Optimizing continuous-time quantum error correction for arbitrary noise

TL;DR

This work addresses the challenge of protecting quantum information under arbitrary space-time noise by optimizing continuous-time quantum error correction (CT-QEC) codes and recovery maps. It introduces a neural-network–based framework that jointly learns the code-space (on the Grassmannian) and a CPTP recovery map, using a cost function based on average logical fidelity and a combination of Markovian and non-Markovian noise models. The approach yields device-tailored recovery schemes that can match or outperform standard stabilizer codes across diverse noise processes, including bit-flip, amplitude damping with correlated dephasing, leakage, and non-Markovian 1/f noise, with demonstrations on qubits and qutrits (e.g., $\![3,1,3]\!$ and $\![5,1,3]\!$). This methodology provides a practical route to adapting quantum error correction to realistic hardware, with potential for concatenation with conventional codes to achieve higher protection levels in small-scale systems.

Abstract

We present a protocol using machine learning (ML) to simultaneously optimize the quantum error-correcting code space and the corresponding recovery map in the framework of continuous-time quantum error correction. Given a Hilbert space and a noise process -- potentially correlated across both space and time -- the protocol identifies the optimal recovery strategy, measured by the average logical state fidelity. This approach enables the discovery of recovery schemes tailored to arbitrary device-level noise.

Figures (5)

• Figure 1: Infidelity as a function of time under the effect of Markovian bit-flip noise and continuous measurements in the $Z$ ($X$) basis and unitary rotation around $X$ ($Y$) axis. The ensemble average is taken over $10^4$ trajectories.
• Figure 2: (Left) Continuously measure in the $Z-$basis, and apply unitary rotations about the $X-$axis. The state does not move towards the target code state on average. (Right) Continuously measure in the $X-$basis, and apply unitary rotations about the $Y-$axis. On average, the purity of the state increases, and it reaches the north pole faster.
• Figure 3: The Zeno effect suppresses the effects of non-Markovian noise. As the measurement rate $\kappa$ is increased, the average infidelity with the initial code state decreases.
• Figure 4: In the forward pass, the code space (represented by a point in $\mathcal{M}$) is input to an MLP, outputting a set of $L$ complex matrices. A nonlinear activation function $f$ then returns a set of Kraus operators for the associated recovery channel.
• Figure 5: Average code state fidelity decay as a function of time in units of $1/\gamma$. Noise process is (a) Markovian bit-flip noise on 3 qubits; (b) Amplitude damping and spatially correlated dephasing on 5 qubits; (c) leakage out of computational subspace in 2 qubits; (d) population transitions in lowest adjacent levels of 2 qutrits; (e) non-Markovian $1/f$ noise of type $X$ and $Z$ on 4 qubits; and (f) constant Hamiltonian errors of type $X$ and $Z$ on 4 qubits. Standard deviation is from 3 initial training seeds.