Subsystem bivariate bicycle codes achieve high-rate BB logical qubits with local four-qubit gauge checks, yielding examples such as [[108,12,6]] that outperform surface-code alternatives.
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Stabilizer Codes and Quantum Error Correction
Canonical reference. 86% of citing Pith papers cite this work as background.
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abstract
Controlling operational errors and decoherence is one of the major challenges facing the field of quantum computation and other attempts to create specified many-particle entangled states. The field of quantum error correction has developed to meet this challenge. A group-theoretical structure and associated subclass of quantum codes, the stabilizer codes, has proved particularly fruitful in producing codes and in understanding the structure of both specific codes and classes of codes. I will give an overview of the field of quantum error correction and the formalism of stabilizer codes. In the context of stabilizer codes, I will discuss a number of known codes, the capacity of a quantum channel, bounds on quantum codes, and fault-tolerant quantum computation.
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representative citing papers
For an explicit prefix/tree family of quantum states, adaptive local Pauli tomography achieves polynomial copy complexity while non-adaptive strategies require exponentially many copies.
Gauss law codes identify the full gauge-invariant sector as the code space while vacuum codes restrict to the matter vacuum, with the two shown to be unitarily equivalent for finite gauge groups.
A new in-situ scheme prepares logical magic states inside arbitrary CSS qLDPC codes using only syndrome-extraction ancillas, with simulations on the [[144,12,12]] BB code and [[225,9,4]] hypergraph-product code showing injection error rates around 10^{-3} or lower under depolarizing and asymmetric噪声
Univariate bicycle codes give an explicit basis for logical operators and distance upper bounds in a restricted class of quantum LDPC codes while matching the performance of less constrained generalized and bivariate bicycle codes in simulations.
Using post-selection to map physical noise to a weaker accepted logical channel and then applying order-K perturbative PEC reduces sampling overhead by 3-4 orders of magnitude for logical GHZ preparation on up to 200 qubits with the Iceberg code.
Punctured surface codes map disjoint or overlapping Z-couplings to a single logical Z for protected distributed estimation of many-body Hamiltonian parameters.
Dual-species Na-Cs Rydberg array enables simultaneous non-destructive readout of multiple Pauli-Z stabilizers on four-qubit plaquettes using a single global pulse sequence after compensating geometric phase errors.
I(n1, n2) is a correlation functional with |I| ≤ 2 for any three-qubit state, saturated only for GHZ-equivalent states under mutually unbiased measurements.
Adding an ancilla qubit to GKP-stabilizer codes reduces Gaussian displacement noise standard deviation from σ to O(σ²) for universal hybrid CV-DV gates.
Harmoniq approximates a quantum-harmonic-analysis data augmentation operator as a mixture of at most quadratic-depth n-qubit circuits, enabling modular combination with other quantum subroutines for signal denoising.
Multi-entropy exhibits a structural obstruction to replica symmetry breaking in random tensor networks due to incompatible boundary permutations in the replica hypercube, unlike entanglement negativity.
Closed-form sector length distributions for recursively definable graph states (paths, cycles, stars, grids) via generating functions, yielding analytical concentratable entanglement, depolarizing fidelity bounds, and multipartite entanglement criteria.
Dismagicker is a non-Clifford unitary that suppresses non-stabilizerness in quantum states, improving simulation accuracy when combined with Clifford disentanglers.
A code-switching protocol in the [[8,3,2]] code yields a universal scheme for postselected fault-tolerant quantum computation with quadratic logical error suppression.
Introduces gauge-invariant QMETTS using mutually unbiased physical bases derived from stabilizer formalism for Z2 LGT at finite T and density, with single-shot sampling shown near-optimal and numerical validation in 1+1D.
Knill error correction reduces circuit-level decoding for quantum LDPC codes to the simpler code-capacity decoder while remaining fault-tolerant under locally decaying noise.
A new Sparse Stabilizer Tensor cost function enables hyper-optimized contraction schedules for Quantum LEGO WEP calculations, delivering orders-of-magnitude improvements over dense tensor baselines for stabilizer codes.
A new framework certifies global quantum properties including multipartite entanglement, circuit complexity, and quantum magic on small subsystems with constant sample complexity via local Pauli measurements.
Introduces a minimal matchgate circuit representation for fermionic Gaussian states together with a Yang-Baxter update algorithm, then maps out entanglement transitions in unitary circuit games under braiding and generic matchgate rules.
Wire codes are a construction that converts any stabilizer code into a local weight-3 subsystem code on an arbitrary graph via low-density Tanner-graph embedding, with overhead governed by the embedding quality.
