Sparse Quantum Codes from Quantum Circuits

Bacon, Dave; Flammia, Steven T.; Harrow, Aram W.; Shi, Jonathan

doi:10.1145/2746539.2746608

Cited by 12 publications

(9 citation statements)

References 44 publications

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“…Present results make two steps in this direction. First, the observation that ML decoding under these conditions amounts to decoding the code [21,22] associated with the circuit EEG, in the absence of correlations. Second, the structure of this latter code can be significantly simplified using row-reduction transformations, while leaving the probability of ML decoding unchanged.…”

Section: Discussionmentioning

confidence: 99%

“…The goal of this work is to give an explicit numerically efficient algorithm for analyzing error correlations resulting from a given qubit-based Clifford measurement circuit, and for constructing decoders optimized for such a circuit. Main result is that such correlations can be accounted for by using phenomenological error model (no error correlations) with the circuit-associated subsystem code constructed by Bacon, Flammia, Harrow, and Shi [21,22]. Thus, any generic decoder capable of dealing with uncorrelated data and syndrome measurement errors in highly-degenerate sparse-generator subsystem codes can be rendered to account for circuit-level error correlations.…”

Section: Introductionmentioning

confidence: 99%

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On maximum-likelihood decoding with circuit-level errors

Pryadko

2020

Quantum

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Error probability distribution associated with a given Clifford measurement circuit is described exactly in terms of the circuit error-equivalence group, or the circuit subsystem code previously introduced by Bacon, Flammia, Harrow, and Shi. This gives a prescription for maximum-likelihood decoding with a given measurement circuit. Marginal distributions for subsets of circuit errors are also analyzed; these generate a family of related asymmetric LDPC codes of varying degeneracy. More generally, such a family is associated with any quantum code. Implications for decoding highly-degenerate quantum codes are discussed.

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Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

On maximum-likelihood decoding with circuit-level errors

Pryadko

2020

Quantum

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“…Using a concatenated base code, the resulting subsystem code appears to have fractal structure. Again, the relationship between these codes and our work is purely superficial, in particular noting that the FPCs are commuting stabilizer code models while the codes of [69] are non-commuting subsystem codes.…”

Section: Relation To Previous Workmentioning

confidence: 98%

“…Recently, a new code construction has appeared that produces local subsystem codes with properties inherited from an arbitrary base stabilizer code [69]. Using a concatenated base code, the resulting subsystem code appears to have fractal structure.…”

Section: Relation To Previous Workmentioning

confidence: 99%

A proposal for self-correcting stabilizer quantum memories in 3 dimensions (or slightly less)

Brell

2016

New J. Phys.

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We propose a family of local CSS stabilizer codes as possible candidates for self-correcting quantum memories in 3D. The construction is inspired by the classical Ising model on a Sierpinski carpet fractal, which acts as a classical self-correcting memory. Our models are naturally defined on fractal subsets of a 4D hypercubic lattice with Hausdorff dimension less than 3. Though this does not imply that these models can be realized with local interactions in 3  , we also discuss this possibility. The X and Z sectors of the code are dual to one another, and we show that there exists a finite temperature phase transition associated with each of these sectors, providing evidence that the system may robustly store quantum information at finite temperature.

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“…The result on subsystem codes was adapted by Bravyi [17] to show that subsystem codes satisfy the bound kd 1/(D−1) ≤ O(n). Quantum stabilizer codes based on the toric code saturate these bounds for D = 2, and subsystem codes that saturate for D = 2 [17] or nearly saturate for D ≥ 3 [18] are also known.…”

Section: Prior Workmentioning

confidence: 99%

Limits on the storage of quantum information in a volume of space

Flammia

Haah

Kastoryano

et al. 2017

Quantum

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We study the fundamental limits on the reliable storage of quantum information in lattices of qubits by deriving tradeoff bounds for approximate quantum error correcting codes. We introduce a notion of local approximate correctability and code distance, and give a number of equivalent formulations thereof, generalizing various exact error-correction criteria. Our tradeoff bounds relate the number of physical qubitsn, the number of encoded qubitsk, the code distanced, the accuracy parameterδthat quantifies how well the erasure channel can be reversed, and the locality parameterℓthat specifies the length scale at which the recovery operation can be done. In a regime where the recovery is successful to accuracyϵthat is exponentially small inℓ, which is the case for perturbations of local commuting projector codes, our bound readskd2D−1≤O(n(log⁡n)2DD−1)for codes onD-dimensional lattices of Euclidean metric. We also find that the code distance of any local approximate code cannot exceedO(ℓn(D−1)/D)ifδ≤O(ℓn−1/D). As a corollary of our formulation of correctability in terms of logical operator avoidance, we show that the code distancedand the sized~of a minimal region that can support all approximate logical operators satisfiesd~d1D−1≤O(nℓDD−1), where the logical operators are accurate up toO((nδ/d)1/2)in operator norm. Finally, we prove that for two-dimensional systems if logical operators can be approximated by operators supported on constant-width flexible strings, then the dimension of the code space must be bounded. This supports one of the assumptions of algebraic anyon theories, that there exist only finitely many anyon types.

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Sparse Quantum Codes from Quantum Circuits

Cited by 12 publications

References 44 publications

On maximum-likelihood decoding with circuit-level errors

On maximum-likelihood decoding with circuit-level errors

A proposal for self-correcting stabilizer quantum memories in 3 dimensions (or slightly less)

Limits on the storage of quantum information in a volume of space

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