Compilation of Fault-Tolerant Quantum Heuristics for Combinatorial Optimization

TL;DR

This paper evaluates the practicality of various fault-tolerant quantum heuristics for combinatorial optimization on small quantum computers, highlighting significant resource requirements and challenging the expectation of near-term quantum advantage.

Contribution

It compiles and analyzes fault-tolerant implementations of multiple quantum heuristics, providing resource estimates and practical limitations for their use on near-term quantum hardware.

Findings

Quantum heuristics require extensive resources, often exceeding current capabilities.

Achieving quantum advantage with quadratic speedups on modest hardware is unlikely without significant improvements.

Simulated annealing on quantum hardware could take days and millions of qubits, compared to minutes classically.

Abstract

Here we explore which heuristic quantum algorithms for combinatorial optimization might be most practical to try out on a small fault-tolerant quantum computer. We compile circuits for several variants of quantum accelerated simulated annealing including those using qubitization or Szegedy walks to quantize classical Markov chains and those simulating spectral gap amplified Hamiltonians encoding a Gibbs state. We also optimize fault-tolerant realizations of the adiabatic algorithm, quantum enhanced population transfer, the quantum approximate optimization algorithm, and other approaches. Many of these methods are bottlenecked by calls to the same subroutines; thus, optimized circuits for those primitives should be of interest regardless of which heuristic is most effective in practice. We compile these bottlenecks for several families of optimization problems and report for how long and for what size systems one can perform these heuristics in the surface code given a range of resource budgets. Our results discourage the notion that any quantum optimization heuristic realizing only a quadratic speedup will achieve an advantage over classical algorithms on modest superconducting qubit surface code processors without significant improvements in the implementation of the surface code. For instance, under quantum-favorable assumptions (e.g., that the quantum algorithm requires exactly quadratically fewer steps), our analysis suggests that quantum accelerated simulated annealing would require roughly a day and a million physical qubits to optimize spin glasses that could be solved by classical simulated annealing in about four CPU-minutes.