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Optimal state discrimination and unstructured search in nonlinear quantum mechanics

Physical Review A 2016年第2期93卷 022314-022314页

作者： Andrew M. Childs Joshua Young Department of Computer Science Institute for Advanced Computer Studies and Joint Center for Quantum Information and Computer Science University of Maryland College Park Maryland 20742 USA Department of Combinatorics & Optimization University of Waterloo Waterloo Ontario Canada N2L 3G1 Institute for Quantum Computing University of Waterloo Waterloo Ontario Canada N2L 3G1 Cheriton School of Computer Science University of Waterloo Waterloo Ontario Canada N2L 3G1

Nonlinear variants of quantum mechanics can solve tasks that are impossible in standard quantum theory, such as perfectly distinguishing nonorthogonal states. Here we derive the optimal protocol for distinguishing two states of a qubit using the Gross-Pitaevskii equation, a model of nonlinear quantum mechanics that arises as an effective description of Bose-Einstein condensates. Using this protocol, we present an algorithm for unstructured search in the Gross-Pitaevskii model, obtaining an exponential improvement over a previous algorithm of Meyer and Wong. This result establishes a limitation on the effectiveness of the Gross-Pitaevskii approximation. More generally, we demonstrate similar behavior under a family of related nonlinearities, giving evidence that the ability to quickly discriminate nonorthogonal states and thereby solve unstructured search is a generic feature of nonlinear quantum mechanics.

关键词： Network searches quantum algorithms

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Universal Adiabatic quantum Computation via the Space-Time Circuit-to-Hamiltonian Construction

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Physical Review Letters 2015年第14期114卷 140501-140501页

作者： David Gosset Barbara M. Terhal Anna Vershynina Institute for Quantum Computing and Dept. of Combinatorics and Optimization University of Waterloo Ontario N2L 3G1 Canada JARA Institute for Quantum Information RWTH Aachen University 52056 Aachen North Rhine-Westphalia Germany

We show how to perform universal adiabatic quantum computation using a Hamiltonian which describes a set of particles with local interactions on a two-dimensional grid. A single parameter in the Hamiltonian is adiabatically changed as a function of time to simulate the quantum circuit. We bound the eigenvalue gap above the unique ground state by mapping our model onto the ferromagnetic XXZ chain with kink boundary conditions; the gap of this spin chain was computed exactly by Koma and Nachtergaele using its q-deformed version of SU(2) symmetry. We also discuss a related time-independent Hamiltonian which was shown by Janzing to be capable of universal computation. We observe that in the limit of large system size, the time evolution is equivalent to the exactly solvable quantum walk on Young’s lattice.

关键词： HAMILTONIAN systems FERROMAGNETIC materials DYNAMICAL systems MAGNETIC materials quantum theory

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Corrigendum to “Coined quantum walks lift the cospectrality of graphs and trees” [Pattern Recognition 42 (9) (2009) 1988–2002]

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Pattern Recognition 2015年第4期48卷 1574-1575页

作者： David Emms Simone Severini Richard C. Wilson Edwin R. Hancock Department of Computer Science University of York York YO10 5DD UK Institute for Quantum Computing Department of Combinatorics and Optimization University of Waterloo 2133 Davis Centre Waterloo Ontario Canada N2L 3G1

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A note on quantum security for post-quantum cryptography

Lecture Notes in Computer Science (including subseries Lectu...

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Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) 2014年 8772卷 246-265页

作者： Song, Fang Department of Combinatorics and Optimization Institute for Quantum Computing University of Waterloo Canada

Shor‘s quantum factoring algorithm and a few other efficient quantum algorithms break many classical crypto-systems. In response, people proposed post-quantum cryptography based on computational problems that are believed hard even for quantum computers. However, security of these schemes against quantum attacks is elusive. This is because existing security analysis (almost) only deals with classical attackers and arguing security in the presence of quantum adversaries is challenging due to unique quantum features such as *** work proposes a general framework to study which classical security proofs can be restored in the quantum setting. Basically, we split a security proof into (a sequence of) classical security reductions, and investigate what security reductions are ‘quantum-friendly’. We characterize sufficient conditions such that a classical reduction can be ‘lifted’ to the quantum *** then apply our lifting theorems to post-quantum signature schemes. We are able to show that the classical generic construction of hash-tree based signatures from one-way functions and and a more efficient variant proposed in [10] carry over to the quantum setting. Namely, assuming existence of (classical) oneway functions that are resistant to efficient quantum inversion algorithms, there exists a quantum-secure signature scheme. We note that the scheme in [10] is a promising (post-quantum) candidate to be implemented in practice and our result further justifies it. Actually, to obtain these result, we formalize a simple criteria, which is motivated by many classical proofs in the literature and is straightforward to check. This makes our lifting theorem easier to apply, and it should be useful elsewhere to prove quantum security of proposed post-quantum cryptographic schemes. Finally we demonstrate the generality of our framework by showing that several existing works (Full-Domain hash in the quantum random-oracle model [47] and the simple hybrid arguments

