The Quantum Approximate Optimization Algorithm and the Sherrington-Kirkpatrick Model at Infinite Size

Abstract

The Quantum Approximate Optimization Algorithm (QAOA) is a general-purpose algorithm for combinatorial optimization problems whose performance can only improve with the number of layers p. While QAOA holds promise as an algorithm that can be run on near-term quantum computers, its computational power has not been fully explored. In this work, we study the QAOA applied to the Sherrington-Kirkpatrick (SK) model, which can be understood as energy minimization of n spins with all-to-all random signed couplings. There is a recent classical algorithm by Montanari that, assuming a widely believed conjecture, can efficiently find an approximate solution for a typical instance of the SK model to within (1−ϵ) times the ground state energy. We hope to match its performance with the QAOA.Our main result is a novel technique that allows us to evaluate the typical-instance energy of the QAOA applied to the SK model. We produce a formula for the expected value of the energy, as a function of the 2p QAOA parameters, in the infinite size limit that can be evaluated on a computer with O(16p) complexity. We evaluate the formula up to p=12, and find that the QAOA at p=11 outperforms the standard semidefinite programming algorithm. Moreover, we show concentration: With probability tending to one as n→∞, measurements of the QAOA will produce strings whose energies concentrate at our calculated value. As an algorithm running on a quantum computer, there is no need to search for optimal parameters on an instance-by-instance basis since we can determine them in advance. What we have here is a new framework for analyzing the QAOA, and our techniques can be of broad interest for evaluating its performance on more general problems where classical algorithms may fail.

Variational Quantum Approximate Support Vector Machine With Inference Transfer

“…Although VQASVM have freedom to choose upto O(M ) number of parameters using hardware-efficient ansatz, we can choose a PQC design with O(polylog(M )) parameters at the cost of accuracy or training loss performance degradation. 37 Taking this as an assumption with a given accuracy bound of ǫ, the training time complexity of VQASVM drops to O(M N polylog(M )/ǫ), suggesting potential asymptotic speed-up compared to SVM. Our numerical experiments with various PQC designs 35 show an interesting observation that the number of parameters can be smaller than the number of training data as shown in Figs.…”

“…We propose a Variational Quantum Approximate Support Vector Machine (VQASVM) algorithm inspired by quantum approximate optimization algorithm that can reduce the number of parameters PQC designs for certain problems. 37 Fig. 2a summarizes the process of VQASVM.…”

Random Max-CSPs Inherit Algorithmic Hardness from Spin Glasses

“…There has been some study of a near-term quantum algorithm (the QAOA [FGG14]) optimizing spin glasses, with recently-proven rigorous performance bounds [CvD21,FGGZ22]. In fact, for large enough clause density, the performance of the QAOA is identical on a random instance of Max-kXOR and on its corresponding spin glass [BM21, BFM + 21, BGMZ22].…”

Adaptive Problem Solving Dynamics in Gate-Model Quantum Computers

“…As the development of quantum computers evolves significantly [ 1 , 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , 11 , 12 , 13 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , 23 , 24 , 25 , 26 , 27 ], a fundamental need to characterize the attributes of problem solving in quantum computers has arisen. Gate-model quantum computers have particular relevance since most of these architectures allow for practical solutions to be implemented on near-term settings.…”

“…In a gate-model quantum computer, the computational steps are realized via unitary gates. The gates are associated with a gate parameter value, while the computational problem fed into the quantum computer identifies an objective function [ 6 , 7 , 8 , 9 , 10 , 29 ] (objective function examples can be found in [ 7 , 9 , 10 , 11 , 14 , 15 ]). The aim of problem solving is to maximize the objective function value via several iteration steps.…”