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arXiv:1601.03030 · Annual Symposium on Foundations of Computer Science

Simulated Quantum Annealing Can Be Exponentially Faster Than Classical Simulated Annealing

Elizabeth Crosson, Aram W. Harrow

Published Oct 1, 2016• 76 citations
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Abstract

Can quantum computers solve optimization problems much more quickly than classical computers? One major piece of evidence for this proposition has been the fact that Quantum Annealing (QA) finds the minimum of some cost functions exponentially more quickly than classical Simulated Annealing (SA). One such cost function is the simple “Hamming weight with a spike” function in which the input is an n-bit string and the objective function is simply the Hamming weight, plus a tall thin barrier centered around Hamming weight n/4. While the global minimum of this cost function can be found by inspection, it is also a plausible toy model of the sort of local minima that arise in realworld optimization problems. It was shown by Farhi, Goldstone and Gutmann [1] that for this example SA takes exponential time and QA takes polynomial time, and the same result was generalized by Reichardt [2] to include barriers with width nζ and height nαfor ζ + α ≤ 1/2. This advantage could be explained in terms of quantummechanical “tunneling.” Our work considers a classical algorithm known as Simulated Quantum Annealing (SQA) which relates certain quantum systems to classical Markov chains. By proving that these chains mix rapidly, we show that SQA runs in polynomial time on the Hamming weight with spike problem in much of the parameter regime where QA achieves an exponential advantage over SA. While our analysis only covers this toy model, it can be seen as evidence against the prospect of exponential quantum speedup using tunneling. Our technical contributions include extending the canonical path method for analyzing Markov chains to cover the case when not all vertices can be connected by low-congestion paths. We also develop methods for taking advantage of warm starts and for relating the quantum state in QA to the probability distribution in SQA. These techniques may be of use in future studies of SQA or of rapidly mixing Markov chains in general.

