Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Binary Quantum Control Optimization with Uncertain Hamiltonians

Xinyu Fei, Lucas T. Brady, Jeffrey Larson, Sven Leyffer, Siqian Shen

TL;DR

This work addresses robust quantum control under uncertain Hamiltonians by formulating a stochastic mixed-integer binary-control problem and solving it via a sample-based reformulation. A risk-aware objective combining expectation and CVaR is introduced, and gradient-based methods solve the continuous relaxation followed by sum-up rounding to binary controls. Theoretical results establish differentiability of the objective and bounds on the gap between binary and continuous solutions, with numerical experiments on energy minimization and circuit compilation showing improved quality and robustness over deterministic controls across uncertainty levels. The findings highlight the potential of stochastic optimization for robust quantum pulse design and point to future directions including hardware-time evolution and model-free approaches.

Abstract

Optimizing the controls of quantum systems plays a crucial role in advancing quantum technologies. The time-varying noises in quantum systems and the widespread use of inhomogeneous quantum ensembles raise the need for high-quality quantum controls under uncertainties. In this paper, we consider a stochastic discrete optimization formulation of a binary optimal quantum control problem involving Hamiltonians with predictable uncertainties. We propose a sample-based reformulation that optimizes both risk-neutral and risk-averse measurements of control policies, and solve these with two gradient-based algorithms using sum-up-rounding approaches. Furthermore, we discuss the differentiability of the objective function and prove upper bounds of the gaps between the optimal solutions to binary control problems and their continuous relaxations. We conduct numerical studies on various sized problem instances based of two applications of quantum pulse optimization; we evaluate different strategies to mitigate the impact of uncertainties in quantum systems. We demonstrate that the controls of our stochastic optimization model achieve significantly higher quality and robustness compared to the controls of a deterministic model.

Binary Quantum Control Optimization with Uncertain Hamiltonians

TL;DR

This work addresses robust quantum control under uncertain Hamiltonians by formulating a stochastic mixed-integer binary-control problem and solving it via a sample-based reformulation. A risk-aware objective combining expectation and CVaR is introduced, and gradient-based methods solve the continuous relaxation followed by sum-up rounding to binary controls. Theoretical results establish differentiability of the objective and bounds on the gap between binary and continuous solutions, with numerical experiments on energy minimization and circuit compilation showing improved quality and robustness over deterministic controls across uncertainty levels. The findings highlight the potential of stochastic optimization for robust quantum pulse design and point to future directions including hardware-time evolution and model-free approaches.

Abstract

Optimizing the controls of quantum systems plays a crucial role in advancing quantum technologies. The time-varying noises in quantum systems and the widespread use of inhomogeneous quantum ensembles raise the need for high-quality quantum controls under uncertainties. In this paper, we consider a stochastic discrete optimization formulation of a binary optimal quantum control problem involving Hamiltonians with predictable uncertainties. We propose a sample-based reformulation that optimizes both risk-neutral and risk-averse measurements of control policies, and solve these with two gradient-based algorithms using sum-up-rounding approaches. Furthermore, we discuss the differentiability of the objective function and prove upper bounds of the gaps between the optimal solutions to binary control problems and their continuous relaxations. We conduct numerical studies on various sized problem instances based of two applications of quantum pulse optimization; we evaluate different strategies to mitigate the impact of uncertainties in quantum systems. We demonstrate that the controls of our stochastic optimization model achieve significantly higher quality and robustness compared to the controls of a deterministic model.
Paper Structure (22 sections, 5 theorems, 50 equations, 8 figures, 7 tables, 2 algorithms)

This paper contains 22 sections, 5 theorems, 50 equations, 8 figures, 7 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Modeling Uncertain Hamiltonians
  3. Deterministic Optimal Control Model
  4. Stochastic Optimal Control Model
  5. Risk Measures and Objective Functions
  6. Gradient-Based Algorithm
  7. Solution Methods for Continuous Relaxation
  8. L-BFGS-B algorithm
  9. Adam method
  10. Sum-Up Rounding Technique
  11. Numerical Studies
  12. Uncertainty Design
  13. Energy Minimization Problem
  14. Results of Scenarios
  15. Results of Weight Parameter
  16. ...and 7 more sections

Key Result

Theorem 1

For a given control variable $u$, define $s^*(u)$ as the scenario number with the largest original objective value $F^{s^*(u)}(u)$ such that Then the closed-form expression of $\min_\zeta F_\mathrm{CVaR}(u,\zeta)$ at point $u$ is given by

Figures (8)

  • Figure 1: Illustration for CVaR function with risk level $\eta=0.05$. The histograms are the distribution of $f(\xi)$. The blue line represents the 95 percentile of $f(\xi)$. The red dashed line represents the CVaR value.
  • Figure 2: Sampled values of $\xi_{0}$ with $10$ scenarios. The x-axis is time step $k=1,\ldots,T$, and the y-axis is the value of $\xi_{0}$. The lines represent values of corresponding samples for each scenario $s=0,\ldots,9$.
  • Figure 3: Objective values in out-of-sample tests with various weight parameters $\alpha$. The blue line marked by dots represents the mean value. The orange line marked by triangles represents the CVaR function value. Red lines, box edges, and caps represent medians, first and third quartiles, and whiskers wickham201140, respectively.
  • Figure 4: Average objective values among samples of uncertainty $\xi$ as a function of uncertainty offsets $\mu_1,\ \mu_2\in [-0.5,0.5]$. The control solutions are obtained from the stochastic optimization model with $\alpha=0,\ 1$ and variance as $0.05$.
  • Figure 5: Histograms of out-of-sample tests for the deterministic and stochastic optimization model with offset variance $0.05$ for both controllers. Blue and yellow histograms represent the results of the deterministic and the stochastic optimization model, respectively.
  • ...and 3 more figures

Theorems & Definitions (14)

  • Theorem 1
  • proof
  • Remark 1
  • Remark 2
  • Theorem 2
  • proof
  • Remark 3
  • Theorem 3
  • proof
  • Proposition 1
  • ...and 4 more