关键词： quantum computers

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On the robustness of bucket brigade quantum ram 10

On the robustness of bucket brigade quantum ram

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10th Conference on the Theory of quantum Computation, Communication and Cryptography, TQC 2015

作者： Arunachalam, Srinivasan Gheorghiu, Vlad Jochym-O'Connor, Tomas Mosca, Michele Srinivasan, Priyaa Varshinee Amsterdam Netherlands Institute for Quantum Computing University of Waterloo WaterlooONN2L 3G1 Canada Department of Combinatorics and Optimization University of Waterloo WaterlooONN2L 3G1 Canada Department of Physics and Astronomy University of Waterloo WaterlooONN2L 3G1 Canada Perimeter Institute for Theoretical Physics WaterlooONN2L 6B9 Canada Canadian Institute for Advanced Research TorontoONM5G 1Z8 Canada David R. Cheriton School of Computer Science University of Waterloo WaterlooONN2L 3G1 Canada Institute for Quantum Science and Technology University of Calgary CalgaryABT2N 1N4 Canada

ISBN: (纸本)9783939897965

We study the robustness of the bucket brigade quantum random access memory model introduced by Giovannetti, Lloyd, and Maccone [Phys. Rev. Lett. 100, 160501 (2008)]. Due to a result of Regev and Schiff [ICALP'08 pp. 773], we show that for a class of error models the error rate per gate in the bucket brigade quantum memory has to be of order o(2-n/2) (where N = 2n is the size of the memory) whenever the memory is used as an oracle for the quantum searching problem. We conjecture that this is the case for any realistic error model that will be encountered in practice, and that for algorithms with super-polynomially many oracle queries the error rate must be super-polynomially small, which further motivates the need for quantum error correction. We introduce a circuit model for the quantum bucket brigade architecture and argue that quantum error correction for the circuit causes the quantum bucket brigade architecture to lose its primary advantage of a small number of "active" gates, since all components have to be actively error corrected. © Srinivasan Arunachalam, Vlad Gheorghiu, Tomas Jochym-O'Connor, Michele Mosca, and Priyaa Varshinee Srinivasan.

关键词： quantum theory

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Spatial search by continuous-time quantum walks on crystal lattices

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Physical Review A 2014年第5期89卷 052337-052337页

作者： Andrew M. Childs Yimin Ge Department of Combinatorics & Optimization and Institute for Quantum Computing University of Waterloo Waterloo Ontario Canada Perimeter Institute for Theoretical Physics Waterloo Ontario Canada

We consider the problem of searching a general d-dimensional lattice of N vertices for a single marked item using a continuous-time quantum walk. We demand locality, but allow the walk to vary periodically on a small scale. By constructing lattice Hamiltonians exhibiting Dirac points in their dispersion relations and exploiting the linear behavior near a Dirac point, we develop algorithms that solve the problem in a time of O(N) for d>2 and O(NlogN) in d=2. In particular, we show that such algorithms exist even for hypercubic lattices in any dimension. Unlike previous continuous-time quantum walk algorithms on hypercubic lattices in low dimensions, our approach does not use external memory.

关键词： CRYSTAL lattices ALGORITHMS quantum computing quantum computers PROBABILITY theory

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Asymptotic Bound for Heat-Bath Algorithmic Cooling

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Physical Review Letters 2015年第10期114卷 100404-100404页

作者： Sadegh Raeisi Michele Mosca Institute for Quantum Computing University of Waterloo Ontario N2L 3G1 Canada Department of Physics and Astronomy University of Waterloo Ontario N2L 3G1 Canada Department of Combinatorics and Optimization University of Waterloo Ontario N2L 3G1 Canada Perimeter Institute for Theoretical Physics Waterloo Ontario N2L 2Y5 Canada Canadian Institute for Advanced Research Toronto Ontario M5G 1Z8 Canada

The purity of quantum states is a key requirement for many quantum applications. Improving the purity is limited by fundamental laws of thermodynamics. Here, we are probing the fundamental limits for a natural approach to this problem, namely, heat-bath algorithmic cooling (HBAC). The existence of the cooling limit for HBAC techniques was proved by Schulman, Mor, and Weinstein. A bound for this value was found by Elias et al. and numerical testing supported the hypothesis that their bound may be the actual limit. A proof or disproof of whether their bound was the actual limit remained open for the past decade. Here, for the first time, we prove this limit. In the context of quantum thermodynamics, this corresponds to the maximum extractable work from the quantum system. We also establish, in the case of higher dimensional reset systems, how the performance of HBAC depends on the energy spectrum of the reset system.