DOI
10.1109/focs.2016.81

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Can quantum computers solve optimization problems much more quickly than classical computers? One major piece of evidence for this proposition has been the fact that Quantum Annealing (QA) finds the minimum of some cost functions exponentially more quickly than classical Simulated Annealing (SA). One such cost function is the simple “Hamming weight with a spike” function in which the input is an n-bit string and the objective function is simply the Hamming weight, plus a tall thin barrier centered around Hamming weight n/4. While the global minimum of this cost function can be found by inspection, it is also a plausible toy model of the sort of local minima that arise in realworld optimization problems. It was shown by Farhi, Goldstone and Gutmann [1] that for this example SA takes exponential time and QA takes polynomial time, and the same result was generalized by Reichardt [2] to include barriers with width nζ and height nαfor ζ + α ≤ 1/2. This advantage could be explained in terms of quantummechanical “tunneling.” Our work considers a classical algorithm known as Simulated Quantum Annealing (SQA) which relates certain quantum systems to classical Markov chains. By proving that these chains mix rapidly, we show that SQA runs in polynomial time on the Hamming weight with spike problem in much of the parameter regime where QA achieves an exponential advantage over SA. While our analysis only covers this toy model, it can be seen as evidence against the prospect of exponential quantum speedup using tunneling. Our technical contributions include extending the canonical path method for analyzing Markov chains to cover the case when not all vertices can be connected by low-congestion paths. We also develop methods for taking advantage of warm starts and for relating the quantum state in QA to the probability distribution in SQA. These techniques may be of use in future studies of SQA or of rapidly mixing Markov chains in general.
Can quantum computers solve optimization problems much more quickly than classical computers? One major piece of evidence for this proposition has been the fact that Quantum Annealing (QA) finds the minimum of some cost functions exponentially more quickly than classical Simulated Annealing (SA). One such cost function is the simple “Hamming weight with a spike” function in which the input is an n-bit string and the objective function is simply the Hamming weight, plus a tall thin barrier centered around Hamming weight n/4. While the global minimum of this cost function can be found by inspection, it is also a plausible toy model of the sort of local minima that arise in realworld optimization problems. It was shown by Farhi, Goldstone and Gutmann [1] that for this example SA takes exponential time and QA takes polynomial time, and the same result was generalized by Reichardt [2] to include barriers with width n^ζ and height n^α for ζ + α ≤ 1/2. This advantage could be explained in terms of quantummechanical “tunneling.” Our work considers a classical algorithm known as Simulated Quantum Annealing (SQA) which relates certain quantum systems to classical Markov chains. By proving that these chains mix rapidly, we show that SQA runs in polynomial time on the Hamming weight with spike problem in much of the parameter regime where QA achieves an exponential advantage over SA. While our analysis only covers this toy model, it can be seen as evidence against the prospect of exponential quantum speedup using tunneling. Our technical contributions include extending the canonical path method for analyzing Markov chains to cover the case when not all vertices can be connected by low-congestion paths. We also develop methods for taking advantage of warm starts and for relating the quantum state in QA to the probability distribution in SQA. These techniques may be of use in future studies of SQA or of rapidly mixing Markov chains in general.
In this work we study the convergence of a classical algorithm called Simulated Quantum Annealing (SQA) on the Spike Hamiltonian, a specific toy model Hamiltonian for quantum-mechanical tunneling introduced by [FGG02]. This toy model Hamiltonian encodes a simple bit-symmetric cost function f in the computational basis, and is used to emulate local minima in more complex optimization problems. In previous work [CH16] showed that SQA runs in polynomial time in much of the regime of spikes that QA does, pointing to evidence against an exponential speedup through tunneling. In this paper we extend their analysis to the remaining polynomial regime of energy gaps of the spike Hamiltonian, to show that indeed QA presents no exponential speedup with respect to SQA on this family of toy models.
Combinatorial optimization problems are frequently classified as NP-hard, which means that the time needed to find the optimal solution generally increases exponentially with the problem size. However, the existing research gap highlights the necessity to develop combinatorial optimization methods capable of scaling to large problems while maintaining low solution latency. In this paper, we introduce stochastic simulated quantum annealing (SSQA) for fast solution of combinatorial optimization problems. SSQA is designed based on stochastic computing, which can simulate quantum annealing (QA) by using multiple replicas of spins (probabilistic bits) in classical computing. The use of stochastic computing leads to an efficient parallel spin-state update algorithm, enabling quick search for a solution around the global minimum energy. Therefore, SSQA realizes quantum-like annealing for large-scale problems and can handle fully connected models in combinatorial optimization, unlike QA. The proposed method is evaluated in MATLAB on graph isomorphism problems, which are typical combinatorial optimization problems. It can handle a 100-times larger problem size compared to QA and a 25-times larger problem size compared to a traditional SA method, respectively, for similar convergence probabilities. The time to solution of SSQA is 11.5 times and 423 times smaller than that of the conventional stochastic simulated annealing method and traditional SA, respectively, when solving a 2,025-node combinatorial optimization problem.
Spike Hamiltonians arise from optimization instances for which the adiabatic algorithm provably out performs classical simulated annealing. In this work, we study the efficiency of the adiabatic algorithm for solving the “the Hamming weight with a spike” problem by analyzing the scaling of the spectral gap at the critical point for various sizes of the barrier. Our main result is a rigorous lower bound on the minimum spectral gap for the adiabatic evolution when the bit-symmetric cost function has a thin but polynomially high barrier, which is based on a comparison argument and an improved variational ansatz for the ground state. We also adapt the discrete WKB method for the case of abruptly changing potentials and compare it with the predictions of the spin coherent instanton method which was previously used by Farhi, Goldstone and Gutmann. Finally, our improved ansatz for the ground state leads to a method for predicting the location of avoided crossings in the excited energy states of the thin spike Hamiltonian, and we use a recursion relation to understand the ordering of some of these avoided crossings as a step towards analyzing the previously observed diabatic cascade phenomenon.