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Efficient Quantum Optimization via Dynamical Simulation

Ahmet Burak Catli, Sophia Simon, Nathan Wiebe

Abstract

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. We focus on the setting where the objective function can \emph{only} be accessed via a phase oracle. 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 $O(N^2κ^2/h_x^2ε)$ queries to a phase oracle to find an $ε$-approximate local optimum, where $κ$ 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 $O(N^{3/2}(1/ε)^{κ\log(3)/4})$ 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.

Efficient Quantum Optimization via Dynamical Simulation

Abstract

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. We focus on the setting where the objective function can \emph{only} be accessed via a phase oracle. Our first two algorithms can find local optima of a differentiable function 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 queries to a phase oracle to find an -approximate local optimum, where is the condition number of the Hessian matrix and is the discretization spacing. In contrast, we show that methods based on gradient descent require 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 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.

Paper Structure

This paper contains 30 sections, 39 theorems, 472 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Coherent Optimization
  3. Example of Bit and Phase Oracles
  4. Main Results
  5. Quantum Algorithms for Coherent Local Optimization
  6. Liouvillian Quantum Optimization Algorithm
  7. Hamiltonian Quantum Optimization Algorithm
  8. Convex Quadratic Optimization
  9. Classical Liouvillian Approach
  10. Quantum Hamiltonian Approach
  11. Extension to Smooth and Strongly Convex Functions
  12. Quantum Algorithm for Coherent Global Optimization
  13. Nosé Hamiltonian in a Discrete Setting
  14. Approach to Microcanonical Equilibrium
  15. Block Encoding of Nosé Liouvillian
  16. ...and 15 more sections

Key Result

Theorem 5

Let $\epsilon > 0$ be an error tolerance, let $A \in \mathbb{R}^{N \times N}$ be positive with eigenvalues $0 < \lambda_0 =: \lambda_{\min} \leq \lambda_1 \leq \dots \leq \lambda_{N-2} \leq \lambda_{N-1} =: \lambda_{\max}$ and let $f(\textbf{x}) = \left( \textbf{x}- \textbf{x}^* \right)^\top A \left queries to controlled-$O_f^{(p)}$ and its inverse. If instead $f$ can be accessed via a bit oracle

Figures (6)

  • Figure 1: Illustration of the Nosé Hamiltonian's thermal state in the extended coordinates $(x,p,s,p_s)$ and the corresponding thermal distribution in the original phase space coordinates $(x,p)$. The idea behind our global optimization method is to equilibrate over the extended microcanonical distribution using Koopman-von Neumann and then trace over $s,p_s$ to construct a low temperature thermal distribution over the parameters $(x,p)$.
  • Figure 2: Mean condition numbers for $1000$ Wishart Matrices of size $N\times N$ for $N$ varying exponentially from $2^0$ to $2^{9}$. Data is consistent with an $N^{3/2}$ powerlaw dependence on the dimension $N$. Note that the condition number for $N=1$ is always $1$ for any non-zero matrix.
  • Figure 3: Trajectories for local optimization with a localized wave packet with hard wall boundaries. The target function is quadratic with the minimum at $(0.5,0.5)$ and $\kappa=100$. The green circle indicates the starting point, while the minimum is marked with a yellow pentagon and the final point of the trajectory is marked with a red star.
  • Figure 4: The evolution of probability density under our local optimization process for a target function of Gaussian wells (Equation \ref{['eq:numerics_gauss']}) and hard wall boundaries. The blue point indicates the global minimum.
  • Figure 5: The evolution of probability density under our local optimization process for a Rastrigin target function (Equation \ref{['eq:numerics_rastrigin']}) and hard wall boundaries. The blue point indicates the global minimum.
  • ...and 1 more figures

Theorems & Definitions (84)

  • Definition 1: Continuous Quantum Optimization Problem
  • Definition 2: Bit oracle for the objective function
  • Definition 3: Phase oracle for the objective function
  • Definition 4: Phase oracles for the partial derivatives of the objective function
  • Theorem 5: Coherent convex quadratic optimization; informal version of Theorem \ref{['thm:main']}
  • Theorem 6: Global Optimization Theorem; informal version of Theorem \ref{['thm:mainGlobal']}
  • Definition 7: Hamiltonian with friction
  • Lemma 8: Equations of motion for the friction Hamiltonian
  • proof
  • Definition 9: Liouvillian with friction
  • ...and 74 more