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. The proof system relies on a new variant of the QMA-complete local Hamiltonian problem in which the local terms are described by Clifford operations and standard basis measurements. We believe that the QMA-completeness of this problem may have other uses in quantum complexity.

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 prove limitations on LOCC and separable measurements in bipartite state discrimination problems using techniques from convex optimization. Specific results that we prove include: an exact formula for the optimal probability of correctly discriminating any set of either three or four Bell states via LOCC or separable measurements when the parties are given an ancillary partially entangled pair of qubits; an easily checkable characterization of when an unextendable product set is perfectly discriminated by separable measurements, along with the first known example of an unextendable product set that cannot be perfectly discriminated by separable measurements; and an optimal bound on the success probability for any LOCC or separable measurement for the recently proposed state discrimination problem of Yu, Duan, and Ying.

This paper proves that the computational power of quantum interactive proof systems, with a double-exponentially small gap in acceptance probability between the completeness and soundness cases, is precisely characterized by EXP, the class of problems solvable in exponential time by deterministic Turing machines. This fact, and our proof of it, has implications concerning quantum and classical interactive proof systems in the setting of unbounded error that include the following:

This section of SIAM Journal on Computing contains extended versions of selected papers from the 43rd ACM Symposium on Theory of Computing (STOC), held June 6--8, 2011, in San Jose, California, as part of the fifth Federated Computing Research Conference (FCRC). The STOC proceedings contained 84 papers, which were selected from 304 submissions by the program committee, consisting of Ittai Abraham, Alexandr Andoni, Avrim Blum, Allan Borodin, Kousha Etessami, Lisa Fleischer, Venkatesan Guruswami, David Kempe, Frederic Magniez, Dieter van Melkebeek, Daniele Micciancio, Moni Naor, Kobbi Nissim, Seth Pettie, Ronitt Rubinfeld, Amir Shpilka, Ravi Sundaram, Eva Tardos, Prasad Tetali, Salil Vadhan (chair), Kasturi Varadarajan, Nisheeth Vishnoi, John Watrous, and Ryan Williams. Five of the STOC papers appear in this special section, each one expanded and fully refereed according to the high standards of the journal. They cover a diverse collection of topics: In “Distributed Verification and Hardness of Distributed Approximation,” Das Sarma, Holzer, Kor, Korman, Nanongkai, Pandurangan, Peleg, and Wattenhofer prove strong lower bounds on the power of distributed networks to verify their own properties (such as connectivity) and solve optimization problems such as computing approximate shortest paths or approximate min-cuts. They establish new connections between distributed computation and two-party communication complexity. The paper “Pareto Optimal Solutions for Smoothed Analysts” by Moitra and O'Donnell considers the smoothed complexity of discrete multi-objective optimization problems with $d+1$ linear objectives and with a solution space consisting of binary $n$-vectors. The authors show that, in a suitable smoothed analysis framework for such problems, the expected number of Pareto optimal solutions is at most $n^{2d}$. This improves greatly, as a function of the dimension d, an earlier upper bound established by Roeglin and Teng, which had roughly the form $n^{d^d}$. The paper “Blackbox Identity Testing for Bounded Top-Fanin Depth-3 Circuits: The Field Doesn't Matter” by Saxena and Seshadhri provides the first deterministic polynomial-time identity test for depth-3 arithmetic circuits with bounded top-fanin that only needs blackbox access to the circuit. Their construction has the feature that it works for arbitrary fields. In their paper “An Optimal Lower Bound on the Communication Complexity of Gap-Hamming-Distance,” Chakrabarti and Regev prove a lower bound establishing that the randomized communication complexity of the gap-Hamming-distance problem is linear. In obtaining this result, they have resolved an important and well-studied communication complexity problem having a fundamental connection to the data stream model of computation. Svensson's paper “Santa Claus Schedules Jobs on Unrelated Machines” breaks the barrier of 2 for efficiently approximating the minimum makespan for scheduling jobs on unrelated machines in the setting where all machines on which a given job can run take the same amount of time for that job. We thank the authors, the referees, and the full program committee for all their work, which made this special section possible.

