Discrete optimization using quantum annealing on sparse Ising models

Bian, Zhengbing; Chudak, Fabián A.; Israel, Robert B.; Lackey, Brad; Macready, William G.; Roy, Aidan

doi:10.3389/fphy.2014.00056

Cited by 104 publications

(134 citation statements)

References 30 publications

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“…Although the main focus of the quantum annealing computational paradigm [17][18][19] has been on solving discrete optimization problems in a wide variety of application domains [20][21][22][23][24][25][26][27], it has been also introduced as a potential candidate to speed up computations in sampling applications. Indeed, it is an important open research question whether or not quantum annealers can sample from Boltzmann distributions more efficiently than traditional techniques [4,5,9].…”

Section: Introductionmentioning

confidence: 99%

Estimation of effective temperatures in quantum annealers for sampling applications: A case study with possible applications in deep learning

et al. 2016

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An increase in the efficiency of sampling from Boltzmann distributions would have a significant impact on deep learning and other machine-learning applications. Recently, quantum annealers have been proposed as a potential candidate to speed up this task, but several limitations still bar these state-of-the-art technologies from being used effectively. One of the main limitations is that, while the device may indeed sample from a Boltzmann-like distribution, quantum dynamical arguments suggest it will do so with an instance-dependent effective temperature, different from its physical temperature. Unless this unknown temperature can be unveiled, it might not be possible to effectively use a quantum annealer for Boltzmann sampling. In this work, we propose a strategy to overcome this challenge with a simple effective-temperature estimation algorithm. We provide a systematic study assessing the impact of the effective temperatures in the learning of a special class of a restricted Boltzmann machine embedded on quantum hardware, which can serve as a building block for deep-learning architectures. We also provide a comparison to k-step contrastive divergence (CD-k) with k up to 100. Although assuming a suitable fixed effective temperature also allows us to outperform one step contrastive divergence (CD-1), only when using an instance-dependent effective temperature do we find a performance close to that of CD-100 for the case studied here.

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Section: Introductionmentioning

confidence: 99%

Estimation of effective temperatures in quantum annealers for sampling applications: A case study with possible applications in deep learning

et al. 2016

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“…For examples of how this can be done in practice, see ref. [19][20][21][22]. However, there is no indication that this approach is optimal.…”

Section: Introductionmentioning

confidence: 99%

“…19,20, would require the construction of a penalty function (up to an unimportant energy offset) on a set of logical qubits, where the penalty is 0 if the clause is satisfied and greater than or equal to a penalty weight g if the clause is violated. In the ideal case g should be infinite, but in practice its maximal value is limited by the hardware.…”

Section: Introductionmentioning

confidence: 99%

“…For problems which require higher order interactions a mapping must be found from a Higher Order Binary Optimization to a Quadratic Unconstrained Binary Optimization problem. 20 Typically this is done iteratively, with an N-body interaction being reduced to a two-body interaction using a complete graph on N logical bits and (N−2) ancilla bits. 23 This necessitates (N−1) (2N−3) two-body couplers.…”

Section: Introductionmentioning

confidence: 99%

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Circuit design for multi-body interactions in superconducting quantum annealing systems with applications to a scalable architecture

2017

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Quantum annealing provides a way of solving optimization problems by encoding them as Ising spin models which are implemented using physical qubits. The solution of the optimization problem then corresponds to the ground state of the system. Quantum tunneling is harnessed to enable the system to move to the ground state in a potentially high non-convex energy landscape. A major difficulty in encoding optimization problems in physical quantum annealing devices is the fact that many real world optimization problems require interactions of higher connectivity, as well as multi-body terms beyond the limitations of the physical hardware. In this work we address the question of how to implement multi-body interactions using hardware which natively only provides two-body interactions. The main result is an efficient circuit design of such multi-body terms using superconducting flux qubits in which effective N-body interactions are implemented using N ancilla qubits and only two inductive couplers. It is then shown how this circuit can be used as the unit cell of a scalable architecture by applying it to a recently proposed embedding technique for constructing an architecture of logical qubits with arbitrary connectivity using physical qubits which have nearest-neighbor four-body interactions. It is further shown that this design is robust to non-linear effects in the coupling loops, as well as mismatches in some of the circuit parameters.npj Quantum Information (2017) 3:21 ; doi:10.1038/s41534-017-0022-6 INTRODUCTION Solving machine learning and optimization problems by casting them as an Ising spin glass, and then using a physical device to take advantage of quantum fluctuations has been a subject of much recent interest. [1][2][3][4][5][6][7][8][9] This interest is due in a large part to demonstration of the underlying principles of quantum annealing in condensed matter systems 10 and the more recent development of a programmable annealing device by D-Wave Systems Inc.

