Nathan Wiebe

We provide several quantum algorithms for continuous optimization that do not require gradient estimation. Instead, we encode the optimization problem into the dynamics of a physical system and coherently simulate the time evolution. Our first two algorithms can find local optima of a differentiable function $f: \mathbb{R}^N \rightarrow \mathbb{R}$ by simulating either classical or quantum dynamics with friction via a time-dependent Hamiltonian. We show that for the benchmark problem of optimizing a locally quadratic objective function, these methods require a total of $O(N^2\kappa^2/h_x^2\epsilon)$ queries to a phase oracle to find an $\epsilon$-approximate local optimum, where $\kappa$ is the condition number of the Hessian matrix and $h_x$ is the discretization spacing. In contrast, we show that methods based on gradient descent require $O(N^{3/2}(1/\epsilon)^{\kappa \log(3)/4})$ queries. This corresponds to an exponential separation between the query upper bounds for the benchmark problem. Our third algorithm can find the global optimum of $f$ by preparing a classical low-temperature thermal state via simulation of the classical Liouvillian operator associated with the Nosé Hamiltonian. We use results from the quantum thermodynamics literature to bound the thermalization time for the discrete system. Additionally, we analyze barren plateau effects that commonly plague quantum optimization algorithms and observe that our approach is vastly less sensitive to this problem than standard gradient-based optimization. Our results suggests that these dynamical optimization approaches may be far more scalable for future quantum machine learning, optimization and variational experiments than was widely believed.

Abstract We compile explicit circuits and evaluate the computational cost for heuristic-based quantum algorithms for combinatorial optimization. We consider several variants of quantum-accelerated simulated annealing as well as adiabatic algorithms, quantum-enhanced population transfer, the quantum approximate optimization algorithm, and other approaches. We provide novel methods for executing the bottleneck subroutines for these heuristics, and our methods can easily be applied to other algorithms where numerical performance matters. We estimate how quickly the subroutines could be executed on a modestly sized superconducting-qubit-based quantum computer with surface code error correction. We conclude that quadratic speedups for heuristic-based quantum optimization algorithms are insufficient for early quantum computers to beat present day classical computers.

Abstract We show how to achieve the highest efficiency yet for simulations with arbitrary basis sets by using a representation of the Coulomb operator known as tensor hypercontraction (THC). We use THC to express the Coulomb operator in a non-orthogonal basis, which we are able to block encode by separately rotating each term with angles that are obtained via QROM. Our algorithm has the best complexity scaling for an arbitrary basis, as well as the best complexity for the specific case of FeMoCo. By optimising the surface code resources, we show that FeMoCo can be simulated using about 4 million physical qubits and 3.5 days of runtime, assuming 1 s cycle times and physical gate error rates no worse than 0.1%.