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

Beyond Pure Dephasing: A Quantitative Analysis of Two-Level Error Correction for Multi-Spin Molecular Qubits

Rowan Brad Quni-Gudzinas

Published Oct 5, 2026• 0 citations
Publisher

Abstract

Molecular spin systems exhibit a natural hierarchy of decoherence channels: fast pure dephasing (characterized by the transverse relaxation time T₂) and comparatively slow energy relaxation (characterized by T₁). A recent proposal argues that fault-tolerant quantum computing with molecules requires a hybrid two-level encoding: a first-level *dephasing-tolerant unit* (DTU) that suppresses the leading pure-dephasing channel, followed by a second-level multi-qubit code that corrects residual off-diagonal (relaxation-like) errors [1,2]. This paper provides an independent quantitative assessment of that scheme. Using assumed representative parameters (T₂ = 100 μs, T₁ = 10 ms, within ranges reported in the experimental molecular-spin-qubit literature), a projected dephasing suppression factor S = 100 (an assumption, not a value reported in [1]), and a 1 μs correction cycle, we derive an explicit logical error probability of 2.9998 × 10⁻⁸ per cycle (Γ_L = 2.9998 × 10⁻² s⁻¹, six physical spins per logical qubit) for a three-unit repetition code, valid only when S ≫ R, because the repetition code leaves residual dephasing uncorrected; at S = R the uncorrected single-Z channel contributes p_L = 1 − (1 − p_Z)³ ≈ 2.9997 × 10⁻⁴ per cycle (Γ_L ≈ 2.9997 × 10² s⁻¹). We further derive the *hierarchy condition*: the first-level suppression must satisfy S ≥ γ_φ/γ₁; with our parameters the equality S = γ_φ/γ₁ = 100 holds exactly, placing the scheme at the threshold where residual dephasing and relaxation contribute equally in the correctable relaxation channel. A computed comparison with a five-qubit outer code, which also corrects single Z errors (P_fail ≈ 4.5 × 10⁻⁷ per cycle, Γ_L ≈ 0.45 s⁻¹, 10 physical spins), shows the five-qubit code is preferable at S = R. We identify failure modes, falsifiable predictions, and open questions regarding synthesis and correlated noise. ## 1. Introduction Quantum information processing demands physical platforms combining long coherence times wit Full text and updates: papers.qnfo.org/papers/beyond-pure-dephasing-a-quantitative-analysis-of-two-level-error-correction-for-/

DOI
10.5281/zenodo.23161482

Related papers

Beyond Pure Dephasing: Quantum Error Correction in Single Molecules Requires Multiple Spins

Matteo Mezzadri, Silvia Macedonio, Luca Lepori +5 more

We propose multi-spin molecules as a viable architecture for fault-tolerant quantum computing. To this aim, we introduce a correction protocol handling both diagonal and off-diagonal errors, typically associated with dephasing and relaxation. The scheme is based on a hybrid encoding which combines dephasing-tolerant units suppressing the leading pure dephasing error into a multi-spin molecule implementing a multi-qubit code for residual off-diagonal errors. Our proposal leverages peculiar properties of molecular spins, i.e. the strong hierarchy between different errors and the possibility to engineer multi-spin molecules at the synthetic level. Moreover, it addresses the important issue of the loss of coherences in anharmonic systems subject to off-diagonal errors, which hampers their correction at the single spin level. Thanks to the huge suppression of dephasing by the first-level code, we numerically demonstrate the potential performance of this strategy even with a limited number of spins per logical unit.

cond-mat.mes-hallCross-listedarXiv:2610.03318•Oct 5, 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

