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The Knill-Laflamme condition is sufficient for the autonomous protection of logical qubits by strong engineered dissipation

Jae-Mo Lihm, Kyungjoo Noh, Uwe R. Fischer

Published Nov 8, 2017• 0 citations
OpenAlex

Abstract

Autonomous quantum error correction utilizes the engineered coupling of a quantum system to a dissipative ancilla to protect quantum logical states from decoherence. We show that the Knill-Laflamme condition, stating that the environmental error operators should act trivially on a subspace, which then becomes the code subspace, is sufficient for logical qubits to be protected against Markovian noise. It is proven that the error caused by the total Lindbladian evolution in the code subspace can be suppressed up to very long times in the limit of large engineered dissipation, by explicitly deriving how the error scales with both time and engineered dissipation strength. As an important application, we demonstrate, in the same limit, that when a linearly parameter-dependent Hamiltonian does not act trivially on the code subspace, Heisenberg-limited precision of parameter estimation with respect to in principle arbitrarily large interrogation time can be achieved.

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Autonomous quantum error correction utilizes the engineered coupling of a quantum system to a dissipative ancilla to protect quantum logical states from decoherence. We show that the Knill-Laflamme condition, stating that the environmental error operators should act trivially on a subspace, which then becomes the code subspace, is sufficient for logical qubits to be protected against Markovian noise. It is proven that the error caused by the total Lindbladian evolution in the code subspace can be suppressed up to very long times in the limit of large engineered dissipation, by explicitly deriving how the error scales with both time and engineered dissipation strength. As an important application, we demonstrate, in the same limit, that when a linearly parameter-dependent Hamiltonian does not act trivially on the code subspace, Heisenberg-limited precision of parameter estimation with respect to in principle arbitrarily large interrogation time can be achieved.
Autonomous quantum error correction utilizes the engineered coupling of a quantum system to a dissipative ancilla to protect quantum logical states from decoherence. We show that the Knill-Laflamme condition, stating that the environmental error operators should act trivially on a subspace, which then becomes the code subspace, is sufficient for logical qudits to be protected against Markovian noise. It is proven that the error caused by the total Lindbladian evolution in the code subspace can be suppressed up to very long times in the limit of large engineered dissipation, by explicitly deriving how the error scales with both time and engineered dissipation strength. To demonstrate the potential of our approach for applications, we implement our general theory with binomial codes, a class of bosonic error-correcting codes, and outline how they can be implemented in a fully autonomous manner to protect against photon loss in a microwave cavity.
In this work, we present a general theoretical framework for the study of autonomously corrected quantum devices. First, we identify a necessary and sufficient revised version of the Knill-Laflamme conditions for the existence of an engineered Lindbladian providing protection against at most ccc consecutive errors of natural dissipation, giving rise to an effective logical decoherence rate suppressed to order ccc. Moreover, we demonstrate that such engineered dissipation can be combined with generalized realizations of error-transparent Hamiltonians (ETH) in order to perform a quantum computation in the logical space while maintaining the same degree of suppression of decoherence. Finally, we introduce a formalism predicting with precision the emergent dynamics in the logical code space resulting from the interplay of natural, engineered dissipations sources and the generalized ETH.
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.
Autonomous quantum memories are a way to passively protect quantum information using engineered dissipation that creates an “always-on'' decoder. We analyze Markovian autonomous decoders that can be implemented with a wide range of qubit and bosonic error-correcting codes, and derive several upper bounds and a lower bound on the logical error rate in terms of correction and noise rates. These bounds suggest that, in general, there is always a correction rate, possibly size-dependent, above which autonomous memories exhibit arbitrarily long coherence times. For any given autonomous memory, size dependence of this correction rate is difficult to rule out: we point to common scenarios where autonomous decoders that stochastically implement active error correction must operate at rates that grow with code size. For codes with a threshold, we show that it is possible to achieve faster-than-polynomial decay of the logical error rate with code size by using superlogarithmic scaling of the correction rate. We illustrate our results with several examples. One example is an exactly solvable global dissipative toric code model that can achieve an effective logical error rate that decreases exponentially with the linear lattice size, provided that the recovery rate grows proportionally with the linear lattice size.
We propose a novel measurement-free scheme for stabilizing a spin-oscillator hybrid qubit via autonomous quantum error correction. The engineered Lindbladian renders the code space into an attractive steady-state subspace, realized by coupling the storage mode to a rapidly cooled bath through a controlled beam-splitter and spin-dependent displacement interactions. The continuous variable-discrete variable hybrid approach to autonomous quantum error correction preserves the hardware efficiency of conventional dissipation engineering while simplifying the required system-bath coupling. The construction is compatible with simple logical gates and leverages primitives already demonstrated in experimental platforms, such as trapped-ion systems, suggesting a practical route to hardware-efficient, noise-biased logical qubits without repeated syndrome measurements and feedforward.
Dissipative quantum error correction (QEC) autonomously protects quantum information using engineered dissipation and offers a promising alternative to error correction via measurement and feedback. However, scalability remains a challenge, as correcting high-weight errors typically requires increasing dissipation rates and exponentially many correction operators. Here, we present a scalable dissipative QEC protocol for discrete-variable codes, correcting multi-qubit errors via a trickle-down mechanism that sequentially reduces errors weight. Our construction exploits redundancy in the Knill-Laflamme conditions to design correction operators that act on multiple error subspaces simultaneously, thereby reducing the overhead from exponential to polynomial in the number of required operators. We illustrate our approach with repetition codes under biased noise, showing a fourfold improvement in the exponential suppression factor at realistic physical error rates. Our approach connects autonomous QEC for discrete-variable codes with demonstrated dissipative protocols for bosonic codes and opens up new avenues for traditional measurement-feedback QEC and fault-tolerant quantum operations.