arXiv subjects

John Watrous

Publications and source records attributed to John Watrous.

At least 19 recordsLinked to original sources

arXiv2025

Understanding Quantum Information and Computation

This is a course on the theory of quantum computing. It consists of 16 lessons, each with a video and written component, covering the basics of quantum information, quantum algorithms (including query algorithms, Shor's algorithm for integer factorization, and Grover's algorithm), the general formulation of quantum information (including density matrices, quantum channels, and general measurements), and quantum error correction (including the basics, the stabilizer formalism, CSS codes, the toric code, and fault-tolerant quantum computation).

quant-ph↗

arXiv2025

A Framework for Quantum Advantage

As quantum computing approaches the threshold where certain tasks demonstrably outpace their classical machines, the need for a precise, clear, consensus-driven definition of quantum advantage becomes essential. Rapid progress in the field has blurred this term across companies, architectures, and application domains. Here, we aim to articulate an operational definition for quantum advantage that is both platform-agnostic and empirically verifiable. Building on this framework, we highlight the algorithmic families most likely to achieve early advantage. Finally, we outline our vision for the near future, in which quantum computers enhance existing high-performance computing platforms, enabling new frontiers in chemistry, materials discovery, optimization, and beyond.

quant-ph↗

arXiv2021

Quantum game theory and the complexity of approximating quantum Nash equilibria

This paper is concerned with complexity theoretic aspects of a general formulation of quantum game theory that models strategic interactions among rational agents that process and exchange quantum information. In particular, we prove that the computational problem of finding an approximate Nash equilibrium in a broad class of quantum games is, like the analogous problem for classical games, included in (and therefore complete for) the complexity class PPAD. Our main technical contribution, which facilitates this inclusion, is an extension of prior methods in computational game theory to strategy spaces that are characterized by semidefinite programs.

cs.CC↗

arXiv2020

On the mixed-unitary rank of quantum channels

In the theory of quantum information, the mixed-unitary quantum channels, for any positive integer dimension $n$, are those linear maps that can be expressed as a convex combination of conjugations by $n\times n$ complex unitary matrices. We consider the mixed-unitary rank of any such channel, which is the minimum number of distinct unitary conjugations required for an expression of this form. We identify several new relationships between the mixed-unitary rank~$N$ and the Choi rank~$r$ of mixed-unitary channels, the Choi rank being equal to the minimum number of nonzero terms required for a Kraus representation of that channel. Most notably, we prove that the inequality $N\leq r^2-r+1$ is satisfied for every mixed-unitary channel (as is the equality $N=2$ when $r=2$), and we exhibit the first known examples of mixed-unitary channels for which $N>r$. Specifically, we prove that there exist mixed-unitary channels having Choi rank $d+1$ and mixed-unitary rank $2d$ for infinitely many positive integers $d$, including every prime power $d$. We also examine the mixed-unitary ranks of the mixed-unitary Werner--Holevo channels.

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arXiv2020

Detecting mixed-unitary quantum channels is NP-hard

A quantum channel is said to be a mixed-unitary channel if it can be expressed as a convex combination of unitary channels. We prove that, given the Choi representation of a quantum channel, it is NP-hard with respect to polynomial-time Turing reductions to determine whether or not that channel is a mixed-unitary channel. This hardness result holds even under the assumption that the channel is not within an inverse-polynomial distance (in the dimension of the space upon which it acts) of the boundary of the mixed-unitary channels.

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arXiv2020

Complexity limitations on one-turn quantum refereed games

This paper studies complexity theoretic aspects of quantum refereed games, which are abstract games between two competing players that send quantum states to a referee, who performs an efficiently implementable joint measurement on the two states to determine which of the player wins. The complexity class $\mathrm{QRG}(1)$ contains those decision problems for which one of the players can always win with high probability on yes-instances and the other player can always win with high probability on no-instances, regardless of the opposing player's strategy. This class trivially contains $\mathrm{QMA} \cup \text{co-}\mathrm{QMA}$ and is known to be contained in $\mathrm{PSPACE}$. We prove stronger containments on two restricted variants of this class. Specifically, if one of the players is limited to sending a classical (probabilistic) state rather than a quantum state, the resulting complexity class $\mathrm{CQRG}(1)$ is contained in $\exists\cdot\mathrm{PP}$ (the nondeterministic polynomial-time operator applied to $\mathrm{PP}$); while if both players send quantum states but the referee is forced to measure one of the states first, and incorporates the classical outcome of this measurement into a measurement of the second state, the resulting class $\mathrm{MQRG}(1)$ is contained in $\mathrm{P}\cdot\mathrm{PP}$ (the unbounded-error probabilistic polynomial-time operator applied to $\mathrm{PP}$).

