Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:2308.16233 · Quantum

Bounds on Autonomous Quantum Error Correction

Oles Shtanko, Yu-Jie Liu, Simon Lieu, Alexey Vyacheslavovich Gorshkov, Victor V. Albert

Published Jul 22, 2025• 5 citations
arXivHTMLPDF

Abstract

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.

DOI
10.22331/q-2025-07-22-1804

Related papers

Quantum Memory and Autonomous Computation in Two Dimensions

Gesa Dünnweber, Georgios Styliaris, Rahul Trivedi

Quantum error correction (QEC) normally requires active maintenance using measurements, classical processing and externally controlled correction schedules. Passive QEC, by contrast, was previously established only in unphysical spatial dimensions. Here, we give an explicit scheme for autonomous quantum error correction and computation in two dimensions, formulated as a dissipative quantum cellular automaton with a fixed, local and translation-invariant update rule. The construction uses hierarchical, self-simulating control elements based on ideas from the seminal classical results of Gács (1986, 1989) together with a measurement-free concatenated quantum code. We prove the existence of a nonzero noise threshold under a local noise model. Below this threshold, logical errors on encoded initial states are suppressed exponentially with increasing system size and the memory lifetime diverges in the thermodynamic limit. As an extension, we outline how this mechanism can also be implemented in continuous time by a time-independent, translation-invariant local Lindbladian. Quantum circuits can moreover be encoded in the automaton's initial state and executed fault-tolerantly with polylogarithmic overhead under the same circuit-independent rule. Our scheme thus realizes a self-correcting quantum computer capable of universal computation.

quant-pharXiv:2601.20818•Jan 29, 2026•0 citations

Performance of quantum error correction with coherent errors

Eric Huang, Andrew C. Doherty, Steven T. Flammia

We compare the performance of quantum error correcting codes when memory errors are unitary with the more familiar case of dephasing noise. For a wide range of codes, we analytically compute the effective logical channel that results when the error correction steps are performed noiselessly. Our examples include the entire family of repetition codes, the five-qubit, Steane, Shor, and surface codes. When errors are measured in terms of the diamond norm, we find that the error correction is typically much more effective for unitary errors than for dephasing. We observe this behavior for a wide range of codes after a single level of encoding, and in the thresholds of concatenated codes using hard decoders. We show that this holds with great generality by proving a bound on the performance of any stabilizer code when the noise at the physical level is unitary. By comparing the diamond norm error ${D}_{\ensuremath{\diamond}}^{\ensuremath{'}}$ of the logical qubit with the same quantity at the physical level ${D}_{\ensuremath{\diamond}}$, we show that ${D}_{\ensuremath{\diamond}}^{\ensuremath{'}}\ensuremath{\le}c{D}_{\ensuremath{\diamond}}^{d}$ where $d$ is the distance of the code and $c$ is a constant that depends on the code but not on the error. This bound compares very favorably to the performance of error correction for dephasing noise and other Pauli channels, where an error correcting code of odd distance $d$ will exhibit a scaling ${D}_{\ensuremath{\diamond}}^{\ensuremath{'}}\ensuremath{\sim}{D}_{\ensuremath{\diamond}}^{(d+1)/2}$.

arXiv:1805.08227•Physical Review A•Feb 12, 2019•76 citations

Implementation-independent sufficient condition of the Knill-Laflamme type for the autonomous protection of logical qudits by strong engineered dissipation

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

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.

arXiv:1711.02999•Physical Review A•Jul 16, 2018•42 citations

Cellular automaton decoders of topological quantum memories in the fault tolerant setting

Michael Herold, Michael J. Kastoryano, Earl T. Campbell +1 more

Active error decoding and correction of topological quantum codes—in particular the toric code—remains one of the most viable routes to large scale quantum information processing. In contrast, passive error correction relies on the natural physical dynamics of a system to protect encoded quantum information. However, the search is ongoing for a completely satisfactory passive scheme applicable to locally interacting two-dimensional systems. Here, we investigate dynamical decoders that provide passive error correction by embedding the decoding process into local dynamics. We propose a specific discrete time cellular-automaton decoder in the fault tolerant setting and provide numerical evidence showing that the logical qubit has a survival time extended by several orders of magnitude over that of a bare unencoded qubit. We stress that (asynchronous) dynamical decoding gives rise to a Markovian dissipative process. We hence equate cellular-automaton decoding to a fully dissipative topological quantum memory, which removes errors continuously. In this sense, uncontrolled and unwanted local noise can be corrected for by a controlled local dissipative process. We analyze the required resources, commenting on additional polylogarithmic factors beyond those incurred by an ideal constant resource dynamical decoder.

