Gus Gutoski

Toward a general theory of quantum games

We study properties of quantum strategies, which are complete specifications of a given partys 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 Kitaevs 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.

Quantum one-time programs

A one-time program is a hypothetical device by which a user may evaluate a circuit on exactly one input of his choice, before the device self-destructs. One-time programs cannot be achieved by software alone, as any software can be copied and re-run. However, it is known that every circuit can be compiled into a one-time program using a very basic hypothetical hardware device called a one-time memory. At first glance it may seem that quantum information, which cannot be copied, might also allow for one-time programs. But it is not hard to see that this intuition is false: one-time programs for classical or quantum circuits based solely on quantum information do not exist, even with computational assumptions.

Quantum interactive proofs with competing provers

This paper studies quantum refereed games, which are quantum interactive proof systems with two competing provers: one that tries to convince the verifier to accept and the other that tries to convince the verifier to reject. We prove that every language having an ordinary quantum interactive proof system also has a quantum refereed game in which the verifier exchanges just one round of messages with each prover. A key part of our proof is the fact that there exists a single quantum measurement that reliably distinguishes between mixed states chosen arbitrarily from disjoint convex sets having large minimal trace distance from one another. We also show how to reduce the probability of error for some classes of quantum refereed games.

On a measure of distance for quantum strategies

The present paper studies an operator norm that captures the distinguishability of quantum strategies in the same sense that the trace norm captures the distinguishability of quantum states or the diamond norm captures the distinguishability of quantum channels. Characterizations of its unit ball and dual norm are established via strong duality of a semidefinite optimization problem. A full, formal proof of strong duality is presented for the semidefinite optimization problem in question. This norm and its properties are employed to generalize a state discrimination result of Gutoski and Watrous [In Proceedings of the 22nd Symposium on Theoretical Aspects of Computer Science (STACS’05), Lecture Notes in Computer Science, Vol. 3404 (Springer, 2005), pp. 605–616. The generalized result states that for any two convex sets S0, S1 of strategies there exists a fixed interactive measurement scheme that successfully distinguishes any choice of S0 ∈ S0 from any choice of S1 ∈ S1 with bias proportional to the minim...

Parallel Approximation of Min-Max Problems

AbstractThis paper presents an efficient parallel approximation scheme for a new class of min-max problems. The algorithm is derived from the matrix multiplicative weights update method and can be used to find near-optimal strategies for competitive two-party classical or quantum interactions in which a referee exchanges any number of messages with one party followed by any number of additional messages with the other. It considerably extends the class of interactions which admit parallel solutions, demonstrating for the first time the existence of a parallel algorithm for an interaction in which one party reacts adaptively to the other. As a consequence, we prove that several competing-provers complexity classes collapse to PSPACE, such as QRG(2), SQG and two new classes called DIP and DQIP. A special case of our result is a parallel approximation scheme for a specific class of semidefinite programs whose feasible region consists of lists of semidefinite matrices that satisfy a transcript-like consistency condition. Applied to this special case, our algorithm yields a direct polynomial-space simulation of multi-message quantum interactive proofs resulting in a first-principles proof of QIP =PSPACE.

Upper bounds for quantum interactive proofs with competing provers

Refereed games are interactive proof systems with two competing provers: one that tries to convince the verifier to accept and another that tries to convince the verifier to reject. In quantum refereed games, the provers and verifier may perform quantum computations and exchange quantum messages. One may consider games with a bounded or unbounded number of rounds of messages between the verifier and provers. In this paper, we prove classical upper bounds on the power of both one-round and many-round quantum refereed games. In particular, we use semidefinite programming to show that many-round quantum refereed games are contained in NEXP. It then follows from the symmetric nature of these games that they are also contained in coNEXP. We also show that one-round quantum refereed games are contained in EXP by supplying a separation oracle for use with the ellipsoid method for convex feasibility.

Quantum Interactive Proofs and the Complexity of Separability Testing

We identify a formal connection between physical problems related to the detection of separable (unentangled) quantum states and complexity classes in theoretical computer science. In particular, we show that to nearly every quantum interactive proof complexity class (including BQP, QMA, QMA(2), and QSZK), there corresponds a natural separability testing problem that is complete for that class. Of particular interest is the fact that the problem of determining whether an isometry can be made to produce a separable state is either QMA-complete or QMA(2)-complete, depending upon whether the distance between quantum states is measured by the one-way LOCC norm or the trace norm. We obtain strong hardness results by employing prior work on entanglement purification protocols to prove that for each n-qubit maximally entangled state there exists a fixed one-way LOCC measurement that distinguishes it from any separable state with error probability that decays exponentially in n.

Process tomography for unitary quantum channels

We study the number of measurements required for quantum process tomography under prior information, such as a promise that the unknown channel is unitary. We introduce the notion of an interactive observable and we show that any unitary channel acting on a d-level quantum system can be uniquely identified among all other channels (unitary or otherwise) with only O(d2) interactive observables, as opposed to the O(d4) required for tomography of arbitrary channels. This result generalizes to the problem of identifying channels with at most q Kraus operators, and slight improvements can be obtained if we wish to identify such a channel only among unital channels or among other channels with q Kraus operators. These results are proven via explicit construction of large subspaces of Hermitian matrices with various conditions on rank, eigenvalues, and partial trace. Our constructions are built upon various forms of totally nonsingular matrices.

Hierarchical Deterministic Bitcoin Wallets that Tolerate Key Leakage

A Bitcoin wallet is a set of private keys known to a user and which allow that user to spend any Bitcoin associated with those keys. In a hierarchical deterministic (HD) wallet, child private keys are generated pseudorandomly from a master private key, and the corresponding child public keys can be generated by anyone with knowledge of the master public key. These wallets have several interesting applications including Internet retail, trustless audit, and a treasurer allocating funds among departments. A specification of HD wallets has even been accepted as Bitcoin standard BIP32.

Optimal bounds for quantum weak oblivious transfer

Oblivious transfer is a fundamental cryptographic primitive in which Bob transfers one of two bits to Alice in such a way that Bob cannot know which of the two bits Alice has learned. We present an optimal security bound for quantum oblivious transfer protocols under a natural and demanding definition of what it means for Alice to cheat. Our lower bound is a smooth tradeoff between the probability B with which Bob can guess Alices bit choice and the probability A with which Alice can guess both of Bobs bits given that she learns one of the bits with certainty. We prove that 2B + A is greater than or equal to 2 in any quantum protocol for oblivious transfer, from which it follows that one of the two parties must be able to cheat with probability at least 2/3. We prove that this bound is optimal by exhibiting a family of protocols whose cheating probabilities can be made arbitrarily close to any point on the tradeoff curve.