关键词： quantum states quantum mechanics THERMODYNAMICS quantum theory PHYSICAL & theoretical chemistry

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Linear programming hierarchies suffice for directed steiner tree

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17th International Conference on Integer Programming and Combinatorial optimization, IPCO 2014

作者： Friggstad, Zachary Könemann, Jochen Kun-Ko, Young Louis, Anand Shadravan, Mohammad Tulsiani, Madhur Department of Computing Science University of Alberta Canada Department of Combinatorics and Optimization University of Waterloo Canada Department of Computer Science Princeton University United States College of Computing Georgia Tech. United States Toyota Technical Institute at Chicago United States

ISBN: (纸本)9783319075563

We demonstrate that rounds of the Sherali-Adams hierarchy and 2 rounds of the Lovász-Schrijver hierarchy suffice to reduce the integrality gap of a natural LP relaxation for Directed Steiner Tree in -layered graphs from ω (√k) to O(·logk) where k is the number of terminals. This is an improvement over Rothvoss' result that 2 rounds of the considerably stronger Lasserre SDP hierarchy reduce the integrality gap of a similar formulation to O(·logk). We also observe that Directed Steiner Tree instances with 3 layers of edges have only an O(logk) integrality gap in the standard LP relaxation, complementing the known fact that the gap can be as large as ω (√k) in graphs with 4 layers. © 2014 Springer International Publishing Switzerland.

关键词： Linear programming

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When the asymptotic limit offers no advantage in the local-operations-and-classical-communication paradigm

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Physical Review A 2014年第5期89卷 052310-052310页

作者： Honghao Fu Debbie Leung Laura Mančinska Institute for Quantum Computing and Department of Combinatorics and Optimization University of Waterloo Waterloo Ontario N2L 3G1 Canada Centre for Quantum Technologies National University of Singapore Singapore 119077

We consider bipartite LOCC, the class of operations implementable by local quantum operations and classical communication between two parties. Surprisingly, there are operations that can be approximated to arbitrary precision but are impossible to implement exactly if only a finite number of messages are exchanged. This significantly complicates the analysis of what can or cannot be approximated with LOCC. Toward alleviating this problem, we exhibit two scenarios in which allowing vanishing error does not help. The first scenario is implementation of projective measurements with product measurement operators. The second scenario is the discrimination of unextendable product bases on two three-dimensional systems.

关键词： quantum communication quantum operators quantum systems quantum states quantum gates quantum information theory

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Simulating Hamiltonian Dynamics with a Truncated Taylor Series

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Physical Review Letters 2015年第9期114卷 090502-090502页

作者： Dominic W. Berry Andrew M. Childs Richard Cleve Robin Kothari Rolando D. Somma Department of Physics and Astronomy Macquarie University Sydney New South Wales 2109 Australia Institute for Quantum Computing University of Waterloo Ontario N2L 3G1 Canada Department of Combinatorics and Optimization University of Waterloo Ontario N2L 3G1 Canada Department of Computer Science Institute for Advanced Computer Studies and Joint Center for Quantum Information and Computer Science University of Maryland College Park Maryland 20910 USA Canadian Institute for Advanced Research Toronto Ontario M5G 1Z8 Canada School of Computer Science University of Waterloo Ontario N2L 3G1 Canada Center for Theoretical Physics Massachusetts Institute of Technology Cambridge Massachusetts 02139 USA Theoretical Division Los Alamos National Laboratory Los Alamos New Mexico 87545 USA

We describe a simple, efficient method for simulating Hamiltonian dynamics on a quantum computer by approximating the truncated Taylor series of the evolution operator. Our method can simulate the time evolution of a wide variety of physical systems. As in another recent algorithm, the cost of our method depends only logarithmically on the inverse of the desired precision, which is optimal. However, we simplify the algorithm and its analysis by using a method for implementing linear combinations of unitary operations together with a robust form of oblivious amplitude amplification.

关键词： HAMILTONIAN systems TAYLOR'S series quantum theory quantum computers LOGARITHMIC functions

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