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Improving Variational Quantum Circuit Optimization via Hybrid Algorithms and Random Axis Initialization

Joona V. Pankkonen, Lauri Ylinen, Matti Raasakka, Ilkka Tittonen

TL;DR

This work addresses the optimization of variational quantum circuits on NISQ devices by enhancing a gradient-free method (Rotosolve) with Haar-randomized initialization (Rotosolve-Haar) and combining it with the Free Quaternion Selection (FQS) approach. It introduces two hybrid algorithms—iteration-based and gate-specific hybrids—that leverage the complementary strengths of fast convergence and high expressivity. Through extensive simulations on 1D/2D Heisenberg Hamiltonians and the H$_2$ molecule, the hybrids generally outperform their constituent methods, with performance depending on system size, circuit depth, and the chosen hybrid configuration. The findings offer practical guidance on when to favor rapid convergence versus expressivity and suggest avenues for applying hybrid strategies to larger, more complex quantum circuits.

Abstract

Variational quantum circuits (VQCs) are an essential tool in applying noisy intermediate-scale quantum computers to practical problems. VQCs are used as a central component in many algorithms, for example, in quantum machine learning, optimization, and quantum chemistry. Several methods have been developed to optimize VQCs. In this work, we enhance the performance of the well-known Rotosolve method, a gradient-free optimization algorithm specifically designed for VQCs. We develop two hybrid algorithms that combine an improved version of Rotosolve with the free quaternion selection (FQS) algorithm, which is the main focus of this study. Through numerical simulations, we observe that these hybrid algorithms achieve higher accuracy and better average performance across different ansatz circuit sizes and cost functions. For shallow variational circuits, we identify a trade-off between the expressivity of the variational ansatz and the speed of convergence to the optimum: a more expressive ansatz ultimately reaches a closer approximation to the true minimum, but at the cost of requiring more circuit evaluations for convergence. By combining the less expressive but fast-converging Rotosolve with the more expressive FQS, we construct hybrid algorithms that benefit from the rapid initial convergence of Rotosolve while leveraging the superior expressivity of FQS. As a result, these hybrid approaches outperform either method used independently.

Improving Variational Quantum Circuit Optimization via Hybrid Algorithms and Random Axis Initialization

TL;DR

This work addresses the optimization of variational quantum circuits on NISQ devices by enhancing a gradient-free method (Rotosolve) with Haar-randomized initialization (Rotosolve-Haar) and combining it with the Free Quaternion Selection (FQS) approach. It introduces two hybrid algorithms—iteration-based and gate-specific hybrids—that leverage the complementary strengths of fast convergence and high expressivity. Through extensive simulations on 1D/2D Heisenberg Hamiltonians and the H molecule, the hybrids generally outperform their constituent methods, with performance depending on system size, circuit depth, and the chosen hybrid configuration. The findings offer practical guidance on when to favor rapid convergence versus expressivity and suggest avenues for applying hybrid strategies to larger, more complex quantum circuits.

Abstract

Variational quantum circuits (VQCs) are an essential tool in applying noisy intermediate-scale quantum computers to practical problems. VQCs are used as a central component in many algorithms, for example, in quantum machine learning, optimization, and quantum chemistry. Several methods have been developed to optimize VQCs. In this work, we enhance the performance of the well-known Rotosolve method, a gradient-free optimization algorithm specifically designed for VQCs. We develop two hybrid algorithms that combine an improved version of Rotosolve with the free quaternion selection (FQS) algorithm, which is the main focus of this study. Through numerical simulations, we observe that these hybrid algorithms achieve higher accuracy and better average performance across different ansatz circuit sizes and cost functions. For shallow variational circuits, we identify a trade-off between the expressivity of the variational ansatz and the speed of convergence to the optimum: a more expressive ansatz ultimately reaches a closer approximation to the true minimum, but at the cost of requiring more circuit evaluations for convergence. By combining the less expressive but fast-converging Rotosolve with the more expressive FQS, we construct hybrid algorithms that benefit from the rapid initial convergence of Rotosolve while leveraging the superior expressivity of FQS. As a result, these hybrid approaches outperform either method used independently.

Paper Structure

This paper contains 16 sections, 1 theorem, 35 equations, 11 figures, 4 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Optimization of variational quantum circuits
  3. Variational quantum circuits
  4. Rotosolve
  5. Free Quaternion Selection
  6. Random axis initialization method and hybrid algorithms
  7. Results
  8. Heisenberg model
  9. 1D Heisenberg model
  10. 2D Heisenberg model on a rectangular lattice
  11. Hydrogen molecular Hamiltonian
  12. Maximizing overlap between the quantum states
  13. Conclusions
  14. Coefficients for the Pauli terms of the molecular Hamiltonians
  15. Proof for unitary representation
  16. ...and 1 more sections

Key Result

Proposition 1

Let $H\in \mathbb{C} ^{2\times 2}$ be a Hermitian matrix with eigenvalues $-1$ and $1$, and let $U\in SU(2)$.

Figures (11)

  • Figure 1: An example of the circuit ansatz architecture for the 5-qubit structure optimization. Here $R_i \in \{R_X, R_Y, R_Z \}$ is a single-qubit rotation gate and $\theta_i$ the corresponding angle parameter for $i=1,\ldots,Ln$.
  • Figure 2: A comparison of algorithm performance on the 6-qubit Heisenberg model Hamiltonian with 10 layers as a function of circuit evaluations for (a) gate-specific hybrids and (b) iteration-specific hybrids. Each line shows the fitted mean of the 20 trials. The ground state of the system is approximately $E_g=-11.2111$.
  • Figure 3: A comparison of algorithm performance on the 10-qubit Heisenberg model Hamiltonian with 10 layers as a function of circuit evaluations for (a) gate-specific hybrids and (b) iteration-specific hybrids. Each line shows the fitted mean of the 20 trials. The ground state of the system is approximately $E_g=-18.3688$.
  • Figure 4: A comparison of algorithm performance on the 5-qubit Heisenberg model Hamiltonian as a function of circuit evaluations with $J = h = 1$. Each line shows the fitted mean of the 20 trials and represents the number of shots used to approximate each Hamiltonian term. The ground state of the system is approximately $E_g=-8.4721$.
  • Figure 5: A comparison of algorithm performance on the 6-qubit two-dimensional Heisenberg model Hamiltonian with 10 layers as a function of circuit evaluations for (a) gate-specific hybrids and (b) iteration-specific hybrids. Lattice dimension is set to $2\times3$ and each line represents the fitted mean of the 20 trials. The ground state of the system is approximately $E_g=-12.5175$.
  • ...and 6 more figures

Theorems & Definitions (2)

  • Proposition 1
  • proof