This work considers the quantum interactive proof system model of computation, which is the (classical) interactive proof system model’s natural quantum computational analogue. An exact characterization of the expressive power of quantum interactive proof systems is obtained: the collection of computational problems having quantum interactive proof systems consists precisely of those problems solvable by deterministic Turing machines that use at most a polynomial amount of space (or, more succinctly, QIP = PSPACE). This characterization is proved through the use of a parallelized form of the matrix multiplicative weights update method, applied to a class of semidefinite programs that captures the computational power of quantum interactive proof systems. One striking implication of this characterization is that quantum computing provides no increase in computational power whatsoever over classical computing in the context of interactive proof systems, for it is well known that the collection of computational problems having classical interactive proof systems coincides with those problems solvable by polynomial-space computations.

We prove that the complexity class QIP, which consists of all problems having quantum interactive proof systems, is contained in PSPACE. This containment is proved by applying a parallelized form of the matrix multiplicative weights update method to a class of semidefinite programs that captures the computational power of quantum interactive proofs. As the containment of PSPACE in QIP follows immediately from the well-known equality IP = PSPACE, the equality QIP = PSPACE follows.

This paper studies a simple class of zero-sum games played by two competing quantum players: each player sends a mixed quantum state to a referee, who performs a joint measurement on the two states to determine the players' payoffs. We prove that an equilibrium point of any such game can be approximated by means of an efficient parallel algorithm, which implies that one-turn quantum refereed games, wherein the referee is specified by a quantum circuit, can be simulated in polynomial space.

We prove that QIP(2), the class of problems having two-message quantum interactive proof systems, is a subset of PSPACE. This relationship is obtained by means of an efficient parallel algorithm, based on the matrix multiplicative weights update method, for approximately solving a certain class of semidefinite programs.

This paper proves that several interactive proof systems are zero-knowledge against general quantum attacks. This includes the well-known Goldreich–Micali–Wigderson classical zero-knowledge protocols for graph isomorphism and graph 3-coloring (assuming the existence of quantum computationally concealing commitment schemes in the second case). Also included is a quantum interactive proof system for a complete problem for the complexity class of problems having honest verifier quantum statistical zero-knowledge proofs, which therefore establishes that honest verifier and general quantum statistical zero-knowledge are equal: $\mathrm{QSZK}= \mathrm{QSZK}_{\mathrm{HV}}$. Previously no nontrivial interactive proof systems were known to be zero-knowledge against quantum attacks, except in restricted settings such as the honest verifier and common reference string models. This paper therefore establishes for the first time that true zero-knowledge is indeed possible in the presence of quantum information and computation.

We study properties of quantum strategies, which are complete specifications of a given party's actions in any multiple-round interaction involving the exchange of quantum information with one or more other parties. In particular, we focus on a representation of quantum strategies that generalizes the Choi-Jamio{\l}kowski representation of quantum operations. This new representation associates with each strategy a positive semidefinite operator acting only on the tensor product of its input and output spaces. Various facts about such representations are established, and two applications are discussed: the first is a new and conceptually simple proof of Kitaev's lower bound for strong coin-flipping, and the second is a proof of the exact characterization QRG = EXP of the class of problems having quantum refereed games.

This paper proves that several interactive proof systems are zero-knowledge against general quantum attacks. This includes the well-known Goldreich-Micali-Wigderson classical zero-knowledge protocols for Graph Isomorphism and Graph 3-Coloring (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 zero-knowledge proofs, which therefore establishes that honest verifier and general quantum statistical zero-knowledge are equal: QSZK = QSZKHV. Previously no non-trivial proof systems were known to be zero-knowledge against quantum attacks, except in restricted settings such as the honest-verifier and common reference string models. This paper therefore establishes for the first time that true zero-knowledge is indeed possible in the presence of quantum information and computation.

This paper studies quantum Arthur–Merlin games, which are Arthur–Merlin games in which Arthur and Merlin can perform quantum computations and Merlin can send Arthur quantum information. As in the classical case, messages from Arthur to Merlin are restricted to be strings of uniformly generated random bits. It is proved that for one-message quantum Arthur–Merlin games, which correspond 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 power over BQP and a simple proof that $$ {\text{QMA}} \subseteq {\text{PP}}. $$ Other facts that are proved include the equivalence of three (or more) message quantum Arthur–Merlin games with ordinary quantum interactive proof systems and some basic properties concerning two-message quantum Arthur–Merlin games.

