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Quadratic Continuous Quantum Optimization

Sascha Mücke, Thore Gerlach, Nico Piatkowski

TL;DR

The paper addresses the gap between quantum annealing's native QUBO formulation and the need for continuous optimization in tasks like regression. It introduces Quadratic Continuous Quantum Optimization (QCQO), an iterative method that solves small QUBO subproblems derived from updates w' = w + R^T z^*, with Q(w,R)=R A R^T + diag[R(2 A w + a)], where R encodes continuous information via binary selections. Theoretical results establish convergence behaviors for convex losses and demonstrate substantial qubit-efficiency advantages through implicit continuous encoding; a step-size scheduling scheme further improves convergence. Empirical evaluation on synthetic linear regression data and real quantum hardware shows QCQO can achieve competitive results with fewer qubits, though hardware noise can slow convergence and affect larger update matrices. Together, these findings broaden the applicability of quantum annealing to continuous optimization and regression, offering a flexible, anytime algorithm suitable for current hardware constraints.

Abstract

Quantum annealers can solve QUBO problems efficiently but struggle with continuous optimization tasks like regression due to their discrete nature. We introduce Quadratic Continuous Quantum Optimization (QCQO), an anytime algorithm that approximates solutions to unconstrained quadratic programs via a sequence of QUBO instances. Rather than encoding real variables as binary vectors, QCQO implicitly represents them using continuous QUBO weights and iteratively refines the solution by summing sampled vectors. This allows flexible control over the number of binary variables and adapts well to hardware constraints. We prove convergence properties, introduce a step size adaptation scheme, and validate the method on linear regression. Experiments with simulated and real quantum annealers show that QCQO achieves accurate results with fewer qubits, though convergence slows on noisy hardware. Our approach enables quantum annealing to address a wider class of continuous problems.

Quadratic Continuous Quantum Optimization

TL;DR

The paper addresses the gap between quantum annealing's native QUBO formulation and the need for continuous optimization in tasks like regression. It introduces Quadratic Continuous Quantum Optimization (QCQO), an iterative method that solves small QUBO subproblems derived from updates w' = w + R^T z^*, with Q(w,R)=R A R^T + diag[R(2 A w + a)], where R encodes continuous information via binary selections. Theoretical results establish convergence behaviors for convex losses and demonstrate substantial qubit-efficiency advantages through implicit continuous encoding; a step-size scheduling scheme further improves convergence. Empirical evaluation on synthetic linear regression data and real quantum hardware shows QCQO can achieve competitive results with fewer qubits, though hardware noise can slow convergence and affect larger update matrices. Together, these findings broaden the applicability of quantum annealing to continuous optimization and regression, offering a flexible, anytime algorithm suitable for current hardware constraints.

Abstract

Quantum annealers can solve QUBO problems efficiently but struggle with continuous optimization tasks like regression due to their discrete nature. We introduce Quadratic Continuous Quantum Optimization (QCQO), an anytime algorithm that approximates solutions to unconstrained quadratic programs via a sequence of QUBO instances. Rather than encoding real variables as binary vectors, QCQO implicitly represents them using continuous QUBO weights and iteratively refines the solution by summing sampled vectors. This allows flexible control over the number of binary variables and adapts well to hardware constraints. We prove convergence properties, introduce a step size adaptation scheme, and validate the method on linear regression. Experiments with simulated and real quantum annealers show that QCQO achieves accurate results with fewer qubits, though convergence slows on noisy hardware. Our approach enables quantum annealing to address a wider class of continuous problems.
Paper Structure (10 sections, 2 theorems, 19 equations, 5 figures, 2 algorithms)

This paper contains 10 sections, 2 theorems, 19 equations, 5 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Quadratic Continuous Quantum Optimization
  3. Sampling the Matrix $\bm{R}$
  4. Theoretical Analysis
  5. Number of Qubits
  6. Application: Linear Regression
  7. Experimental Evaluation
  8. Step Size Scheduling
  9. Evaluation on Quantum Annealer
  10. Conclusion

Key Result

Theorem 1

For an arbitrary but fixed $n\in\mathbb{N}$, if $\bm{R}_i$ is sampled i.i.e. from $\mathcal{N}(\bm{\mu},\bm{\Sigma})$ for each row $i\in\lbrace 1,\dots,n\rbrace$, the expected update step $\bm{U}$ in each iteration of alg:qcqo follows the distribution $\mathcal{N}(n/2\bm{\mu},n/4\bm{\Sigma})$.

Figures (5)

  • Figure 1: Convergence behavior of QCQO on Linear Regression tasks with different choices of $n$ (number of rows of update matrix $\bm{R}$) and $\sigma$ (standard deviation used for sampling the rows of $\bm{R}$); MSE is averaged over 10 runs.
  • Figure 2: Same experiment as \ref{['fig:lr']}, but showing the Euclidean distance between current solution $\bm{w}$ of each iteration and the globally optimal solution $\bm{w}^*$, averaged over 10 runs.
  • Figure 3: Convergence behavior of QCQO on Linear Regression tasks with different choices of $n$ (number of rows of update matrix $\bm{R}$) and $\sigma$ (standard deviation used for sampling the rows of $\bm{R}$), using the step size update rule described by \ref{['eq:stepsize']}; MSE is averaged over 10 runs.
  • Figure 4: Same experiment as \ref{['fig:lr_stepsize']}, but showing the Euclidean distance between current solution $\bm{w}_t$ of each iteration and the globally optimal solution $\bm{w}^*$, averaged over 10 runs.
  • Figure 5: Same experiment as \ref{['fig:lr_stepsize']}, but run on a D-Wave quantum annealer, averaged over 10 runs.

Theorems & Definitions (5)

  • Definition 1: Qubo
  • Theorem 1
  • proof
  • Theorem 2
  • proof