Quantum Error Correction for Quantum Memories

Abstract

Active quantum error correction using qubit stabilizer codes has emerged as a promising, but experimentally challenging, engineering program for building a universal quantum computer. In this review we consider the formalism of qubit stabilizer and subsystem stabilizer codes and their possible use in protecting quantum information in a quantum memory. We review the theory of fault-tolerance and quantum error-correction, discuss examples of various codes and code constructions, the general quantum error correction conditions, the noise threshold, the special role played by Clifford gates and the route towards fault-tolerant universal quantum computation. The second part of the review is focused on providing an overview of quantum error correction using two-dimensional (topological) codes, in particular the surface code architecture. We discuss the complexity of decoding and the notion of passive or self-correcting quantum memories. The review does not focus on a particular technology but discusses topics that will be relevant for various quantum technologies.

Final version: 47 pages, 17 Figs, 311 references

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• $i$-QER: An Intelligent Approach towards Quantum Error Reduction
• Complexity of Fermionic Dissipative Interactions and Applications to Quantum Computing
• Correcting spanning errors with a fractal code
• Loss-tolerant architecture for quantum computing with quantum emitters
• An exactly solvable dissipative spin liquid
• Variational Quantum Algorithms for Trace Distance and Fidelity Estimation
• Lecture Notes on Quantum Electrical Circuits
• Stability and Loop Models from Decohering Non-Abelian Topological Order
• Machine Learning for Estimation and Control of Quantum Systems
• Demonstrating experimentally the encoding and dynamics of an error-correctable logical qubit on a hyperfine-coupled nuclear spin qudit
• Noise-adapted recovery circuits for quantum error correction
• Zero noise extrapolation on logical qubits by scaling the error correction code distance
• Synthetic ${\mathbb{Z}}_2$ gauge theories based on parametric excitations of trapped ions
• Exponential Lifetime Improvement in Topological Quantum Memories
• Non-Pauli topological stabilizer codes from twisted quantum doubles
• Heralded entangling quantum gate via cavity-assisted photon scattering
• Electric Circuit Realizations of Fracton Physics
• Extracting topological orders of generalized Pauli stabilizer codes in two dimensions
• Adaptive variational low-rank dynamics for open quantum systems
• Emergence of noise-induced barren plateaus in arbitrary layered noise models
• Simulation of open quantum systems via low-depth convex unitary evolutions
• Topologically Protected Quantum Entanglement
• Hybrid quantum computing with ancillas
• Towards a realistic GaAs-spin qubit device for a classical error-corrected quantum memory
• Quantum Reliability
• Iterative Phase Optimisation of Elementary Quantum Error Correcting Codes
• Information recoverability of noisy quantum states
• Greedy versus Map-based Optimized Adaptive Algorithms for random-telegraph-noise mitigation by spectator qubits
• Global Variational Quantum Circuits for Arbitrary Symmetric State Preparation
• Universal resource-efficient topological measurement-based quantum computation via color-code-based cluster states
• Decoding the Projective Transverse Field Ising Model
• Feedback cooled Bose-Einstein condensation: near and far from equilibrium
• Designing fault-tolerant circuits using detector error models
• Robust quantum classifier with minimal overhead
• Asymptotic symmetries and soft charges of fractons
• Using three-partite GHZ states for partial quantum error-detection in entanglement-based protocols
• Quantifying Quantum Computational Advantage on a Processor of Ultracold Atoms
• Virtual distillation with noise dilution
• What determines the ultimate precision of a quantum computer?
• Low-overhead quantum computing with the color code
• Physics-informed tracking of qubit fluctuations
• Error-rate-agnostic decoding of topological stabilizer codes
• Strategies for implementing quantum error correction in molecular rotation
• Non-Pauli errors can be efficiently sampled in qudit surface codes
• A New Mechanism of Open System Evolution and Its Entropy Using Unitary Transformations in Noncomposite Qudit Systems
• Generalized Kraus operators for the one-qubit depolarizing quantum channel
• Error channels in quantum nondemolition measurements on spin systems
• Coherent information as a mixed-state topological order parameter of fermions