Quantum error correction below the surface code threshold

Google Quantum AI and Collaborators, Rajeev Acharya, Dmitry A. Abanin +97 more

Quantum error correction [1, 2, 3, 4] provides a path to reach practical quantum computing by combining multiple physical qubits into a logical qubit, where the logical error rate is suppressed exponentially as more qubits are added. However, this exponential suppression only occurs if the physical error rate is below a critical threshold. Here, we present two below-threshold surface code memories on our newest generation of superconducting processors, Willow: a distance-7 code, and a distance-5 code integrated with a real-time decoder. The logical error rate of our larger quantum memory is suppressed by a factor of $$\varLambda \mathrm{=2.14}\pm 0.02$$ when increasing the code distance by two, culminating in a 101-qubit distance-7 code with 0.143% ± 0.003% error per cycle of error correction. This logical memory is also beyond break-even, exceeding its best physical qubit’s lifetime by a factor of $$2.4\pm 0.3$$ . Our system maintains below-threshold performance when decoding in real time, achieving an average decoder latency of 63 μs at distance-5 up to a million cycles, with a cycle time of 1.1 μs. We also run repetition codes up to distance-29 and find that logical performance is limited by rare correlated error events occurring approximately once every hour, or $$3$$ $$\times $$ $${10}^{9}$$ cycles. Our results present device performance that, if scaled, could realize the operational requirements of large scale fault-tolerant quantum algorithms.

arXiv:2408.13687•Nature•Dec 9, 2024•778 citations

Efficient Quantum Error Correction of Dephasing Induced by a Common Fluctuator

David Layden, Mo Chen, Paola Cappellaro

Quantum error correction is expected to be essential in large-scale quantum technologies. However, the substantial overhead of qubits it requires is thought to greatly limit its utility in smaller, near-term devices. Here we introduce a new family of special-purpose quantum error-correcting codes that offer an exponential reduction in overhead compared to the usual repetition code. They are tailored for a common and important source of decoherence in current experiments, whereby a register of qubits is subject to phase noise through coupling to a common fluctuator, such as a resonator or a spin defect. The smallest instance encodes one logical qubit into two physical qubits, and corrects decoherence to leading-order using a constant number of one- and two-qubit operations. More generally, while the repetition code on n qubits corrects errors to order t^{O(n)}, with t the time between recoveries, our codes correct to order t^{O(2^{n})}. Moreover, they are robust to model imperfections in small- and intermediate-scale devices, where they already provide substantial gains in error suppression. As a result, these hardware-efficient codes open a potential avenue for useful quantum error correction in near-term, pre-fault tolerant devices.

arXiv:1903.01046•Physical Review Letters•Jan 17, 2020•21 citations

Surface-Code Quantum Error Correction for Molecular Tweezer Arrays: Encoding, Layout, and Correlated Noise

Niharika Verma, Utpal Roy

Polar molecules trapped in optical tweezer arrays offer a promising platform for quantum information processing, providing precise control and long-range interactions that enable high-fidelity gate operations. We investigate quantum error correction in this system and show the influence of underlying physical noise. A mapping is constructed from a molecular tweezer array onto a rotated surface code in which a single specification of the array, namely, which code qubits share a molecule and where those molecules are located, determines both the correlated erasure structure and the dipolar exchange graph. We compare molecular encodings with rotational qudit dimensions (D), two and four under heralded molecular loss, coherent dipolar exchange, and imperfect heralding. It is found that a D= 4 encoding with spatially dispersed pairing exhibits a finite distance crossing at approximately the same per molecule loss rate as D= 2, while using 48-49 % fewer molecules, at the cost of a 6-10 % increase in sub-threshold logical error. Fixing the encoding and varying only the spatial embedding produces substantially larger effects: pairing the two co-located qubits along a lattice direction yields a logical-sector asymmetry of approximately fifty times. Over the simulated distances (d = 5, 7, 9), the disfavoured sector shows little or no suppression of logical error with increasing code distance, whereas the favoured sector improves by a factor of 1.5-2.3. We also find that Pauli twirl of the exchange interaction overestimates the logical error rate, which we attribute to the excitation-conserving structure of the interaction. These results reveal that the spatial embedding of correlated loss units is an important design parameter for molecular architectures and that single-sector benchmarks may be insufficient when correlated loss has directional structure.

quant-pharXiv:2608.21961•Aug 25, 2026•0 citations

Modeling coherent errors in quantum error correction

Daniel Greenbaum, Zachary Dutton

Analysis of quantum error correcting codes is typically done using a stochastic, Pauli channel error model for describing the noise on physical qubits. However, it was recently found that coherent errors (systematic rotations) on physical data qubits result in both physical and logical error rates that differ significantly from those predicted by a Pauli model. Here we examine the accuracy of the Pauli approximation for noise containing coherent errors (characterized by a rotation angle ) under the repetition code. We derive an analytic expression for the logical error channel as a function of arbitrary code distance d and concatenation level n , in the small error limit. We find that coherent physical errors result in logical errors that are partially coherent and therefore non-Pauli. However, the coherent part of the logical error is negligible at fewer than error correction cycles when the decoder is optimized for independent Pauli errors, thus providing a regime of validity for the Pauli approximation. Above this number of correction cycles, the persistent coherent logical error will cause logical failure more quickly than the Pauli model would predict, and this may need to be combated with coherent suppression methods at the physical level or larger codes.