cs.CC↗

arXiv2018

Certifying optimality for convex quantum channel optimization problems

We identify necessary and sufficient conditions for a quantum channel to be optimal for any convex optimization problem in which the optimization is taken over the set of all quantum channels of a fixed size. Optimality conditions for convex optimization problems over the set of all quantum measurements of a given system having a fixed number of measurement outcomes are obtained as a special case. In the case of linear objective functions for measurement optimization problems, our conditions reduce to the well-known Holevo-Yuen-Kennedy-Lax measurement optimality conditions. We illustrate how our conditions can be applied to various state transformation problems having non-linear objective functions based on the fidelity, trace distance, and quantum relative entropy.

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arXiv2018

On the complementary quantum capacity of the depolarizing channel

The qubit depolarizing channel with noise parameter $η$ transmits an input qubit perfectly with probability $1-η$, and outputs the completely mixed state with probability $η$. We show that its complementary channel has positive quantum capacity for all $η>0$. Thus, we find that there exists a single parameter family of channels having the peculiar property of having positive quantum capacity even when the outputs of these channels approach a fixed state independent of the input. Comparisons with other related channels, and implications on the difficulty of studying the quantum capacity of the depolarizing channel are discussed.

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arXiv2018

Time-reversal of rank-one quantum strategy functions

The quantum strategy (or quantum combs) framework is a useful tool for reasoning about interactions among entities that process and exchange quantum information over the course of multiple turns. We prove a time-reversal property for a class of linear functions, defined on quantum strategy representations within this framework, that corresponds to the set of rank-one positive semidefinite operators on a certain space. This time-reversal property states that the maximum value obtained by such a function over all valid quantum strategies is also obtained when the direction of time for the function is reversed, despite the fact that the strategies themselves are generally not time reversible. An application of this fact is an alternative proof of a known relationship between the conditional min- and max-entropy of bipartite quantum states, along with generalizations of this relationship.

quant-ph↗

arXiv2018

Revisiting the simulation of quantum Turing machines by quantum circuits

Yao (1993) proved that quantum Turing machines and uniformly generated quantum circuits are polynomially equivalent computational models: $t \geq n$ steps of a quantum Turing machine running on an input of length $n$ can be simulated by a uniformly generated family of quantum circuits with size quadratic in $t$, and a polynomial-time uniformly generated family of quantum circuits can be simulated by a quantum Turing machine running in polynomial time. We revisit the simulation of quantum Turing machines with uniformly generated quantum circuits, which is the more challenging of the two simulation tasks, and present a variation on the simulation method employed by Yao together with an analysis of it. This analysis reveals that the simulation of quantum Turing machines can be performed by quantum circuits having depth linear in $t$, rather than quadratic depth, and can be extended to variants of quantum Turing machines, such as ones having multi-dimensional tapes. Our analysis is based on an extension of a method of Arrighi, Nesme, and Werner (2011) that allows for the localization of causal unitary evolutions.

cs.CC↗

arXiv2018

Oracle Separations for Quantum Statistical Zero-Knowledge

This paper investigates the power of quantum statistical zero knowledge interactive proof systems in the relativized setting. We prove the existence of an oracle relative to which quantum statistical zero-knowledge does not contain UP intersect coUP, and we prove that quantum statistical zero knowledge does not contain UP relative to a random oracle with probability 1. Our proofs of these statements rely on a bound on output state discrimination for relativized quantum circuits based on the quantum adversary method of Ambainis, following a technique similar to one used by Ben-David and Kothari to prove limitations on a query complexity variant of quantum statistical zero-knowledge.

cs.CC↗

arXiv2017

Extended Nonlocal Games from Quantum-Classical Games

Several variants of nonlocal games have been considered in the study of quantum entanglement and nonlocality. This paper concerns two of these variants, called quantum-classical games and extended nonlocal games. We give a construction of an extended nonlocal game from any quantum-classical game that allows one to translate certain facts concerning quantum- classical games to extended nonlocal games. In particular, based on work of Regev and Vidick, we conclude that there exist extended nonlocal games for which no finite-dimensional entangled strategy can be optimal. While this conclusion is a direct consequence of recent work of Slofstra, who proved a stronger, analogous result for ordinary (non-extended) nonlocal games, the proof based on our construction is considerably simpler, and the construction itself might potentially have other applications in the study of entanglement and nonlocality.

