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Short Quantum Games [article]

Gus Gutoski
2005 arXiv   pre-print
In this thesis we introduce quantum refereed games, which are quantum interactive proof systems with two competing provers. We focus on a restriction of this model that we call "short quantum games" and we prove an upper bound and a lower bound on the expressive power of these games. For the lower bound, we prove that every language having an ordinary quantum interactive proof system also has a short quantum game. An important part of this proof is the establishment of a quantum measurement
more » ... reliably distinguishes between quantum states chosen from disjoint convex sets. For the upper bound, we show that certain types of quantum refereed games, including short quantum games, are decidable in deterministic exponential time by supplying a separation oracle for use with the ellipsoid method for convex feasibility.
arXiv:cs/0511017v1 fatcat:cthe6q5jrbgy3o6gwop2c27ora

Quantum Strategies and Local Operations [article]

Gus Gutoski
2012 arXiv   pre-print
This thesis is divided into two parts. In Part I we introduce a new formalism for quantum strategies, which specify the actions of one party in any multi-party interaction involving the exchange of multiple quantum messages among the parties. This formalism associates with each strategy a single positive semidefinite operator acting only upon the tensor product of the input and output message spaces for the strategy. We establish three fundamental properties of this new representation for
more » ... m strategies and we list several applications, including a quantum version of von Neumann's celebrated 1928 Min-Max Theorem for zero-sum games and an efficient algorithm for computing the value of such a game. In Part II we establish several properties of a class of quantum operations that can be implemented locally with shared quantum entanglement or classical randomness. In particular, we establish the existence of a ball of local operations with shared randomness lying within the space spanned by the no-signaling operations and centred at the completely noisy channel. The existence of this ball is employed to prove that the weak membership problem for local operations with shared entanglement is strongly NP-hard. We also provide characterizations of local operations in terms of linear functionals that are positive and "completely" positive on a certain cone of Hermitian operators, under a natural notion of complete positivity appropriate to that cone. We end the thesis with a discussion of the properties of no-signaling quantum operations.
arXiv:1003.0038v2 fatcat:x2n762novvd6tjcwccysfcr5ie

Quantum interactive proofs and the complexity of separability testing [article]

Gus Gutoski, Patrick Hayden, Kevin Milner, Mark M. Wilde
2013 arXiv   pre-print
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
more » ... 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 proving 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.
arXiv:1308.5788v1 fatcat:yh52pgnknvdjdfvfumqegepwji

Fidelity of quantum strategies with applications to cryptography

Gus Gutoski, Ansis Rosmanis, Jamie Sikora
2018 Quantum  
We introduce a definition of the fidelity function for multi-round quantum strategies, which we call the strategy fidelity, that is a generalization of the fidelity function for quantum states. We provide many properties of the strategy fidelity including a Fuchs-van de Graaf relationship with the strategy norm. We also provide a general monotinicity result for both the strategy fidelity and strategy norm under the actions of strategy-to-strategy linear maps. We illustrate an operational
more » ... etation of the strategy fidelity in the spirit of Uhlmann's Theorem and discuss its application to the security analysis of quantum protocols for interactive cryptographic tasks such as bit-commitment and oblivious string transfer. Our analysis is general in the sense that the actions of the protocol need not be fully specified, which is in stark contrast to most other security proofs. Lastly, we provide a semidefinite programming formulation of the strategy fidelity.
doi:10.22331/q-2018-09-03-89 fatcat:rx4g2kqu6nexjegoqrtc7gs7ty

Optimal bounds for semi-honest quantum oblivious transfer [article]

André Chailloux, Gus Gutoski, Jamie Sikora
2016 arXiv   pre-print
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 Alice's bit choice and the probability A with which Alice can guess
more » ... of Bob's 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.
arXiv:1310.3262v2 fatcat:urny4s75xvb3znsr6mlnu36mwm

Fidelity of quantum strategies with applications to cryptography [article]