arXiv:1612.03908•Quantum Science and Technology•Dec 20, 2017•81 citations
Molecular spin systems exhibit a natural hierarchy of decoherence channels: fast pure dephasing (characterized by the transverse relaxation time T₂) and comparatively slow energy relaxation (characterized by T₁). A recent proposal argues that fault-tolerant quantum computing with molecules requires a hybrid two-level encoding: a first-level dephasing-tolerant unit (DTU) that suppresses the leading pure-dephasing channel, followed by a second-level multi-qubit code that corrects residual off-diagonal (relaxation-like) errors [1,2]. This paper provides an independent quantitative assessment of that scheme. Using assumed representative parameters (T₂ = 100 μs, T₁ = 10 ms, within ranges reported in the experimental molecular-spin-qubit literature), a projected dephasing suppression factor S = 100 (an assumption, not a value reported in [1]), and a 1 μs correction cycle, we derive an explicit logical error probability of 2.9998 × 10⁻⁸ per cycle (Γ_L = 2.9998 × 10⁻² s⁻¹, six physical spins per logical qubit) for a three-unit repetition code, valid only when S ≫ R, because the repetition code leaves residual dephasing uncorrected; at S = R the uncorrected single-Z channel contributes p_L = 1 − (1 − p_Z)³ ≈ 2.9997 × 10⁻⁴ per cycle (Γ_L ≈ 2.9997 × 10² s⁻¹). We further derive the hierarchy condition: the first-level suppression must satisfy S ≥ γ_φ/γ₁; with our parameters the equality S = γ_φ/γ₁ = 100 holds exactly, placing the scheme at the threshold where residual dephasing and relaxation contribute equally in the correctable relaxation channel. A computed comparison with a five-qubit outer code, which also corrects single Z errors (P_fail ≈ 4.5 × 10⁻⁷ per cycle, Γ_L ≈ 0.45 s⁻¹, 10 physical spins), shows the five-qubit code is preferable at S = R. We identify failure modes, falsifiable predictions, and open questions regarding synthesis and correlated noise. ## 1. Introduction Quantum information processing demands physical platforms combining long coherence times wit Full text and updates: papers.qnfo.org/papers/beyond-pure-dephasing-a-quantitative-analysis-of-two-level-error-correction-for-/
We propose multi-spin molecules as a viable architecture for fault-tolerant quantum computing. To this aim, we introduce a correction protocol handling both diagonal and off-diagonal errors, typically associated with dephasing and relaxation. The scheme is based on a hybrid encoding which combines dephasing-tolerant units suppressing the leading pure dephasing error into a multi-spin molecule implementing a multi-qubit code for residual off-diagonal errors. Our proposal leverages peculiar properties of molecular spins, i.e. the strong hierarchy between different errors and the possibility to engineer multi-spin molecules at the synthetic level. Moreover, it addresses the important issue of the loss of coherences in anharmonic systems subject to off-diagonal errors, which hampers their correction at the single spin level. Thanks to the huge suppression of dephasing by the first-level code, we numerically demonstrate the potential performance of this strategy even with a limited number of spins per logical unit.
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 ${D}_{\ensuremath{\diamond}}^{\ensuremath{'}}\ensuremath{\sim}{D}_{\ensuremath{\diamond}}^{(d+1)/2}$.
Quantum error correction [1, 2, 3, 4] provides a path to reach practical quantum computing by combining multiple physical qubits into a logical qubit, where the logical error rate is suppressed exponentially as more qubits are added. However, this exponential suppression only occurs if the physical error rate is below a critical threshold. Here, we present two below-threshold surface code memories on our newest generation of superconducting processors, Willow: a distance-7 code, and a distance-5 code integrated with a real-time decoder. The logical error rate of our larger quantum memory is suppressed by a factor of Λ=2.14±0.02\varLambda \mathrm{=2.14}\pm 0.02Λ=2.14±0.02 when increasing the code distance by two, culminating in a 101-qubit distance-7 code with 0.143% ± 0.003% error per cycle of error correction. This logical memory is also beyond break-even, exceeding its best physical qubit’s lifetime by a factor of 2.4±0.32.4\pm 0.32.4±0.3 . Our system maintains below-threshold performance when decoding in real time, achieving an average decoder latency of 63 μs at distance-5 up to a million cycles, with a cycle time of 1.1 μs. We also run repetition codes up to distance-29 and find that logical performance is limited by rare correlated error events occurring approximately once every hour, or 333 ×\times × 109{10}^{9}109 cycles. Our results present device performance that, if scaled, could realize the operational requirements of large scale fault-tolerant quantum algorithms.
