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Interpolation-based coordinate descent method for parameterized quantum circuits

Zhijian Lai, Jiang Hu, Taehee Ko, Jiayuan Wu, Dong An

TL;DR

This work introduces Interpolation-based Coordinate Descent (ICD) to optimize PQCs by exploiting the intrinsic trigonometric structure of single-parameter cost functions. ICD reconstructs a univariate surrogate via Fourier-interpolation from a small number of noisy evaluations and then performs an exact argmin on the surrogate, reducing quantum resource use. For equidistant frequencies, the authors prove that $\frac{2\pi}{n}$-equidistant interpolation nodes simultaneously minimize the mean-squared error, the interpolation matrix condition number, and the average derivative variance, and they show an exact eigenvalue-based solver for the subproblem in this case. Numerical experiments on MaxCut, TFIM, and XXZ demonstrate that ICD often outperforms SGD and RCD under realistic sampling budgets, though barren plateaus in XXZ pose a fundamental challenge to all non-gradient-based methods. Overall, ICD offers a unified, noise-aware framework that leverages Fourier structure to accelerate PQC training with controlled quantum overhead.

Abstract

Parameterized quantum circuits (PQCs) are ubiquitous in the design of hybrid quantum-classical algorithms. In this work, we propose an interpolation-based coordinate descent (ICD) method to address the parameter optimization problem in PQCs. The ICD method provides a unified framework for existing structure optimization techniques such as Rotosolve, sequential minimal optimization, ExcitationSolve, and others. ICD employs interpolation to approximate the PQC cost function, effectively recovering its underlying trigonometric structure, and then performs an argmin update on a single parameter in each iteration. In contrast to previous studies on structure optimization, we determine the optimal interpolation nodes to mitigate statistical errors arising from quantum measurements. Moreover, in the common case of $r$ equidistant frequencies, we show that the optimal interpolation nodes are equidistant nodes with spacing $2π/(2r+1)$ (under constant variance assumption), and that our ICD method simultaneously minimizes the mean squared error, the condition number of the interpolation matrix, and the average variance of the approximated cost function. We perform numerical simulations and test on the MaxCut problem, the transverse field Ising model, and the XXZ model. Numerical results imply that our ICD method is more efficient than the commonly used gradient descent and random coordinate descent method.

Interpolation-based coordinate descent method for parameterized quantum circuits

TL;DR

This work introduces Interpolation-based Coordinate Descent (ICD) to optimize PQCs by exploiting the intrinsic trigonometric structure of single-parameter cost functions. ICD reconstructs a univariate surrogate via Fourier-interpolation from a small number of noisy evaluations and then performs an exact argmin on the surrogate, reducing quantum resource use. For equidistant frequencies, the authors prove that -equidistant interpolation nodes simultaneously minimize the mean-squared error, the interpolation matrix condition number, and the average derivative variance, and they show an exact eigenvalue-based solver for the subproblem in this case. Numerical experiments on MaxCut, TFIM, and XXZ demonstrate that ICD often outperforms SGD and RCD under realistic sampling budgets, though barren plateaus in XXZ pose a fundamental challenge to all non-gradient-based methods. Overall, ICD offers a unified, noise-aware framework that leverages Fourier structure to accelerate PQC training with controlled quantum overhead.

Abstract

Parameterized quantum circuits (PQCs) are ubiquitous in the design of hybrid quantum-classical algorithms. In this work, we propose an interpolation-based coordinate descent (ICD) method to address the parameter optimization problem in PQCs. The ICD method provides a unified framework for existing structure optimization techniques such as Rotosolve, sequential minimal optimization, ExcitationSolve, and others. ICD employs interpolation to approximate the PQC cost function, effectively recovering its underlying trigonometric structure, and then performs an argmin update on a single parameter in each iteration. In contrast to previous studies on structure optimization, we determine the optimal interpolation nodes to mitigate statistical errors arising from quantum measurements. Moreover, in the common case of equidistant frequencies, we show that the optimal interpolation nodes are equidistant nodes with spacing (under constant variance assumption), and that our ICD method simultaneously minimizes the mean squared error, the condition number of the interpolation matrix, and the average variance of the approximated cost function. We perform numerical simulations and test on the MaxCut problem, the transverse field Ising model, and the XXZ model. Numerical results imply that our ICD method is more efficient than the commonly used gradient descent and random coordinate descent method.

