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Comprehensive Library of Variational LSE Solvers

Nico Meyer, Martin Röhn, Jakob Murauer, Axel Plinge, Christopher Mutschler, Daniel D. Scherer

TL;DR

The paper tackles solving linear systems on NISQ-era quantum devices using variational methods and introduces the variational-lse-solver library, which assembles existing variational LSE approaches with practical enhancements. Key contributions include a normalized cost framework ($C_G$, $C_L$) to ensure stable convergence, techniques to reduce quantum-evaluation overhead via unitary decompositions and symmetry, multiple modes for loading the system matrix, a dynamic circuit Ansatz that grows with training, and additional evaluation methods for efficient prototyping. The framework is built on PennyLane with configurable optimizers and early stopping, and it provides concrete usage examples such as reproducing Bravo-Prieto et al.'s experiment and integrating dynamic-depth circuits; code is available via pip and GitHub. By abstracting away low-level implementation details and offering an end-to-end research tool, the library accelerates exploration of quantum linear algebra and quantum-software development for variational solvers on near-term devices.

Abstract

Linear systems of equations can be found in various mathematical domains, as well as in the field of machine learning. By employing noisy intermediate-scale quantum devices, variational solvers promise to accelerate finding solutions for large systems. Although there is a wealth of theoretical research on these algorithms, only fragmentary implementations exist. To fill this gap, we have developed the variational-lse-solver framework, which realizes existing approaches in literature, and introduces several enhancements. The user-friendly interface is designed for researchers that work at the abstraction level of identifying and developing end-to-end applications.

Comprehensive Library of Variational LSE Solvers

TL;DR

The paper tackles solving linear systems on NISQ-era quantum devices using variational methods and introduces the variational-lse-solver library, which assembles existing variational LSE approaches with practical enhancements. Key contributions include a normalized cost framework (, ) to ensure stable convergence, techniques to reduce quantum-evaluation overhead via unitary decompositions and symmetry, multiple modes for loading the system matrix, a dynamic circuit Ansatz that grows with training, and additional evaluation methods for efficient prototyping. The framework is built on PennyLane with configurable optimizers and early stopping, and it provides concrete usage examples such as reproducing Bravo-Prieto et al.'s experiment and integrating dynamic-depth circuits; code is available via pip and GitHub. By abstracting away low-level implementation details and offering an end-to-end research tool, the library accelerates exploration of quantum linear algebra and quantum-software development for variational solvers on near-term devices.

Abstract

Linear systems of equations can be found in various mathematical domains, as well as in the field of machine learning. By employing noisy intermediate-scale quantum devices, variational solvers promise to accelerate finding solutions for large systems. Although there is a wealth of theoretical research on these algorithms, only fragmentary implementations exist. To fill this gap, we have developed the variational-lse-solver framework, which realizes existing approaches in literature, and introduces several enhancements. The user-friendly interface is designed for researchers that work at the abstraction level of identifying and developing end-to-end applications.
Paper Structure (10 sections, 6 equations, 2 figures, 1 table)

This paper contains 10 sections, 6 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction.
  2. Underlying Concept and Modifications
  3. Training with Modified Cost Function
  4. Reduced Evaluation Overhead
  5. Framework Components and Usage
  6. Modes of Loading System Matrix
  7. Usage Example for Recreating Simple Experiment
  8. Dynamic Circuit Ansatz
  9. Additional Evaluation Methods
  10. Concluding Remarks

Figures (2)

  • Figure 1: Results produced with the proposed framework on the described in \ref{['eq:lse_1', 'eq:lse_2']}. The variational ansatz consists of a depth $d=1$ version of \ref{['fig:circuit']}. The training for $50$ steps in the upper plot averaged over $100$ random initializations shows smooth convergence for both loss functions. The evaluation of the final results with $1000$ shots in the lower plot is in good agreement with the normalized ground truth solution. The error bars denote the 25th and 75th percentile over the $100$ trained parameter sets.
  • Figure 2: The dynamic ansatz used in the proposed framework, a modified version of Patil_2022. The parameterized gates are realized as $\text{Rot}_{zyz}(\boldsymbol{\alpha}) = R_z(\alpha_2)R_y(\alpha_1)R_z(\alpha_0)$, entanglement is created with nearest-neighbor $CZ$-gates. The initial parameter set $\boldsymbol{\theta}^{(0)}$ is drawn u.a.r. from $\left[ 0, 2\pi\right]$. For increasing the depth, the new parameters are selected as $\boldsymbol{\theta}_i^{(d+1)}=\left( -\alpha, 0, \alpha \right)$, for all qubits $i$, and with $\alpha$ u.a.r. as above. This ensures that the best solution to the found until that point is not worsened by the modification itself. For depth $d$, the number of trainable parameters therefore is $3n(d+1)$.