This paper investigates various aspects of the nonlocal effects that can arise when entangled quantum information is shared between two parties. A natural framework for studying nonlocality is that of cooperative games with incomplete information, where two cooperating players may share entanglement. Here, nonlocality can be quantified in terms of the values of such games. We review some examples of non-locality and show that it can profoundly affect the soundness of two-prover interactive proof systems. We then establish limits on nonlocal behavior by upper-bounding the values of several of these games. These upper bounds can be regarded as generalizations of the so-called Tsirelson inequality. We also investigate the amount of entanglement required by optimal and nearly optimal quantum strategies.

This paper studies quantum Arthur-Merlin games, which are a restricted form of quantum interactive proof system in which the verifier's messages are given by unbiased coin-flips. The following results are proved. For one-message quantum Arthur-Merlin games, which correspond 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 power over BQP and a simple proof that QMA /spl sube/ PP. In the case of three or more messages, quantum Arthur-Merlin games are equivalent in power to ordinary quantum interactive proof systems. In fact, for any language having a quantum interactive proof system there exists a three-message quantum Arthur-Merlin game in which Arthur's only message consists of just a single coin-flip that achieves perfect completeness and soundness error exponentially close to 1/2. Any language having a two-message quantum Arthur-Merlin game is contained in BP /spl middot/ PP. This gives some suggestion that three messages are stronger than two in the quantum Arthur-Merlin setting.

Arthur does not have a lot of time to spend performing difficult computations. He's recently obtained a quantum computer, but often it seems not to help - he only has a few quantum algorithms, and Merlin maintains that there aren't any other interesting ones, so Merlin is forced to convince the untrusting Arthur of the truth of various facts. However, Arthur and Merlin have a new resource at their disposal: quantum information. Some relationships among complexity classes defined by quantum Arthur-Merlin games and other commonly studied complexity classes are known, but many open questions remain. In this paper, I discuss quantum Arthur-Merlin games in detail, with an emphasis on open problems.

In this paper we propose a definition for (honest verifier) quantum statistical zero-knowledge interactive proof systems and study the resulting complexity class, which we denote QSZK/sub HV/. We prove several facts regarding this class, including: the following problem is a complete promise problem for QSZKHV: given instructions for preparing two mixed quantum states, are the states close together or far apart in the trace norm metric? This problem is a quantum generalization of the complete promise problem of Sahai and Vadhan (1997) for (classical) statistical zero-knowledge; QSZK/sub HV/ is closed under complement; QSZK/sub HV//spl sube/PSPACE. (At present it is not known if arbitrary quantum interactive proof systems can be simulated in PSPACE even for one-round proof systems); any polynomial-round honest verifier quantum statistical zero-knowledge proof system can be simulated by a two-message (i.e., one-round) honest verifier quantum statistical zero-knowledge proof system. Similarly, any polynomial-round honest verifier quantum statistical zero-knowledge proof system can be simulated by a three-message public-coin honest verifier quantum statistical zero-knowledge proof system. These facts establish close connections between classical statistical zero-knowledge and our definition for quantum statistical zero-knowledge, and give some insight regarding the effect of this zero-knowledge restriction on quantum interactive proof systems. The relationship between our definition and possible definitions of general (i.e., not necessarily honest) quantum statistical zero-knowledge are also discussed.

We define and analyze quantum computational variants of \nrandom walks on one-dimensional lattices. In particular, we analyze a quantum analog of the symmetric random \nwalk, which we call the Hadamard walk. Several striking \ndifferences between the quantum and classical cases are ob- served. For example, when unrestricted in either direction, \nthe Hadamard walk has position that is nearly uniformly \ndistributed in the range [-t/√2, t/√2] after t steps, which \nis in sharp contrast to the classical random walk, which has \ndistance O(√t) from the origin with high probability. With \nan absorbing boundary immediately to the left of the starting position, the probability that the walk exits to the left is 2/π, and with an additional absorbing boundary at location n, the probability that the walk exits to the left actually increases, approaching 1/√2 in the limit. In the classical case both values are 1.

ABSTRACT In this paper we give a polynomial-time 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 polynomial-time quantum algorithms as well. Our algorithm works in the setting of black-box groups, wherein none of these problems have polynomial-time 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.

