Compilation of Fault-Tolerant Quantum Heuristics for Combinatorial Optimization

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.

Focus beyond quadratic speedups for error-corrected quantum advantage

“…We will focus on a modest realization of the surface code that would involve enough resources to perform classically intractable calculations but only support a few state distillation factories. Our analysis differs from results such as [7,12] by addressing the prospects for achieving quantum advantage via polynomial speedup for a broad class of algorithms, rather than for specific problems. We will assume that there is some problem which can be solved by a classical computer that makes M d calls to a "classical primitive" circuit or by a quantum computer which makes M calls to a "quantum primitive" circuit (which is often, but not always, related to the classical primitive circuit).…”

“…For example, many quantum algorithms (often based on amplitude amplification [4]) give quadratic speedups for tasks such as search [5], optimization [5][6][7], Monte Carlo [4,8,9] various areas of machine learning [10,11] and more. However, attempts [7,12] to assess the overheads of some such applications within fault-tolerance have come up with discouraging predictions for what would be required to achieve practical advantage against classical algorithms.…”

Low depth mechanisms for quantum optimization

“…Using a more modern interpretation of Grover search and amplitude amplification has lead to asymptotic speedups over structured classical algorithms like annealing and branch-and-bound [4][5][6][7][8], but the overheads for general cases remain challenging when compiled all the way to fault tolerant gate sequences [9]. This is exacerbated by the fact that real use cases have shown that it is likely the case that improved solutions are most needed on problems > 1000 bits, where classical heuristics break down [10,11].…”

“…For problems without any structure, quadratic speedups are the best one can hope for, and indeed this leads to a steep overhead for practical problems that can look insurmountable for near-term devices [9]. However, this opens the questions of if there are structures in classical optimization problems that quantum computers are uniquely posed to take advantage of that could lead to super-polynomial speedups, and if these structures are present in classical optimization problems of interest.…”

Early fault-tolerant simulations of the Hubbard model

“…For instance, for an 8 × 8 Hubbard simulation at 0.5% relative error, they estimated that a transmon-based surface code architecture may require only 62,000 physical qubits and 23 million non-Clifford T gates. While such a device would be much larger than any currently existing, these resource estimates are orders of magnitude smaller than those needed to break RSA [8,9], or even more costly, to perform combinatorial optimisation [10,11].…”