Gus Gutoski and Ansis Rosmanis and Jamie Sikora
2018 arXiv   pre-print
We introduce a definition of the fidelity function for multi-round quantum strategies, which we call the strategy fidelity, that is a generalization of the fidelity function for quantum states. We provide many properties of the strategy fidelity including a Fuchs-van de Graaf relationship with the strategy norm. We also provide a general monotinicity result for both the strategy fidelity and strategy norm under the actions of strategy-to-strategy linear maps. We illustrate an operational
more » ... etation of the strategy fidelity in the spirit of Uhlmann's Theorem and discuss its application to the security analysis of quantum protocols for interactive cryptographic tasks such as bit-commitment and oblivious string transfer. Our analysis is general in the sense that the actions of the protocol need not be fully specified, which is in stark contrast to most other security proofs. Lastly, we provide a semidefinite programming formulation of the strategy fidelity.
arXiv:1704.04033v2 fatcat:6rx3lo7e5zfldo6pru7yumulwq

Quantum One-Time Programs [chapter]

Anne Broadbent, Gus Gutoski, Douglas Stebila
2013 Lecture Notes in Computer Science  
One-time programs are modelled after a black box that allows a single evaluation of a function, and then self-destructs. Because software can, in principle, be copied, general one-time programs exists only in the hardware token model: it has been shown that any function admits a one-time program as long as we assume access to physical devices called one-time memories. Quantum information, with its well-known property of no-cloning, would, at first glance, prevent the basic copying attack for
more » ... ssical programs. We show that this intuition is false: one-time programs for both classical and quantum maps, based solely on quantum information, do not exist, even with computational assumptions. We complement this strong impossibility proof by an equally strong possibility result: assuming the same basic one-time memories as used for classical one-time programs, we show that every quantum map has a quantum one-time program that is secure in the universal composability framework. Our construction relies on a new, simpler quantum authentication scheme and corresponding mechanism for computing on authenticated data.
doi:10.1007/978-3-642-40084-1_20 fatcat:b6umgu4yzvdxzbw4xj2kxu2lwi

Quantum Interactive Proofs with Competing Provers [chapter]

Gus Gutoski, John Watrous
2005 Lecture Notes in Computer Science  
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
more » ... 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.
doi:10.1007/978-3-540-31856-9_50 fatcat:mnf7icnx7ffqhmixjzhwdtdtsm

Entropy, Convex Optimization, and Competitive Quantum Interactions [article]

Gus Gutoski
2006 arXiv   pre-print
This paper has been withdrawn by the author due to errors.
arXiv:cs/0511049v3 fatcat:kvb7kk6vk5dfna3ygxbxzarqc4

Properties of Local Quantum Operations with Shared Entanglement [article]

Gus Gutoski
2009 arXiv   pre-print
Multi-party local quantum operations with shared quantum entanglement or shared classical randomness are studied. The following facts are established: (i) There is a ball of local operations with shared randomness lying within the space spanned by the no-signaling operations and centred at the completely noisy channel. (ii) The existence of the ball of local operations with shared randomness is employed to prove that the weak membership problem for local operations with shared entanglement is
more » ... rongly NP-hard. (iii) Local operations with shared entanglement are characterized in terms of linear functionals that are "completely" positive on a certain cone K of separable Hermitian operators, under a natural notion of complete positivity appropriate to that cone. Local operations with shared randomness (but not entanglement) are also characterized in terms of linear functionals that are merely positive on that same cone K. (iv) Existing characterizations of no-signaling operations are generalized to the multi-party setting and recast in terms of the Choi-Jamiolkowski representation for quantum super-operators. It is noted that the standard nonlocal box is an example of a no-signaling operation that is separable, yet cannot be implemented by local operations with shared entanglement.
arXiv:0805.2209v3 fatcat:g2nyxj6gobayrdifdscy3mbboa

Revisiting TESLA in the Quantum Random Oracle Model [chapter]

