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Correcting quantum errors one gradient step at a time

Manav Seksaria, Anil Prabhakar

TL;DR

This work addresses the problem of optimally encoding quantum information for a given noise channel by directly differentiating fidelity with respect to complex codeword coefficients. It introduces a Wirtinger-calculus-based gradient method with soft orthonormal penalties to enforce physicality, and demonstrates that the approach is deterministic, parallelisable, and scalable. On the [[5,1,3]] Laflamme code under isotropic Pauli noise with Petz recovery, fidelity rises from 0.783 to 0.915 in 100 steps, with supporting symmetry checks on repetition codes. The framework treats fidelity as a black box, enabling larger-code optimization via Monte Carlo, tensor networks, or hardware-based fidelity estimates, potentially accelerating practical quantum error-correction design.

Abstract

In this work, we introduce a general, gradient-based method that optimises codewords for a given noise channel and fixed recovery. We do so by differentiating fidelity and descending on the complex coefficients using finite-difference Wirtinger gradients with soft penalties to promote orthonormalisation. We validate the gradients on symmetry checks (XXX/ZZZ repetition codes) and the $[[5, 1, 3]]$ code, then demonstrate substantial gains under isotropic Pauli noise with Petz recovery: fidelity improves from 0.783 to 0.915 in 100 steps for an isotropic Pauli noise of strength 0.05. The procedure is deterministic, highly parallelisable, and highly scalable.

Correcting quantum errors one gradient step at a time

TL;DR

This work addresses the problem of optimally encoding quantum information for a given noise channel by directly differentiating fidelity with respect to complex codeword coefficients. It introduces a Wirtinger-calculus-based gradient method with soft orthonormal penalties to enforce physicality, and demonstrates that the approach is deterministic, parallelisable, and scalable. On the [[5,1,3]] Laflamme code under isotropic Pauli noise with Petz recovery, fidelity rises from 0.783 to 0.915 in 100 steps, with supporting symmetry checks on repetition codes. The framework treats fidelity as a black box, enabling larger-code optimization via Monte Carlo, tensor networks, or hardware-based fidelity estimates, potentially accelerating practical quantum error-correction design.

Abstract

In this work, we introduce a general, gradient-based method that optimises codewords for a given noise channel and fixed recovery. We do so by differentiating fidelity and descending on the complex coefficients using finite-difference Wirtinger gradients with soft penalties to promote orthonormalisation. We validate the gradients on symmetry checks (XXX/ZZZ repetition codes) and the code, then demonstrate substantial gains under isotropic Pauli noise with Petz recovery: fidelity improves from 0.783 to 0.915 in 100 steps for an isotropic Pauli noise of strength 0.05. The procedure is deterministic, highly parallelisable, and highly scalable.

Paper Structure

This paper contains 7 sections, 15 equations, 3 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Fidelity Differentiation
  3. Calculating Gradients
  4. Stable Gradient Descent (GD)
  5. Conclusion
  6. Acknowledgements
  7. Stable GD

Figures (3)

  • Figure 1: Sanity check for the gradient calculation. As $\Delta x \to 10^{-4}$, gradient values converge to 0.917.
  • Figure 2: $||\nabla f||$ for the $[[5, 1, 3]]$ code under unidirectional Pauli noise. As expected, the gradients are identical for all three cases, regardless of the noise strength. The curves have been shifted by $0.01$ to avoid overlap. An isotropic Pauli noise case is also shown for reference.
  • Figure 3: Stabilised gradient descent on the $[[5, 1, 3]]$ code under $p = [0.05, 0.05, 0.05]$ isotropic Pauli noise with the Petz recovery. Fidelity improves from $0.783$ to $0.915$ in $100$ steps, while orthonormality and normalisation penalties remain controlled.