Quantum error correction is expected to be essential in large-scale quantum technologies. However, the substantial overhead of qubits it requires is thought to greatly limit its utility in smaller, near-term devices. Here we introduce a new family of special-purpose quantum error-correcting codes that offer an exponential reduction in overhead compared to the usual repetition code. They are tailored for a common and important source of decoherence in current experiments, whereby a register of qubits is subject to phase noise through coupling to a common fluctuator, such as a resonator or a spin defect. The smallest instance encodes one logical qubit into two physical qubits, and corrects decoherence to leading-order using a constant number of one- and two-qubit operations. More generally, while the repetition code on n qubits corrects errors to order t^{O(n)}, with t the time between recoveries, our codes correct to order t^{O(2^{n})}. Moreover, they are robust to model imperfections in small- and intermediate-scale devices, where they already provide substantial gains in error suppression. As a result, these hardware-efficient codes open a potential avenue for useful quantum error correction in near-term, pre-fault tolerant devices.
Polar molecules trapped in optical tweezer arrays offer a promising platform for quantum information processing, providing precise control and long-range interactions that enable high-fidelity gate operations. We investigate quantum error correction in this system and show the influence of underlying physical noise. A mapping is constructed from a molecular tweezer array onto a rotated surface code in which a single specification of the array, namely, which code qubits share a molecule and where those molecules are located, determines both the correlated erasure structure and the dipolar exchange graph. We compare molecular encodings with rotational qudit dimensions (D), two and four under heralded molecular loss, coherent dipolar exchange, and imperfect heralding. It is found that a D= 4 encoding with spatially dispersed pairing exhibits a finite distance crossing at approximately the same per molecule loss rate as D= 2, while using 48-49 % fewer molecules, at the cost of a 6-10 % increase in sub-threshold logical error. Fixing the encoding and varying only the spatial embedding produces substantially larger effects: pairing the two co-located qubits along a lattice direction yields a logical-sector asymmetry of approximately fifty times. Over the simulated distances (d = 5, 7, 9), the disfavoured sector shows little or no suppression of logical error with increasing code distance, whereas the favoured sector improves by a factor of 1.5-2.3. We also find that Pauli twirl of the exchange interaction overestimates the logical error rate, which we attribute to the excitation-conserving structure of the interaction. These results reveal that the spatial embedding of correlated loss units is an important design parameter for molecular architectures and that single-sector benchmarks may be insufficient when correlated loss has directional structure.
Analysis of quantum error correcting codes is typically done using a stochastic, Pauli channel error model for describing the noise on physical qubits. However, it was recently found that coherent errors (systematic rotations) on physical data qubits result in both physical and logical error rates that differ significantly from those predicted by a Pauli model. Here we examine the accuracy of the Pauli approximation for noise containing coherent errors (characterized by a rotation angle ) under the repetition code. We derive an analytic expression for the logical error channel as a function of arbitrary code distance d and concatenation level n , in the small error limit. We find that coherent physical errors result in logical errors that are partially coherent and therefore non-Pauli. However, the coherent part of the logical error is negligible at fewer than error correction cycles when the decoder is optimized for independent Pauli errors, thus providing a regime of validity for the Pauli approximation. Above this number of correction cycles, the persistent coherent logical error will cause logical failure more quickly than the Pauli model would predict, and this may need to be combated with coherent suppression methods at the physical level or larger codes.