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.
Highlights
The prospect of quantum-enhanced optimization has driven much interest in quantum technologies over the years
We focus our analysis on four families of combinatorial optimization problems: the L-term spin model, in which the Hamiltonian is specified as a real linear combination of L tensor products of Pauli-Z operators; quadratic unconstrained binary optimization (QUBO), which is an NP-hard special case of a two-local L-term spin model; the Sherrington-Kirkpatrick (SK) model, which is a model of spin-glass physics and an instance of QUBO that has been well studied in the context of simulated annealing [24]; and the low autocorrelation binary sequence (LABS) problem, which is a problem with many terms but significant structure that is known to be extremely challenging in practice
III C, we introduce a heuristic method for adiabatic optimization that is likely to be computationally cheaper for some applications of early quantum computers, we do not expect an asymptotic advantage over other state-of-the-art approaches
Summary
The prospect of quantum-enhanced optimization has driven much interest in quantum technologies over the years. These include variants of Grover’s algorithm [3,4], quantum annealing [5,6], adiabatic quantum computing [7,8], the shortestpath algorithm [9], quantum-enhanced population transfer [10,11], the quantum approximate optimization algorithm [12], quantum versions of classical simulated annealing [13,14], quantum versions of backtracking [15,16] as well as branch and bound techniques [17], among many others While often these works focus on the asymptotic scaling of exact quantum optimization, in many cases one can use these algorithms heuristically through trivial modifications of the approach. We focus on Toffoli complexity since we imagine realizing these algorithms in the surface code [18,19], where non-Clifford gates such as Toffoli or T gates require considerably more time (and physical qubits) to implement than Clifford gates
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