Magic state cultivation prepares high-fidelity T states with an order of magnitude fewer qubit-rounds than prior distillation methods by gradually growing them within a surface code under depolarizing noise.
Defines a resource theory of GPT-contextuality whose free operations are classical systems and univalent simulations, yielding monotones including classical excess (minimal embedding error into infinite classical systems) and parity-oblivious multiplexing success probability, with noncontextual GPTs
Compactification of a single higher-dimensional hypergraph-product fracton model yields a broad family of translation-invariant quantum LDPC codes that includes fracton models and all A2BGA codes such as BB codes.
citing papers explorer
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Topological subsystem bivariate bicycle codes with four-qubit check operators
Subsystem bivariate bicycle codes achieve high-rate BB logical qubits with local four-qubit gauge checks, yielding examples such as [[108,12,6]] that outperform surface-code alternatives.
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An Exponential Advantage for Adaptive Tomography of Structured States under Pauli Basis Measurements
For an explicit prefix/tree family of quantum states, adaptive local Pauli tomography achieves polynomial copy complexity while non-adaptive strategies require exponentially many copies.
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Gauss law codes and vacuum codes from lattice gauge theories
Gauss law codes identify the full gauge-invariant sector as the code space while vacuum codes restrict to the matter vacuum, with the two shown to be unitarily equivalent for finite gauge groups.
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In-Situ Simultaneous Magic State Injection on Arbitrary CSS qLDPC Codes
A new in-situ scheme prepares logical magic states inside arbitrary CSS qLDPC codes using only syndrome-extraction ancillas, with simulations on the [[144,12,12]] BB code and [[225,9,4]] hypergraph-product code showing injection error rates around 10^{-3} or lower under depolarizing and asymmetric噪声
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Univariate Bicycle Quantum LDPC Codes: Explicit Logical Structure and Distance Bounds
Univariate bicycle codes give an explicit basis for logical operators and distance upper bounds in a restricted class of quantum LDPC codes while matching the performance of less constrained generalized and bivariate bicycle codes in simulations.
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Zeno-Enhanced Probabilistic Error Cancellation with Quantum Error Detection Codes
Using post-selection to map physical noise to a weaker accepted logical channel and then applying order-K perturbative PEC reduces sampling overhead by 3-4 orders of magnitude for logical GHZ preparation on up to 200 qubits with the Iceberg code.
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Distributed estimation of many-body Hamiltonians via punctured surface code
Punctured surface codes map disjoint or overlapping Z-couplings to a single logical Z for protected distributed estimation of many-body Hamiltonian parameters.
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Multi-Qubit Stabilizer Readout on a Dual-Species Rydberg Array
Dual-species Na-Cs Rydberg array enables simultaneous non-destructive readout of multiple Pauli-Z stabilizers on four-qubit plaquettes using a single global pulse sequence after compensating geometric phase errors.
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A measure for genuine tripartite entanglement
I(n1, n2) is a correlation functional with |I| ≤ 2 for any three-qubit state, saturated only for GHZ-equivalent states under mutually unbiased measurements.
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Noise Reduction for Universal Hybrid Oscillator-Qubit Quantum Computation
Adding an ancilla qubit to GKP-stabilizer codes reduces Gaussian displacement noise standard deviation from σ to O(σ²) for universal hybrid CV-DV gates.
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Harmoniq: Efficient Data Augmentation on a Quantum Computer Inspired by Harmonic Analysis
Harmoniq approximates a quantum-harmonic-analysis data augmentation operator as a mixture of at most quadratic-depth n-qubit circuits, enabling modular combination with other quantum subroutines for signal denoising.
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Structural Obstruction to Replica Symmetry Breaking for Multi-Entropy in Random Tensor Networks
Multi-entropy exhibits a structural obstruction to replica symmetry breaking in random tensor networks due to incompatible boundary permutations in the replica hypercube, unlike entanglement negativity.
-
Sector length distributions of recursively definable graph states through analytic combinatorics
Closed-form sector length distributions for recursively definable graph states (paths, cycles, stars, grids) via generating functions, yielding analytical concentratable entanglement, depolarizing fidelity bounds, and multipartite entanglement criteria.
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Dismagicker: Unitary Gate for Non-Stabilizerness Reduction
Dismagicker is a non-Clifford unitary that suppresses non-stabilizerness in quantum states, improving simulation accuracy when combined with Clifford disentanglers.
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Universal Weakly Fault-Tolerant Quantum Computation via Code Switching in the [[8,3,2]] Code
A code-switching protocol in the [[8,3,2]] code yields a universal scheme for postselected fault-tolerant quantum computation with quadratic logical error suppression.