quant-ph↗

arXiv2016

Ancilla dimension in quantum channel discrimination

Single-shot quantum channel discrimination is a fundamental task in quantum information theory. It is well known that entanglement with an ancillary system can help in this task, and furthermore that an ancilla with the same dimension as the input of the channels is always sufficient for optimal discrimination of two channels. A natural question to ask is whether the same holds true for the output dimension. That is, in cases when the output dimension of the channels is (possibly much) smaller than the input dimension, is an ancilla with dimension equal to the output dimension always sufficient for optimal discrimination? We show that the answer to this question is "no" by construction of a family of counterexamples. This family contains instances with arbitrary finite gap between the input and output dimensions, and still has the property that in every case, for optimal discrimination, it is necessary to use an ancilla with dimension equal to that of the input. The proof relies on a characterization of all operators on the trace norm unit sphere that maximize entanglement negativity. In the case of density operators we generalize this characterization to a broad class of entanglement measures, which we call weak entanglement measures. This characterization allows us to conclude that a quantum channel is reversible if and only if it preserves entanglement as measured by any weak entanglement measure, with the structure of maximally entangled states being equivalent to the structure of reversible maps via the Choi isomorphism. We also include alternate proofs of other known characterizations of channel reversibility.

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arXiv2016

Quantum Proofs

Quantum information and computation provide a fascinating twist on the notion of proofs in computational complexity theory. For instance, one may consider a quantum computational analogue of the complexity class \class{NP}, known as QMA, in which a quantum state plays the role of a proof (also called a certificate or witness), and is checked by a polynomial-time quantum computation. For some problems, the fact that a quantum proof state could be a superposition over exponentially many classical states appears to offer computational advantages over classical proof strings. In the interactive proof system setting, one may consider a verifier and one or more provers that exchange and process quantum information rather than classical information during an interaction for a given input string, giving rise to quantum complexity classes such as QIP, QSZK, and QMIP* that represent natural quantum analogues of IP, SZK, and MIP. While quantum interactive proof systems inherit some properties from their classical counterparts, they also possess distinct and uniquely quantum features that lead to an interesting landscape of complexity classes based on variants of this model. In this survey we provide an overview of many of the known results concerning quantum proofs, computational models based on this concept, and properties of the complexity classes they define. In particular, we discuss non-interactive proofs and the complexity class QMA, single-prover quantum interactive proof systems and the complexity class QIP, statistical zero-knowledge quantum interactive proof systems and the complexity class \class{QSZK}, and multiprover interactive proof systems and the complexity classes QMIP, QMIP*, and MIP*.

quant-ph↗

arXiv2016

Operational meaning of quantum measures of recovery

Several information measures have recently been defined which capture the notion of "recoverability." In particular, the fidelity of recovery quantifies how well one can recover a system $A$ of a tripartite quantum state, defined on systems $ABC$, by acting on system $C$ alone. The relative entropy of recovery is an associated measure in which the fidelity is replaced by relative entropy. In this paper, we provide concrete operational interpretations of the aforementioned recovery measures in terms of a computational decision problem and a hypothesis testing scenario. Specifically, we show that the fidelity of recovery is equal to the maximum probability with which a computationally unbounded quantum prover can convince a computationally bounded quantum verifier that a given quantum state is recoverable. The quantum interactive proof system giving this operational meaning requires four messages exchanged between the prover and verifier, but by forcing the prover to perform his actions in superposition, we construct a different proof system that requires only two messages. The result is that the associated decision problem is in QIP(2) and another argument establishes it as hard for QSZK (both classes contain problems believed to be difficult to solve for a quantum computer). We finally prove that the regularized relative entropy of recovery is equal to the optimal Type II error exponent when trying to distinguish many copies of a tripartite state from a recovered version of this state, such that the Type I error is constrained to be no larger than a constant.

quant-ph↗

arXiv2016

Zero-knowledge proof systems for QMA

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.

quant-ph↗

arXiv2015

Extended nonlocal games and monogamy-of-entanglement games

We study a generalization of nonlocal games---which we call extended nonlocal games---in which the players, Alice and Bob, initially share a tripartite quantum state with the referee. In such games, the winning conditions for Alice and Bob may depend on outcomes of measurements made by the referee, on its part of the shared quantum state, in addition to Alice and Bob's answers to randomly selected questions. Our study of this class of games was inspired by the monogamy-of-entanglement games introduced by Tomamichel, Fehr, Kaniewski, and Wehner, which they also generalize. We prove that a natural extension of the Navascues--Pironio--Acin hierarchy of semidefinite programs converges to the optimal commuting operator value of extended nonlocal games, and we prove two extensions of results of Tomamichel et al. concerning monogamy-of-entanglement games.

quant-ph↗

arXiv2014

Limitations on separable measurements by convex optimization

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.

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