Quantum variational optimization: The role of entanglement and problem hardness

Abstract

Quantum variational optimization has been posed as an alternative to solve optimization problems faster and at a larger scale than what classical methods allow. In this manuscript we study systematically the role of entanglement, the structure of the variational quantum circuit, and the structure of the optimization problem, in the success and efficiency of these algorithms. For that purpose, our study focuses on the Variational Quantum Quantum Eigensolver (VQE) algorithm, as applied to Quadratic Unconstrained Binary Optimization (QUBO) problems on random graphs with tunable density. Our numerical results indicate an advantage in adapting the distribution of entangling gates to the problem's topology, specially for problems defined on low-dimensional graphs. Furthermore, we find evidence that applying CVaR type cost functions improves the optimization, increasing the probability of overlap with the optimal solutions. However, these techniques also improve the performance of ansatz based on product states (no entanglement), suggesting a new classical optimization method based on these could outperform existing NISQ architectures in certain regimes. Finally, our study also reveals a correlation between the hardness of a problem and the Hamming distance between the ground-and first-excited state, an idea that can be used to engineer benchmarks and understand the performance bottlenecks of optimization methods.

Filtering variational quantum algorithms for combinatorial optimization

“…Owing to the high flexibility of F-VQE, various promising strategies can be considered to further improve the performance. F-VQE can readily be combined with the conditional value-at-risk cost function [16,44,45] to provide new filtering operators with additional capabilities. Local cost functions and shallow ansatz circuits can be used to avoid barren plateaus [46].…”

“…VQE imposes no restrictions on the ansatz circuit and has become a powerful method for quantum chemistry [13], condensed matter [14], and combinatorial optimization [15]. For combinatorial optimization problems, however, it tends to produce sub-optimal solutions [16]. QAOA uses a specific ansatz circuit inspired by adiabatic quantum computation [17] and the Trotterization of the time evolution corresponding to quantum annealing [18].…”

Filtering variational quantum algorithms for combinatorial optimization

“…VQE imposes no restrictions on the ansatz circuit and has become a powerful method for quantum chemistry [12], condensed matter [13], and combinatorial optimization [14]. For combinatorial optimization problems, however, it tends to produce sub-optimal solutions [15]. QAOA uses a specific ansatz circuit inspired by adiabatic quantum computation [16] and the Trotterization of the time evolution corresponding to quantum annealing [17].…”

Mixer-Phaser Ansätze for Quantum Optimization with Hard Constraints

“…It could be straightforwardly computed by accessing the wavefunction of the final state, or its sampling statistics. Metrics that are more focused on the high-quality solutions portion of the probability, such as our (11), or the Conditional Value at Risk (CVaR) [21,22] or Gibbs averages [23], are suspected to have some desirable "trainability" properties to guide parameter setting, as opposed to the more traditional ψ F ( γ, β)|H C |ψ F ( γ, β) [24] (which is simply BEST 1 ). For illustration, we will work with BEST 5 , since R = 5 seems to be a reasonable value to use to reach good approximation ratios for the moderate sizes of problems that we are studying, as we will demonstrate empirically ex-post.…”