We give new bounds on the circuit complexity of the quantum Fourier transform (QFT). We give an upper bound of O(log n+log log(1//spl epsiv/)) on the circuit depth for computing an approximation of the QFT with respect to the modulus 2/sup n/ with error bounded by /spl epsiv/. Thus, even for exponentially small error, our circuits have depth O(log n). The best previous depth bound was O(n), even for approximations with constant error. Moreover, our circuits have size O(n log(n//spl epsiv/)). As an application of this depth bound, we show that P. Shor's (1997) factoring algorithm may be based on quantum circuits with depth only O(log n) and polynomial size, in combination with classical polynomial-time pre- and postprocessing. Next, we prove an /spl Omega/(log n) lower bound on the depth complexity of approximations of the QFT with constant error. This implies that the above upper bound is asymptotically tight (for a reasonable range of values of /spl epsiv/). We also give an upper bound of O(n(log n)/sup 2/ log log n) on the circuit size of the exact QFT modulo 2/sup n/, for which the best previous bound was O(n/sup 2/). Finally, based on our circuits for the QFT with power-of-2 moduli, we show that the QFT with respect to an arbitrary modulus m can be approximated with accuracy /spl epsiv/ with circuits of depth O((log log m)(log log 1//spl epsiv/)) and size polynomial in log m+log(1//spl epsiv/).

The article considers a quantum computational variant of nondeterminism based on the notion of a quantum proof, which is a quantum state that plays a role similar to a certificate in an NP-type proof. Specifically, we consider quantum proofs for properties of black-box groups, which are finite groups whose elements are encoded as strings of a given length and whose group operations are performed by a group oracle. We prove that for an arbitrary group oracle, there exist succinct (polynomial-length) quantum proofs for the Group Non-Membership problem that can be checked with small error in polynomial time on a quantum computer. Classically, this is impossible; it is proved that there exists a group oracle, relative to which this problem does not have succinct proofs that can be checked classically with bounded error in polynomial time (i.e., the problem is not in MA relative to the group oracle constructed). By considering a certain subproblem of the Group Non-Membership problem, we obtain a simple proof that there exists an oracle relative to which BQP is not contained in MA. Finally, we show that quantum proofs for non-membership and classical proofs for various other group properties can be combined to yield succinct quantum proofs for other group properties not having succinct proofs in the classical setting, such as verifying that a number divides the order of a group and verifying that a group is not a simple group.

In this paper we consider quantum interactive proof systems, which are interactive proof systems in which the prover and verier may perform quantum computations and exchange quantum information. We prove that any polynomial-round quantum interactive proof system with two-sided bounded error can be parallelized to a quantum interactive proof system with exponentially small one-sided error in which the prover and verier exchange only 3 messages. This yields a simplied proof that PSPACE has 3-message quantum interactive proof systems. We also prove that any language having a quantum interactive proof system can be decided in deterministic exponential time, implying that single-prover quantum interactive proof systems are strictly less powerful than multiple-prover classical interactive proof systems unless EXP = NEXP. 1. INTRODUCTION Interactive proof systems were introduced by Babai [3] and Goldwasser, Micali, and Racko [17] in 1985. In the same year, Deutsch [10] gave the rst for...

There are a number of questions in quantum complexity that have been resolved in the time-bounded setting, but remain open in the space-bounded setting. For example, it is not currently known if space-bounded probabilistic computations can be simulated by space-bounded quantum machines without allowing measurements during the computation, while it is known that an analogous statement holds in the time-bounded case. A more general question asks if measurements during a quantum computation can allow for more space-efficient solutions to certain problems. In this paper we show that space-bounded quantum Turing machines can efficiently simulate a limited class of random processes-random walks on undirected graphs-without relying on measurements during the computation. By means of such simulations, it is demonstrated that the undirected graph connectivity problem for regular graphs can be solved by one-sided error quantum Turing machines that run in logspace and require a single measurement at the end of their computations. It follows that symmetric logspace is contained in the quantum analogue of randomized logspace, i.e., SL/spl sube/QR/sub H/L.

We introduce quantum interactive proof systems, which are interactive proof systems in which the prover and verifier may perform quantum computations and exchange quantum messages. It is proved that every language in PSPACE has a quantum interactive proof system that requires a total of only three messages to be sent between the prover and verifier and has exponentially small (one-sided) probability of error. It follows that quantum interactive proof systems are strictly more powerful than classical interactive proof systems in the constant-round case unless the polynomial time hierarchy collapses to the second level.

