classical consensus dynamics by studying an induced graph (with @math nodes) of the quantum interaction graph (with @math qubits). A fundamental connection is then shown that quantum consensus over the quantum graph is equivalent to componentwise classical consensus over the induced graph, which allows various existing works on classical consensus to be applicable to the quantum setting. Some basic scaling and structural properties of the quantum induced graph are established via combinatorial analysis. Necessary and sufficient conditions for exponential and asymptotic quantum consensus are obtained, respectively, for switching quantum interaction graphs. As a quantum analogue of classical synchronization of coupled oscillators, quantum synchronization conditions are also presented, in which the reduced states of all qubits tend to a common trajectory. We provide the first two-party protocol allowing Alice and Bob to evaluate privately even against active adversaries any completely positive, trace-preserving map @math @math , given as a quantum circuit, upon their joint quantum input state @math . Our protocol leaks no more to any active adversary than an ideal functionality for @math provided Alice and Bob have the cryptographic resources for active secure two-party classical computation. Our protocol is constructed from the protocol for the same task secure against specious adversaries presented in [4]. We propose a class of qubit networks that admit the perfect state transfer of any quantum state in a fixed period of time. Unlike many other schemes for quantum computation and communication, these networks do not require qubit couplings to be switched on and off. When restricted to @math -qubit spin networks of identical qubit couplings, we show that @math is the maximal perfect communication distance for hypercube geometries. Moreover, if one allows fixed but different couplings between the qubits, then perfect state transfer can be achieved over arbitrarily long distances in a linear chain. 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. First, we show that BQP qpoly is contained in PP poly, where BQP qpoly is the class of problems solvable in quantum polynomial time, given a polynomial-size "quantum advice state" that depends only on the input length. This resolves a question of Buhrman, and means that we should not hope for an unrelativized separation between quantum and classical advice. Underlying our complexity result is a general new relation between deterministic and quantum one-way communication complexities, which applies to partial as well as total functions. Second, we construct an oracle relative to which NP is not contained in BQP qpoly. To do so, we use the polynomial method to give the first correct proof of a direct product theorem for quantum search. This theorem has other applications; for example, it can be used to fix a flawed result of Klauck about quantum time-space tradeoffs for sorting. Third, we introduce a new trace distance method for proving lower bounds on quantum one-way communication complexity. Using this method, we obtain optimal quantum lower bounds for two problems of Ambainis, for which no nontrivial lower bounds were previously known even for classical randomized protocols. Prior work has established that all problems in NP admit classical zero-knowledge proof systems, and under reasonable hardness assumptions for quantum computations, these proof systems can be made secure against quantum attacks. We prove a result representing a further quantum generalization of this fact, which is that every problem in the complexity class QMA has a quantum zero-knowledge proof system. More specifically, assuming the existence of an unconditionally binding and quantum computationally concealing commitment scheme, we prove that every problem in the complexity class QMA has a quantum interactive proof system that is zero-knowledge with respect to efficient quantum computations. Our QMA proof system is sound against arbitrary quantum provers, but only requires an honest prover to perform polynomial-time quantum computations, provided that it holds a quantum witness for a given instance of the QMA problem under consideration. We present a new quantum algorithm that either finds a triangle (a copy of @math ) in an undirected graph @math on @math nodes, or it outputs reject'' if @math is triangle free. The algorithm uses @math queries, and it is based on a new design concept of Ambainis that incorporates the benefits of quantum walks into Grover search. The algorithm both improves on, and is simpler than a recent algorithm of Szegedy which has @math query complexity. The Triangle Problem was first treated in [BDH+01], where an algorithm with @math query complexity was presented (here @math is the number of edges of @math ). We introduce the problem of *shadow tomography*: given an unknown D-dimensional quantum mixed state ρ, as well as known two-outcome measurements E1,…,EM, estimate the probability that Ei accepts ρ, to within additive error e, for each of the M measurements. How many copies of ρ are needed to achieve this, with high probability? Surprisingly, we give a procedure that solves the problem by measuring only O( e−5·log4 M·logD) copies. This means, for example, that we can learn the behavior of an arbitrary n-qubit state, on *all* accepting rejecting circuits of some fixed polynomial size, by measuring only nO( 1) copies of the state. This resolves an open problem of the author, which arose from his work on private-key quantum money schemes, but which also has applications to quantum copy-protected software, quantum advice, and quantum one-way communication. Recently, building on this work, have given a different approach to shadow tomography using semidefinite programming, which achieves a savings in computation time. Freedman, Kitaev, and Wang (2002), and later Aharonov, Jones, and Landau (2009), established a quantum algorithm to "additively" approximate the Jones polynomial V(L;t) at any principal root of unity t. The strength of this additive approximation depends exponentially on the bridge number of the link presentation. Freedman, Larsen, and Wang (2002) established that the approximation is universal for quantum computation at a non- lattice, principal root of unity. We show that any value-distinguishing approximation of the Jones polynomial at these non-lattice roots of unity is #P-hard. Given the power to decide whetherjV(L;t)j b for fixed constants 0 1, T(G; x; y) is #P-hard to approximate within a factor of c even for planar graphs G. Along the way, we clarify and generalize both Aaronson's theorem and the Solovay- Kitaev theorem. We present a protocol that transforms any quantum multi-prover interactive proof into a nonlocal game in which questions consist of logarithmic number of bits and answers of constant number of bits. As a corollary, it follows that the promise problem corresponding to the approximation of the nonlocal value to inverse polynomial accuracy is complete for QMIP*, and therefore NEXP-hard. This establishes that nonlocal games are provably harder than classical games without any complexity theory assumptions. Our result also indicates that gap amplification for nonlocal games may be impossible in general and provides a negative evidence for the feasibility of the gap amplification approach to the multi-prover variant of the quantum PCP conjecture.