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.
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.
We can encode a qubit in the energy levels of a quantum system.Relaxation and other dissipation processes lead to decay of the fidelity of this stored information.Is it possible to preserve the quantum information for a longer time by introducing additional drives and dissipation?The existence of autonomous quantum error correcting codes answers this question in the positive.Nonetheless, discovering these codes for a real physical system, i.e., finding the encoding and the associated driving fields and bath couplings, remains a challenge that has required intuition and inspiration to overcome.In this work, we develop and demonstrate a computational approach based on adjoint optimization for discovering autonomous quantum error correcting codes given a Hamiltonian description of a physical system.We implement an optimizer that searches for a logical subspace and control parameters to better preserve quantum information.We demonstrate our method on a system of a harmonic oscillator coupled to a lossy qubit, and find that varying the Hamiltonian distance in Fock space-a proxy for the control hardware complexity-leads to discovery of different and new error correcting schemes.We discover what we call the √ 3 code, realizable with a Hamiltonian distance d = 2, and propose a hardware-efficient implementation based on superconducting circuits.
Quantum error correction is essential for fault-tolerant quantum computing. However, standard methods relying on active measurements may introduce additional errors. Autonomous quantum error correction (AQEC) circumvents this by utilizing engineered dissipation and drives in bosonic systems, but identifying practical encoding remains challenging due to stringent Knill-Laflamme conditions. In this work, we utilize curriculum learning enabled deep reinforcement learning to discover Bosonic codes under approximate AQEC framework to resist both single-photon and double-photon losses. We present an analytical solution of solving the master equation under approximation conditions, which can significantly accelerate the training process of reinforcement learning. The agent first identifies an encoded subspace surpassing the breakeven point through rapid exploration within a constrained evolutionary time-frame, then strategically fine-tunes its policy to sustain this performance advantage over extended temporal horizons. We find that the two-phase trained agent can discover the optimal set of codewords, i.e., the Fock states ∣4⟩ and ∣7⟩ considering the effect of both single-photon and double-photon loss. We identify that the discovered code surpasses the breakeven threshold over a longer evolution time and achieve the state-of-art performance. We also analyze the robustness of the code against the phase damping and amplitude damping noise. Our work highlights the potential of curriculum learning enabled deep reinforcement learning in discovering the optimal quantum error correct code especially in early fault-tolerant quantum systems.
Quantum error correcting (QEC) plays a crucial role in protecting quantum information against decoherence and enabling scalable, reliable quantum computing. One of the most realistic and ubiquitous sources of noise affecting quantum hardware today is generalized amplitude-damping (GAD) noise. Conventional, deterministic QEC codes struggle to correct for GAD noise because of their inherent structure, leading to fidelity losses that scale linearly with the damping strength. In this work, we introduce the framework of probabilistic approximate quantum error correction (PAQEC), that combines the flexibility of approximate QEC with the potential of post-selected recovery, enabling high-fidelity, resource-efficient error correction. We construct a five-qubit permutation-invariant code that, under probabilistic recovery, achieves a fidelity loss quadratic in the damping strength, thus outperforming existing QEC codes. Formulating PAQEC as an optimization problem, we present a numerical technique based on Charnes-Cooper and semidefinite programming to identify the optimal recovery map for any PAQEC code. Our results establish PAQEC as a powerful tool for developing resource-efficient, high-fidelity quantum codes tailored to realistic noise, with promising implications for near-term quantum devices and future fault-tolerant architectures.
Autonomous quantum error correction (AQEC) protects logical qubits by engineered dissipation and thus circumvents the necessity of frequent, error-prone measurement-feedback loops. Bosonic code spaces, where single-photon loss represents the dominant source of error, are promising candidates for AQEC due to their flexibility and controllability. While existing proposals have demonstrated the in-principle feasibility of AQEC with bosonic code spaces, these schemes are typically based on the exact implementation of the Knill-Laflamme conditions and thus require the realization of Hamiltonian distances d≥2. Implementing such Hamiltonian distances requires multiple nonlinear interactions and control fields, rendering these schemes experimentally challenging. Here, we propose a bosonic code for approximate AQEC by relaxing the Knill-Laflamme conditions. Using reinforcement learning (RL), we identify the optimal bosonic set of code words (denoted here by RL code), which, surprisingly, is composed of the Fock states |2⟩ and |4⟩. As we show, the RL code, despite its approximate nature, successfully suppresses single-photon loss, reducing it to an effective dephasing process that well surpasses the break-even threshold. It may thus provide a valuable building block toward full error protection. The error-correcting Hamiltonian, which includes ancilla systems that emulate the engineered dissipation, is entirely based on the Hamiltonian distance d=1, significantly reducing model complexity. Single-qubit gates are implemented in the RL code with a maximum distance d_{g}=2.
We demonstrate that there exists a universal, near-optimal recovery map---the transpose channel---for approximate quantum error-correcting codes, where optimality is defined using the worst-case fidelity. Using the transpose channel, we provide an alternative interpretation of the standard quantum error correction (QEC) conditions, and generalize them to a set of conditions for approximate QEC (AQEC) codes. This forms the basis of a simple algorithm for finding AQEC codes. Our analytical approach is a departure from earlier work relying on exhaustive numerical search for the optimal recovery map, with optimality defined based on entanglement fidelity. For the practically useful case of codes encoding a single qubit of information, our algorithm is particularly easy to implement.
Approximate quantum error correction (AQEC) extends the framework of discrete- and continuous-variable quantum error correction beyond the Knill-Laflamme (KL) conditions, where the recovery performance is quantified by entanglement fidelity. Recent studies have enabled efficient evaluation of near-optimal entanglement fidelity using transpose-channel recovery. Yet, determining the global optimal recovery map and its entanglement fidelity for general codes beyond the KL conditions remains a major computational challenge. Direct optimization becomes prohibitive as the number of noise Kraus operators grows rapidly with system size, and existing approaches lack rigorous guarantees for reducing this optimization to a tractable dimension. Here, we derive an explicit characterization of the optimal environmental state of complement channel, which transforms the optimization over recovery channels into an equivalent optimization over quotient unitaries. For a broader class of codes that satisfy only the orthogonality part of the KL conditions, we show that the optimal recovery map admits an explicit analytical form. Building on this form, we derive novel rigorous lower bounds of entanglement fidelity that strictly improve upon the transpose-recovery bound. We further develop a novel recovery strategy based on principle components, and derive a rigorous bound on the error introduced by noise truncation. Our approach enables efficient searches for approximate recovery maps for AQEC codes, avoiding the need to optimize over the full Kraus-operator space.