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Gauge-invariant QMETTS with mutually unbiased physical bases for $Z_2$ lattice gauge theories at finite temperature and density
Introduces gauge-invariant QMETTS using mutually unbiased physical bases derived from stabilizer formalism for Z2 LGT at finite T and density, with single-shot sampling shown near-optimal and numerical validation in 1+1D.
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Simplified circuit-level decoding using Knill error correction
Knill error correction reduces circuit-level decoding for quantum LDPC codes to the simpler code-capacity decoder while remaining fault-tolerant under locally decaying noise.
-
Hyper-optimized Quantum Lego Contraction Schedules
A new Sparse Stabilizer Tensor cost function enables hyper-optimized contraction schedules for Quantum LEGO WEP calculations, delivering orders-of-magnitude improvements over dense tensor baselines for stabilizer codes.
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Certifying localizable quantum properties with constant sample complexity
A new framework certifies global quantum properties including multipartite entanglement, circuit complexity, and quantum magic on small subsystems with constant sample complexity via local Pauli measurements.
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Disentangling strategies and entanglement transitions in unitary circuit games with matchgates
Introduces a minimal matchgate circuit representation for fermionic Gaussian states together with a Yang-Baxter update algorithm, then maps out entanglement transitions in unitary circuit games under braiding and generic matchgate rules.
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Wire Codes
Wire codes are a construction that converts any stabilizer code into a local weight-3 subsystem code on an arbitrary graph via low-density Tanner-graph embedding, with overhead governed by the embedding quality.
-
Magic state cultivation: growing T states as cheap as CNOT gates
Magic state cultivation prepares high-fidelity T states with an order of magnitude fewer qubit-rounds than prior distillation methods by gradually growing them within a surface code under depolarizing noise.
-
Resource-theoretic hierarchy of contextuality for general probabilistic theories
Defines a resource theory of GPT-contextuality whose free operations are classical systems and univalent simulations, yielding monotones including classical excess (minimal embedding error into infinite classical systems) and parity-oblivious multiplexing success probability, with noncontextual GPTs
-
Translation-invariant quantum low-density parity-check codes from compactified fracton models
Compactification of a single higher-dimensional hypergraph-product fracton model yields a broad family of translation-invariant quantum LDPC codes that includes fracton models and all A2BGA codes such as BB codes.
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Spatial overhead reduction for 2D hypergraph product codes
A qubit-reduction method for hypergraph product codes preserves dimension, distance, and fault-tolerance properties, producing smaller codes such as [[441,64,6]] from [[610,64,6]] with comparable noise performance and compatibility with logical gates.
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Protocol for Efficient Generation of Fusion-Based Quantum Computing Resource States from Quantum Emitters
Logically encoded 24-photon FBQC resource states can be deterministically produced from 3 quantum emitters and 11 CNOT gates by using symmetries to reduce the search over photon emission orderings.
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Mid-Circuit Measurements for Clifford Noise Reduction in Hamiltonian Simulations
Mid-circuit stabilizer verification in six-qubit GSE-encoded Clifford Trotter steps reduces logical error rates by up to 54% on Barium ion hardware, with the gain vanishing if checks are deferred to circuit end.
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Quantum Magic in early FTQC: From Diagonal Clifford Hierarchy No-Go Theorems to Architecture Design Blueprints
No-go theorems prove hierarchy level and state-independent sequences cannot maximize operational magic in early FTQC, requiring state-aware differentiable optimization and nonlinear phases for scalable magic generation.
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Design and Analysis of Quantum Dual-Containing CSS LDPC Codes based on Quasi-Dyadic Matrices
Two new constructions of quantum dual-containing CSS LDPC codes from quasi-dyadic matrices achieve improved finite-length error performance over existing DC codes.
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Interplay of Nonstabilizerness and Ergotropy in Quantum Batteries
Ergotropy in the battery corresponds one-to-one with total nonstabilizerness under U(1)-symmetric charger-battery interactions, while maximum average charging power in Clifford evolution is achievable even with zero initial magic.
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Defect-Adaptive Lattice Surgery on Irregular Boundary Surface-Code Patches
A defect-adaptive lattice surgery technique reconstructs joint logical parities on irregular surface-code patches via GF(2) binary synthesis from seam measurements and pre-merge constraints, yielding executable rules or failure certificates while preserving effective distance.
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A graph-aware bounded distance decoder for all stabilizer codes
A graph-based bounded distance decoder corrects all errors up to a chosen weight in arbitrary stabilizer codes by representing stabilizers and syndromes as graphs and pruning the search space with a feed-forward structure.
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Coherence dynamics in quantum many-body systems with conservation laws
Conservation laws in quantum circuits and Hamiltonians replace logarithmic coherence saturation with slow hydrodynamic relaxation globally and produce algebraic peak-time growth locally, unlike ergodic cases.