Paper Structure

This paper contains 66 sections, 14 theorems, 188 equations, 13 figures, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Optimization methods
  3. Derivative-free methods.
  4. Gradient-based methods
  5. Structure optimization methods.
  6. Overview of ICD framework
  7. Contribution
  8. Organization
  9. Notations
  10. Restating the problem
  11. Observation I: statistical nature from quantum measurement postulate
  12. Observation II: trigonometric representation from quantum circuit structures
  13. Reformulated problem
  14. Interpolation-based coordinate descent method
  15. Overview
  16. ...and 51 more sections

Key Result

Theorem 1

When assp-varassm-equi holds, the $\frac{2 \pi}{n}$-equidistant nodes $\mathbf{x}^*$ with an arbitrary shift value, as defined in eq-2pi/n-equidistant, globally solves where the global minimum is $2\sigma^2$.

Figures (13)

  • Figure 1: (a) A diagram illustrating the ICD algorithm workflow. (b) Suppose we first update $\theta_1$, then update $\theta_2$, and so on. We move the current value of $\theta_j$ (gray cross) to the origin. The black solid line represents the true curve of $f$ with respect to $\theta_j$, and we aim to find the true minimum (black cross). By using an interpolation method under noisy conditions, we obtain relatively accurate estimates of $a_0$, $a_k$, and $b_k$ in \ref{['eq-653']}. Using these estimated coefficients, we recover an approximate function (blue dashed line). This approximate function can be used in place of the original cost function, and its value at any point can be evaluated using a classical computer. In each update step, ICD finds the global minimum of the approximate function (blue cross), i.e., takes the argmin, which results in a significantly larger descent compared to the RCD method using one-step update (brown cross).
  • Figure 2: Variance between the approximate functions (recovered from different interpolation nodes) and the true function varies. Suppose we consider $\theta_j$ and theoretical true curve (black solid line) is $\theta_j \mapsto f(\boldsymbol{\theta}) = a_0 + a_1 \cos (\theta_j) + b_1 \sin (\theta_j).$ To recover $n = 3$ values ($a_0$, $a_1$, and $b_1$), we simply select $n$ different nodes, evaluate their corresponding $f$ values, and solve a linear equation. The details of the interpolations will be given in \ref{['sec-ICD']}. Since the $f$ values always contain noise, the recovered function only approximate the true function within a certain range. It can be proven that for any positive integer $r_j$ and $\Omega_k^j = k$, the equidistant nodes with spacing $2 \pi /(2 r_j+1)$ are the optimal interpolation nodes, as they yield the closest approximation to the true function.
  • Figure 3: Comparison of the three optimality criteria --- mean squared error (MSE), condition number (Cond) and average variance of derivatives ($h^{(1)}$) (i.e., \ref{['thm-first-veiw', 'thm-second-veiw', 'thm-third-veiw']}, respectively) --- as functions of $k \in (0,3)$, where the interpolation nodes spaced by $k\pi/3$.
  • Figure 4: We test the 4-qubit MaxCut problem in \ref{['sec-pro-maxcut']}. The figures present the convergence of the ICD algorithm in the noiseless setting (infinite shots). (a) Using $k\pi/3$ equidistant interpolation nodes ($k = 0.5, 1, 1.5, 2$); (b) Using randomly selected interpolation nodes. All configurations exhibit identical convergence trajectories, confirming that in the absence of sampling noise, the choice of interpolation nodes has no impact on the optimization performance.
  • Figure 5: We test the 4-qubit MaxCut problem in \ref{['sec-pro-maxcut']}. The figures present the convergence behavior of (a) the reduced ICD algorithm and (b) the standard ICD algorithm under noisy settings (1024 shots), using $k\pi/3$ equidistant interpolation nodes ($k = 0.5, 1, 1.5, 2$).
  • ...and 8 more figures

Theorems & Definitions (29)

  • Remark 1: Constant variance is not realistic
  • Theorem 1: Minimal mean squared error
  • Theorem 2: Minimal condition number
  • Theorem 3: Minimal average variance of derivatives
  • Remark 2
  • Remark 3: Actual frequency is more important
  • Lemma 1: Shift invariance of interpolation nodes
  • proof
  • Remark 4: Beware of error accumulation!
  • Lemma 2
  • ...and 19 more