We define a class of stochastic processes based on evolutions and measurements of quantum systems, and consider the complexity of predicting their long term behavior. It is shown that a very general class of decision problems regarding these stochastic processes can be efficiently solved classically in the space-bounded case. The following corollaries are implied by our main result for any space-constructible space bound s satisfying s(n)=/spl Omega/(log n): (i) any space O(s) uniform family of quantum circuit acting on s qubits and consisting of unitary gates and measurement gates defined in a typical way by matrices of algebraic numbers can be simulated by an unbounded error space O(s) ordinary (i.e., fair-coin flipping) probabilistic Turing machine, and hence by space O(s) uniform classical (deterministic) circuits of depth O(s/sup 2/) and size 2/sup 0/(s); (2) any quantum Turing machine running in space s, having arbitrary algebraic transition amplitudes, allowing unrestricted measurements during its computation, and having no restrictions on running time can be simulated by a space O(s) ordinary probabilistic Turing machine in the unbounded error setting. We also obtain the following classical result: any unbounded error probabilistic Turing machine running in space s that allows algebraic probabilities and algebraic cut-point can be simulated by a space O(s) ordinarily probabilistic Turing machine with cut-point 1/2. Our technique for handling algebraic numbers in the above simulations may be of independent interest. It is shown that any real algebraic number can be accurately approximated by a ratio of GapL functions.

This paper investigates the relative power of space-bounded quantum and classical (probabilistic) computational models. The following relationships are proved. 1. Any probabilistic Turing machine (PTM) which runs in space s and which halts absolutely (i.e. halts with certainty after a finite number of steps) can be simulated in space O(s) by a quantum Turing machine (QTM). If the PTM operates with bounded error, then the QTM may be taken to operate with bounded error as well, although the QTM may not halt absolutely in this case. In the unbounded error case, the QTM may be taken to halt absolutely. 2. Any QTM running in space s can be simulated by an unbounded error PTM running in space O(s). No assumptions on the probability of error or Turing time for the QTM are required, but it is assumed that all transition amplitudes of the quantum machine are rational. It follows that unbounded error, space O(s) bounded quantum Turing machines and probabilistic Turing machines are equivalent in power. This implies that any space s QTM can be simulated deterministically in space O(s/sup 2/), and further that any (unbounded-error) QTM running in log-space can be simulated in NC/sup 2/ We also consider quantum analogues of nondeterministic and one-sided error probabilistic space-bounded classes, and prove some simple relationships regarding these classes.

In this paper, we introduce 1-way and 2-way quantum finite state automata (1qfa's and 2qfa's), which are the quantum analogues of deterministic, nondeterministic and probabilistic 1-way and 2-way finite state automata. We prove the following facts regarding 2qfa's. 1. For any /spl epsiv/>0, there is a 2qfa M which recognizes the non-regular language L={a/sup m/b/sup m/|m/spl ges/1} with (one-sided) error bounded by E, and which halts in linear time. Specifically, M accepts any string in L with probability 1 and rejects any string not in L with probability at least 1-/spl epsiv/. 2. For every regular language L, there is a reversible (and hence quantum) 2-way finite state automaton which recognizes L and which runs in linear time. In fact, it is possible to define 2qfar's which recognize the non-context-free language {a/sup m/b/sup m/c/sup m/|m/spl ges/1}, based on the same technique used for 1. Consequently, the class of languages recognized by linear time, bounded error 2qfa's properly includes the regular languages. Since it is known that 2-way deterministic, nondeterministic and polynomial expected time, bounded error probabilistic finite automata can recognize only regular languages, it follows that 2qfa's are strictly more powerful than these "classical" models. In the case of 1-way automata, the situation is reversed. We prove that the class of languages recognizable by bounded error 1qfa's is properly contained in the class of regular languages.

Since Richard Feynman introduced the notion of quantum computation in 1982, various models of "quantum computers" have been proposed (R. Feynman, 1992). These models include quantum Turing machines and quantum circuits. We define another quantum computational model, one dimensional quantum cellular automata, and demonstrate that any quantum Turing machine can be efficiently simulated by a one dimensional quantum cellular automaton with constant slowdown. This can be accomplished by consideration of a restricted class of one dimensional quantum cellular automata called one dimensional partitioned quantum cellular automata. We also show that any one dimensional partitioned quantum cellular automaton can be simulated by a quantum Turing machine with linear slowdown, but the problem of efficiently simulating an arbitrary one dimensional quantum cellular automaton with a quantum Turing machine is left open. From this discussion, some interesting facts concerning these models are easily deduced.