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Deterministic generation of grid states with programmable nonlinear bosonic circuits
Programmable nonlinear bosonic circuits can deterministically produce phased-comb states that serve as a scalable bosonic quantum error-correcting code with near-optimal performance against boson loss.
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GreenPeas: Unlocking Adaptive Quantum Error Correction with Just-in-Time Decoding Hypergraphs
GreenPeas delivers a just-in-time GPU compiler for decoding hypergraphs that achieves >10x speedup on surface and bivariate bicycle codes, unlocking circuit-level decoding for adaptive quantum error correction.
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The Rotation Gap Is Not An Error: Ternary Structure in IBM Quantum Hardware
IBM quantum hardware has ternary error structure causing sub-Poissonian syndromes; a classifier decoder that skips correcting them cuts logical errors by 7-19%.
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Error Correction in Lattice Quantum Electrodynamics with Quantum Reference Frames
Lattice QED is established as a quantum error-correcting code beyond stabilizers, with explicit recovery operations constructed via quantum reference frames for gauge and fermionic sectors.
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Catalytic Quantum Error Correction: Theory, Efficient Catalyst Preparation, and Numerical Benchmarks
Catalytic Quantum Error Correction recovers known target states from noisy copies with F > 0.96 using only eight copies by preserving coherent modes and applying a CPMG-Clifford-swap-test pipeline, bypassing magnitude thresholds of standard QEC.
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Transversal gates of the ((3,3,2)) qutrit code and local symmetries of the absolutely maximally entangled state of four qutrits
A bijection maps LU orbits of AME states (n even) to orbits of ((n-1,D,n/2))_D codes, with the 4-qutrit AME state and ((3,3,2))_3 code both unique up to LU and their symmetry/transversal groups explicitly generated via Vinberg theory.
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Stabilizer Code-Generic Universal Fault-Tolerant Quantum Computation
Ancilla-mediated protocols enable deterministic universal logical gates on any stabilizer code without ancilla consumption or code modification.
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A Concatenated Dual Displacement Code for Continuous-Variable Quantum Error Correction
The concatenated dual displacement code suppresses Gaussian displacement error variance by up to 50% under infinite squeezing while correcting lattice-crossing events in CV quantum error correction.
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Entanglement boosting: Low-volume logical Bell pair preparation for distributed fault-tolerant quantum computation
Entanglement boosting protocol prepares logical Bell pairs in rotated surface codes with orders-of-magnitude lower link-limited volume, reaching 10^{-10} logical error from 86 physical pairs at 1% error using soft decoders and postselection within one patch.
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Dense packing of the surface code: code deformation procedures and hook-error-avoiding gate scheduling
The paper introduces concrete code deformation procedures for dense surface code packing, proposes hook-error-avoiding CNOT scheduling for syndrome extraction, and reports Monte Carlo simulations showing lower logical error rates than standard surface codes at large distances and low physical error,
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Rigorous estimation of error thresholds of transversal Clifford logical circuits
Generalizes stat-mech mapping from toric code memories to transversal Clifford circuits, mapping tCNOT to random Ashkin-Teller and 4-body Ising models and estimating reduced thresholds of p=0.080 and p>=0.028.
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Dynamical error reshaping for dual-rail erasure qubits
Dynamical control schemes for dual-rail erasure qubits suppress transmon-induced noise, reducing erasure check errors by three orders of magnitude and logical two-qubit gate infidelities by up to three orders of magnitude.
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Composite-Dimensional Topological Codes with Boundaries and Defects
Algorithms construct stabilizer models for boundaries and 0D defects in composite-dimensional twisted quantum double codes, with examples like Z4 double coupled to double semion phase and threshold comparisons to surface codes.
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Entanglement witnesses for stabilizer states and subspaces beyond qubits
Generalizes entanglement witnesses from qubit stabilizer states to multi-qudit versions, showing better noise robustness in some cases.
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Constrained free energy minimization for the design of thermal states and stabilizer thermodynamic systems
Benchmarks gradient-ascent algorithms for constrained free energy minimization on quantum Heisenberg models and stabilizer codes, with applications to thermal state design and fixed-temperature quantum encoding.
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Frustration graph formalism for qudit observables
A frustration graph formalism for prime-d qudit observables yields a unitary map to generalized Pauli form plus bounds on observable sums used for qudit entanglement quantification.
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Geometrical constructions of purity testing protocols and their applications to quantum communication
Geometrical constructions map classical linear error correcting codes to purity testing protocols whose properties are fully determined by the codes, enabling applications in quantum communication.