arXiv:1511.05579•New Journal of Physics•Jun 6, 2017•36 citations

Single-Shot Error Correction at Optimal Spacetime Cost

Kishor Bharti, Tobias Haug, Derek Khu +1 more

Recent work established that, for independent erasures, storing $K$ logical qubits for $S$ time steps with error at most $\varepsilon$ requires a spacetime cost of $Ω(S(K+\log(S/\varepsilon)))$. A matching construction was also given, but assumes ideal error correction. Here we show that the same scaling can be achieved with an explicit noisy error-correction circuit and efficient decoding, counting all state-preparation, gate, measurement, and wait locations, provided that the hardware supports long-range qubit connectivity and fast, reliable classical processing. Thus, reliability adds only a logarithmic overhead shared by all stored qubits. The construction works for sufficiently weak but otherwise general circuit noise, including faults during syndrome extraction and correlated faults, and for independent erasures it matches the best known lower bound up to constant factors. We attain this scaling using quantum Tanner codes, with each memory step consisting of one round of syndrome extraction followed by a fixed number of parallel classical decoder steps that reduce the residual error enough to keep later faults correctable, regardless of memory size, storage time, or target accuracy. Together, this gives exponentially small failure probability per step while keeping the cost of each memory step linear in the code size. The same analysis also extends to certain constant-depth logical Clifford operations.

quant-pharXiv:2610.03684•Oct 5, 2026•0 citations

Passive correction of quantum logical errors in a driven, dissipative system: A blueprint for an analog quantum code fabric

Eliot Kapit, J. T. Chalker, Steven H. Simon

A physical realization of self-correcting quantum code would be profoundly useful for constructing a quantum computer. In this theoretical work, we provide a partial solution to major challenges preventing self-correcting quantum code from being engineered in realistic devices. We consider a variant of Kitaev's toric code coupled to propagating bosons, which induce a ranged interaction between anyonic defects. By coupling the primary quantum system to an engineered dissipation source through resonant energy transfer, we demonstrate a ``rate barrier'' which leads to a potentially enormous increase in the system's quantum-state lifetime through purely passive quantum error correction, even when coupled to an infinite-temperature bath. While our mechanism is not scalable to infinitely large systems, the maximum effective size can be very large, and it is fully compatible with active error-correction schemes. Our model uses only on-site and nearest-neighbor interactions and could be implemented in superconducting qubits. We sketch one such implementation at the end of this work.

