John Watrous

FOCS Conference 2016 Conference Paper

Zero-Knowledge Proof Systems for QMA

• Anne Broadbent
• Zhengfeng Ji
• Fang Song 0001
• John Watrous

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.

STOC Conference 2010 Conference Paper

QIP = PSPACE

• Rahul Jain 0001
• Zhengfeng Ji
• Sarvagya Upadhyay
• John Watrous

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.

FOCS Conference 2009 Conference Paper

Two-Message Quantum Interactive Proofs Are in PSPACE

• Rahul Jain 0001
• Sarvagya Upadhyay
• John Watrous

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.

STOC Conference 2007 Conference Paper

Toward a general theory of quantum games

• Gus Gutoski
• John Watrous

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-Jamiolkowski representation of quantum, with respect to which each strategy is described by a single 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.

STOC Conference 2006 Conference Paper

Zero-knowledge against quantum attacks

• John Watrous

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 = QSZK HV . 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.

TCS Journal 2003 Journal Article

PSPACE has constant-round quantum interactive proof systems

• John Watrous

In this paper 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.

FOCS Conference 2002 Conference Paper

imits on the Power of Quantum Statistical Zero-Knowledge

• John Watrous

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.

TCS Journal 2002 Journal Article

Two-way finite automata with quantum and classical states

• Andris Ambainis
• John Watrous

We introduce 2-way finite automata with quantum and classical states (2qcfa's). This is a variant on the 2-way quantum finite automata (2qfa) model which may be simpler to implement than unrestricted 2qfa's; the internal state of a 2qcfa may include a quantum part that may be in a (mixed) quantum state, but the tape head position is required to be classical. We show two languages for which 2qcfa's are better than classical 2-way automata. First, 2qcfa's can recognize palindromes, a language that cannot be recognized by 2-way deterministic or probabilistic finite automata. Second, in polynomial time 2qcfa's can recognize {anbn|n∈ N}, a language that can be recognized classically by a 2-way probabilistic automaton but only in exponential time.

STOC Conference 2001 Conference Paper

One-dimensional quantum walks

• Andris Ambainis
• Eric Bach 0001
• Ashwin Nayak 0001
• Ashvin Vishwanath
• John Watrous

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

STOC Conference 2001 Conference Paper

Quantum algorithms for solvable groups

• John Watrous

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.

FOCS Conference 2000 Conference Paper

Fast parallel circuits for the quantum Fourier transform

• Richard Cleve
• John Watrous

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/).

STOC Conference 2000 Conference Paper

Parallelization, amplification, and exponential time simulation of quantum interactive proof systems

• Alexei Y. Kitaev
• John Watrous

FOCS Conference 2000 Conference Paper

Succinct quantum proofs for properties of finite groups

• John Watrous

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.

FOCS Conference 1999 Conference Paper

On Quantum and Classical Space-bounded Processes with Algebraic Transition Amplitudes

• John Watrous

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.

FOCS Conference 1999 Conference Paper

PSPACE Has Constant-Round Quantum Interactive Proof Systems

• John Watrous

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.

FOCS Conference 1997 Conference Paper

On the Power of Quantum Finite State Automata

• Attila Kondacs
• John Watrous

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.

FOCS Conference 1995 Conference Paper

On One-Dimensional Quantum Cellular Automata

• John Watrous

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.