Erdem Alkim, Nina Bindel, Johannes Buchmann, Özgür Dagdelen, Edward Eaton, Gus Gutoski, Juliane Krämer, Filip Pawlega
2017 Lecture Notes in Computer Science  
We study a scheme of Bai and Galbraith (CT-RSA'14), also known as TESLA. TESLA was thought to have a tight security reduction from the learning with errors problem (LWE) in the random oracle model (ROM). Moreover, a variant using chameleon hash functions was lifted to the quantum random oracle model (QROM). However, both reductions were later found to be flawed and hence it remained unresolved until now whether TESLA can be proven to be tightly secure in the (Q)ROM. In the present paper we
more » ... de an entirely new, tight security reduction for TESLA from LWE in the QROM (and thus in the ROM). Our security reduction involves the adaptive re-programming of a quantum oracle. Furthermore, we propose parameter sets targeting 128 bits of security against both classical and quantum adversaries and compare TESLA's performance with state-of-the-art signature schemes.
doi:10.1007/978-3-319-59879-6_9 fatcat:qd3gs6qknban5jbkq2fq45fz6i

Interactive proofs with competing teams of no-signaling provers [article]

Gus Gutoski
2011 arXiv   pre-print
This paper studies a generalization of multi-prover interactive proofs in which a verifier interacts with two competing teams of provers: one team attempts to convince the verifier to accept while the other attempts to convince the verifier to reject. Each team consists of two provers who jointly implement a no-signaling strategy. No-signaling strategies are a curious class of joint strategy that cannot in general be implemented without communication between the provers, yet cannot be used as a
more » ... black box to establish communication between them. Attention is restricted in this paper to two-turn interactions in which the verifier asks questions of each of the four provers and decides whether to accept or reject based on their responses. We prove that the complexity class of decision problems that admit two-turn interactive proofs with competing teams of no-signaling provers is a subset of PSPACE. This upper bound matches existing PSPACE lower bounds on the following two disparate and weaker classes of interactive proof: 1. Two-turn multi-prover interactive proofs with only one team of no-signaling provers. 2. Two-turn competing-prover interactive proofs with only one prover per team. Our result implies that the complexity of these two models is unchanged by the addition of a second competing team of no-signaling provers in the first case and by the addition of a second no-signaling prover to each team in the second case. Moreover, our result unifies and subsumes prior PSPACE upper bounds on these classes.
arXiv:1012.0821v2 fatcat:j64w7mzqk5f57ppymplowwgnke

Parallel Approximation of Min-max Problems with Applications to Classical and Quantum Zero-Sum Games

Gus Gutoski, Xiaodi Wu
2012 2012 IEEE 27th Conference on Computational Complexity  
This paper presents an efficient parallel algorithm for a new class of min-max problems based on the matrix multiplicative weights update method. Our algorithm can be used to find near-optimal strategies for competitive two-player classical or quantum games in which a referee exchanges any number of messages with one player followed by any number of additional messages with the other. This algorithm considerably extends the class of games which admit parallel solutions, demonstrating for the
more » ... st time the existence of a parallel algorithm for a game in which one player 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 new 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. We also describe parallel implementations of this oracle for certain sets P, yielding an unconditionally efficient parallel approximation algorithm for the min-max problem (2) for those choices of P. Applications to zero-sum games: This algorithm can be used to find near-optimal strategies for a new class of competitive two-player games that are moderated by a referee and obey the following protocol: 1) The referee exchanges several messages only with Alice.
doi:10.1109/ccc.2012.12 dblp:conf/coco/GutoskiW12 fatcat:cnomniipjffrjatt32qllqvm6a

Parallel Approximation of Min-Max Problems

Gus Gutoski, Xiaodi Wu
2013 Computational Complexity  
This 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,
more » ... onstrating 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.
doi:10.1007/s00037-013-0065-9 fatcat:y2mdtakesbhtjfl7ujuj5xr4cu

Process tomography for unitary quantum channels

Gus Gutoski, Nathaniel Johnston
2014 Journal of Mathematical Physics  
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(d^2) interactive observables, as opposed to the O(d^4) required for tomography of arbitrary channels. This result generalizes, so
more » ... at channels with at most q Kraus operators can be identified with only O(qd^2) interactive observables. 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.
doi:10.1063/1.4867625 fatcat:sufuuxiy6vg4pl5bfpftltz4p4
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