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“…Given a set of cities and the directed distances between each pair of cities, the TSP asks for a shortest route that visits each city exactly once and returns to the starting city. There are excellent, non-quantum TSP surveys [5,7,10].Quantum computing, particularly quantum annealing [3], is a new paradigm for discrete optimization that could use TSP benchmarks for the hardware. Quantum computing opens the possibility of large speedup with high probability of optimal answers, but requires new techniques for solving the TSP [11,19].…”

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confidence: 99%

Small Traveling Salesman Problems

Warren¹

2017

JAAM

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Abstract. This paper illustrates fundamental principles about small traveling salesman problems (TSPs) which are a current application for quantum annealing computers. The 2048 qubit, quantum annealing computer manufactured by D-Wave Systems is estimated to be able to solve all TSPs on 8 cities, which advances a recent 4-city result on a quantum simulator. Additionally, the D-Wave quantum computer is expected to find all optimal tours for each TSP. To prepare for this, we show the expected quantum output for 5,000 randomly generated TSPs on 6, 8 and 10 cities. The examples in the TSPLIB have 14 or more cities. These are too large for the current quantum annealing processors. We have included an annotated bibliography about solving the TSP on a quantum computer. Keywords:Traveling salesman, optimal tour, combinatorial analysis, discrete optimization, quantum annealing AMS Classification Codes: 68R05, 81P68, 90C27 IntroductionRecently first steps have been taken to solve the traveling salesman problem (TSP) using the computational advantages promised by quantum algorithms [13]. As with any emerging effort, the initial steps have been limited and focused on a small number of cities that fit on the hardware. The major contributions of this paper are for 6, 8 and 10 city TSPs. We present empirical evidence based on 15,000 random samples, 5,000 for each number of cities. The majority of evidence indicates that the number of optimal tours and the length of an optimal tour increase together. All of the data shows that as the number of cities increases, the number of optimal tours increases. In all cases, the distances between cities are random integers in the interval [1,21].Outside the quantum world, the traveling salesman problem continues to be a hot topic that attracts wide interest. The problem is easy to explain and has several diverse applications [9,18], as well as theoretical studies [2,20]. There are many fascinating ways to solve it, usually approximately. Given a set of cities and the directed distances between each pair of cities, the TSP asks for a shortest route that visits each city exactly once and returns to the starting city. There are excellent, non-quantum TSP surveys [5,7,10].Quantum computing, particularly quantum annealing [3], is a new paradigm for discrete optimization that could use TSP benchmarks for the hardware. Quantum computing opens the possibility of large speedup with high probability of optimal answers, but requires new techniques for solving the TSP [11,19].The paper is organized as follows: Section 2 has references for other reports that use quantum computing algorithms for solving the TSP. Section 3 contains the results from a study of the expected output for TSPs from current quantum annealing computers. The output is envisioned to be all optimal tours, since these are obtained on a quantum annealing simulator. We have classified this output for 6, 8 and 10-city TSPs which is the estimated size that can be solved currently. The last section points to future quantum work for the TS...

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Discrete optimization using quantum annealing on sparse Ising models

Cited by 104 publications

References 30 publications

Estimation of effective temperatures in quantum annealers for sampling applications: A case study with possible applications in deep learning

Estimation of effective temperatures in quantum annealers for sampling applications: A case study with possible applications in deep learning

Circuit design for multi-body interactions in superconducting quantum annealing systems with applications to a scalable architecture

Small Traveling Salesman Problems

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