Automated discovery and optimization of autonomous quantum error correction codes for a general open quantum system

TL;DR

The paper develops a gradient-based framework to automatically discover and optimize autonomous quantum error correction codes for general open quantum systems described by Lindblad dynamics, jointly optimizing the code space, induced decay, and control Hamiltonian to maximize the code-space fidelity after a finite evolution time. By formulating the problem in terms of a vectorized Liouvillian and a code-space projector, the authors implement an alternating optimization that updates the encoding, induced-decay, and coherent control components to drive the system toward decoherence-free evolution within the code space. Demonstrations on qudits of dimension 4–6 show that the method can recover intuitive AQEC codes in simple cases (e.g., a 13-code for a 4-level system) and, in more complex or perturbed settings, can discover nontrivial codes or bound the achievable fidelity; the results also reveal practical convergence bottlenecks and the sensitivity to starting conditions. Overall, the work provides a computational pathway to tailor AQEC protocols for real devices, with implications for extending quantum information lifetimes in heterogeneous open-system environments through reservoir engineering and optimized control.

Abstract

We develop a method to search for the optimal code space, induced decay rates and control Hamiltonian to implement autonomous quantum error correction (AQEC) for a general open quantum system. The system is defined by a free-evolution Lindbladian superoperator, which contains the free Hamiltonian and naturally occurring decoherence terms, as well as the control superoperators. The performance metric for optimization in our algorithm is the fidelity between the projector onto the code space and the same projector after Lindbladian evolution for a specified time. We use a gradient-based search to update the code words, induced decay matrix and control Hamiltonian matrix. We apply our algorithm to optimize AQEC codes for a variety of few-level systems. The four-level system with uniform decay rates offers a simple example for testing and illustrating the operation of our approach. The algorithm reliably succeeds in finding the optimal code in this case, while success becomes probabilistic for more complicated cases. For a five-level system with photon loss decay, the algorithm finds good AQEC codes, but these codes are not as good as the well-known binomial code. We use the binomial code as a starting point to search for the optimal code for a perturbed five-level system. In this case, the algorithm finds a code that is better than both the original binomial code and any other code obtained numerically when starting from a random initial guess. Our results demonstrate the promise of using computational techniques to discover and optimize AQEC codes in future real-world quantum computers.

Figures (9)

• Figure 1: Flowchart that summarizes the algorithm. First, the three components of the AQEC code are initialized, possibly to random initial values. These three components are then updated by following the gradients to improve the fidelity defined in Eq. (\ref{['Eq:Fidelity']}). After every iteration, the two termination condition are tested. When neither condition is satisfied, the AQEC code is updated further. When one of the termination conditions is satisfied, the algorithm outputs the optimized AQEC code.
• Figure 2: Progression of AQEC code with optimization iteration number $k$ for 4-level system with uniform decay rates. The infidelity $(1-F)$, plotted in Panel (a), decreases steadily and approaches zero as a function of $k$, indicating that the algorithm is successful in finding a good AQEC code. Panel (b) shows that the code space approaches $\left\{\left| 1 \right\rangle,\left| 3 \right\rangle\right\}$. In Panel (c) the legend specifies only the highest two lines. In this specific run, by coincidence, the real parts of $b_{1,0}$ and $b_{3,2}$ did not grow significantly up to $k=10^5$. In Panel (d), the inset shows the real (cyan) and imaginary (gray) parts of $O_{1,3}$. This matrix element is expected to converge to zero in the optimal code. Most other matrix elements in $\hat{O}$ are expected to have a small effect on the fidelity, as explained in Appendix \ref{['Sec:AppendixOptimization']}. All plots in this figure indicate that the algorithm is progressing towards the 13 code but that it has not converged yet.
• Figure 3: Progression of AQEC code with optimization iteration number $k$ for 5-level system with photon loss decay. This figure shows an instance in which the algorithm failed to produce a nontrivial AQEC code. The fidelity converges to the free-evolution value $F_0$ (shown as the horizontal dotted line in Panel a). Panel (b) shows the probability for the code to be outside two code spaces that can be intuitively guessed as candidate code spaces, namely the space of the binomial code and the space $\left\{ \left| 0 \right\rangle, \left| 1 \right\rangle \right\}$. The code space gradually approaches $\left\{ \left| 0 \right\rangle, \left| 1 \right\rangle \right\}$. All the plots in this figure indicate that the algorithm is converging towards the trivial code of effectively doing nothing. This figure is representative of most runs of the algorithm. Figure \ref{['Fig:Convergence5']} shows one of the relatively rare instances where the algorithm found a nontrivial code.
• Figure 4: Same as Fig. \ref{['Fig:Convergence501']}, but for an instance in which the algorithm succeeded in finding an AQEC code with $F>F_0$. Such successful instances occurred in a few percent of the runs, each starting from a random seed. Although the fidelity is quite high ($F=0.998$) after $10^5$ iterations, Panel (b) clearly shows that the code is significantly different from the binomial code, which we expect is the optimal code in this model. Panel (a) suggests that there is still plenty of room to further optimize the code. On the other hand, the high fidelity indicates that this code is already a very good AQEC code. In some sense, this instance represents a case in which the algorithm discovered a new code that is very good, although it is still not as good as the binomial code, which has $1-F=6\times 10^{-6}$. As can be seen in Panel (c), this code requires that a large number of transitions are induced with varying transition rates. The legend specifies only the highest eight lines.
• Figure 5: Optimizing AQEC code for a slightly perturbed system. The infidelity $(1-F)$ and the decay rate suppression factor $\kappa$ are plotted as functions of the exponent $\alpha$ in Eq. (\ref{['Eq:FiveLevelGeneralizedAnnihilationOp']}). The data labeled "binomial" are obtained by applying the binomial code in its original form to the perturbed system. The data labeled "optimized" are obtained by optimizing the AQEC code starting from the binomial code as an initial guess. For the data labeled "random seed", we took the best result out of ten runs using random seeds for each value of $\alpha$. When $\alpha=0.5$ (unperturbed case), the binomial code performs better than any AQEC code that we obtained numerically starting from a random seed. Furthermore, numerical optimization did not improve on the binomial code. When $\alpha=0.45$, applying the binomial code in its original form gives a relatively low fidelity. Starting from the binomial code and numerically optimizing it gives a better fidelity than any code obtained from a random seed. When $\alpha=0.4$ the best AQEC obtained from a random seed performs better than the code obtained by optimizing the binomial code, indicating that the perturbation is so large that the binomial code is no longer a useful starting point for optimization. These results demonstrate how our algorithm can find optimal AQEC codes that cannot be obtained by simple intuition about the system.