arXiv:1408.0959•Physical Review A•Jun 24, 2015•38 citations
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.
Quantum error correction (QEC) normally requires active maintenance using measurements, classical processing and externally controlled correction schedules. Passive QEC, by contrast, was previously established only in unphysical spatial dimensions. Here, we give an explicit scheme for autonomous quantum error correction and computation in two dimensions, formulated as a dissipative quantum cellular automaton with a fixed, local and translation-invariant update rule. The construction uses hierarchical, self-simulating control elements based on ideas from the seminal classical results of Gács (1986, 1989) together with a measurement-free concatenated quantum code. We prove the existence of a nonzero noise threshold under a local noise model. Below this threshold, logical errors on encoded initial states are suppressed exponentially with increasing system size and the memory lifetime diverges in the thermodynamic limit. As an extension, we outline how this mechanism can also be implemented in continuous time by a time-independent, translation-invariant local Lindbladian. Quantum circuits can moreover be encoded in the automaton's initial state and executed fault-tolerantly with polylogarithmic overhead under the same circuit-independent rule. Our scheme thus realizes a self-correcting quantum computer capable of universal computation.
We compare the performance of quantum error correcting codes when memory errors are unitary with the more familiar case of dephasing noise. For a wide range of codes, we analytically compute the effective logical channel that results when the error correction steps are performed noiselessly. Our examples include the entire family of repetition codes, the five-qubit, Steane, Shor, and surface codes. When errors are measured in terms of the diamond norm, we find that the error correction is typically much more effective for unitary errors than for dephasing. We observe this behavior for a wide range of codes after a single level of encoding, and in the thresholds of concatenated codes using hard decoders. We show that this holds with great generality by proving a bound on the performance of any stabilizer code when the noise at the physical level is unitary. By comparing the diamond norm error ${D}_{\ensuremath{\diamond}}^{\ensuremath{'}}$ of the logical qubit with the same quantity at the physical level ${D}_{\ensuremath{\diamond}}$, we show that ${D}_{\ensuremath{\diamond}}^{\ensuremath{'}}\ensuremath{\le}c{D}_{\ensuremath{\diamond}}^{d}$ where ddd is the distance of the code and ccc is a constant that depends on the code but not on the error. This bound compares very favorably to the performance of error correction for dephasing noise and other Pauli channels, where an error correcting code of odd distance ddd will exhibit a scaling ${D}_{\ensuremath{\diamond}}^{\ensuremath{'}}\ensuremath{\sim}{D}_{\ensuremath{\diamond}}^{(d+1)/2}$.
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.
Active error decoding and correction of topological quantum codes—in particular the toric code—remains one of the most viable routes to large scale quantum information processing. In contrast, passive error correction relies on the natural physical dynamics of a system to protect encoded quantum information. However, the search is ongoing for a completely satisfactory passive scheme applicable to locally interacting two-dimensional systems. Here, we investigate dynamical decoders that provide passive error correction by embedding the decoding process into local dynamics. We propose a specific discrete time cellular-automaton decoder in the fault tolerant setting and provide numerical evidence showing that the logical qubit has a survival time extended by several orders of magnitude over that of a bare unencoded qubit. We stress that (asynchronous) dynamical decoding gives rise to a Markovian dissipative process. We hence equate cellular-automaton decoding to a fully dissipative topological quantum memory, which removes errors continuously. In this sense, uncontrolled and unwanted local noise can be corrected for by a controlled local dissipative process. We analyze the required resources, commenting on additional polylogarithmic factors beyond those incurred by an ideal constant resource dynamical decoder.
Recent work established that, for independent erasures, storing KKK logical qubits for SSS time steps with error at most ε\varepsilonε requires a spacetime cost of Ω(S(K+log⁡(S/ε)))Ω(S(K+\log(S/\varepsilon)))Ω(S(K+log(S/ε))). A matching construction was also given, but assumes ideal error correction. Here we show that the same scaling can be achieved with an explicit noisy error-correction circuit and efficient decoding, counting all state-preparation, gate, measurement, and wait locations, provided that the hardware supports long-range qubit connectivity and fast, reliable classical processing. Thus, reliability adds only a logarithmic overhead shared by all stored qubits. The construction works for sufficiently weak but otherwise general circuit noise, including faults during syndrome extraction and correlated faults, and for independent erasures it matches the best known lower bound up to constant factors. We attain this scaling using quantum Tanner codes, with each memory step consisting of one round of syndrome extraction followed by a fixed number of parallel classical decoder steps that reduce the residual error enough to keep later faults correctable, regardless of memory size, storage time, or target accuracy. Together, this gives exponentially small failure probability per step while keeping the cost of each memory step linear in the code size. The same analysis also extends to certain constant-depth logical Clifford operations.
A physical realization of self-correcting quantum code would be profoundly useful for constructing a quantum computer. In this theoretical work, we provide a partial solution to major challenges preventing self-correcting quantum code from being engineered in realistic devices. We consider a variant of Kitaev's toric code coupled to propagating bosons, which induce a ranged interaction between anyonic defects. By coupling the primary quantum system to an engineered dissipation source through resonant energy transfer, we demonstrate a ``rate barrier'' which leads to a potentially enormous increase in the system's quantum-state lifetime through purely passive quantum error correction, even when coupled to an infinite-temperature bath. While our mechanism is not scalable to infinitely large systems, the maximum effective size can be very large, and it is fully compatible with active error-correction schemes. Our model uses only on-site and nearest-neighbor interactions and could be implemented in superconducting qubits. We sketch one such implementation at the end of this work.