• XYZ ruby code: Making a case for a three-colored graphical calculus for quantum error correction in spacetime
• Improved Belief Propagation Decoding Algorithms for Surface Codes
• Three-qubit direct dispersive parity measurement with Tunable Coupling Qubits
• Multivariate Trace Inequalities, p-Fidelity, and Universal Recovery Beyond Tracial Settings
• Preserving Entanglement in a Solid-Spin System Using Quantum Autoencoders
• Weak Hopf symmetry and tube algebra of the generalized multifusion string-net model
• Decoherence and wavefunction deformation of $D_4$ non-Abelian topological order
• Accelerating Quantum Algorithms with Precomputation
• Analytical percolation theory for topological color codes under qubit loss
• Tunable exciton valley-pseudospin orders in moiré Bose-Hubbard model
• Interleaving: Modular architectures for fault-tolerant photonic quantum computing
• Experimental signatures of Hilbert-space ergodicity: Universal bitstring distributions and applications in noise learning
• Dressed Interference in Giant Superatoms: Entanglement Generation and Transfer
• Quantum Error-Detection at Low Energies
• Error estimation in current noisy quantum computers
• FragQC: An Efficient Quantum Error Reduction Technique using Quantum Circuit Fragmentation
• Gate-set evaluation metrics for closed-loop optimal control on nitrogen-vacancy center ensembles in diamond
• Exploiting Anyonic Behavior of Quasicrystals for Topological Quantum Computing
• Parity measurement of remote qubits using dispersive coupling and photodetection
• Quantum continual learning on a programmable superconducting processor
• Near-Term Spin-Qubit Architecture Design via Multipartite Maximally-Entangled States
• Modeling error correction with Lindblad dynamics and approximate channels
• Local Clustering Decoder as a fast and adaptive hardware decoder for the surface code
• Non-perturbative Floquet engineering of the toric-code Hamiltonian and its ground state
• Graph Optimization Perspective for Low-Depth Trotter-Suzuki Decomposition
• Benchmarking Quantum Computers: Towards a Standard Performance Evaluation Approach
• Dynamical subset sampling of quantum error correcting protocols
• Quantum Two-Way Protocol Beyond Superdense Coding: Joint Transfer of Data and Entanglement
• Bulk-to-boundary anyon fusion from microscopic models
• Quantum Back-action Limits in Dispersively Measured Bose-Einstein Condensates
• Spanning Tree Matching Decoder for Quantum Surface Codes
• Noise-adapted qudit codes for amplitude-damping noise
• Efficient Learning for Linear Properties of Bounded-Gate Quantum Circuits
• Tunable three-body loss in a nonlinear Rydberg medium
• Realizing Lattice Surgery on Two Distance-Three Repetition Codes with Superconducting Qubits
• Resource Analysis of Low-Overhead Transversal Architectures for Reconfigurable Atom Arrays
• Trapped-ion quantum error-correcting protocols using only global operations
• Protection of Logical Qubits via Optimal State Transfers
• Greenberger-Horne-Zeilinger state generation in qubit-chains via a single $\pi /2$-pulse
• A bilayer Double Semion Model with Symmetry-Enriched Topological Order
• Strictly local one-dimensional topological quantum error correction with symmetry-constrained cellular automata
• Engineering the Kitaev spin liquid in a quantum dot system
• Low-Depth Flag-Style Syndrome Extraction for Small Quantum Error-Correction Codes
• A lower bound on the space overhead of fault-tolerant quantum computation
• Phase diagram of the three-dimensional subsystem toric code
• Demonstrating quantum error mitigation on logical qubits
• Spatially-Coupled QLDPC Codes
• Layer-Wise Security Framework and Analysis for the Quantum Internet
• Excitations in the Higher Lattice Gauge Theory Model for Topological Phases I: Overview
• Thermal state entanglement entropy on a quantum graph
• Protecting logical qubits with dynamical decoupling
• Fault-Tolerant Stabilizer Measurements in Surface Codes with Three-Qubit Gates
• Noise-tolerant quantum speedups in quantum annealing without fine tuning
• Entanglement Protection via Periodic Environment Resetting in Continuous Time Quantum Dynamical Processes
• Spatially parallel decoding for multi-qubit lattice surgery
• Secure authentication via Quantum Physical Unclonable Functions: a review
• Topological Protection of Two-photon Quantum Correlation on a Photonic Chip
• Feedback-induced interactive dynamics: unitary but dissipative evolution
• eQASM: An Executable Quantum Instruction Set Architecture
• Transport probe of nonadiabatic transition caused by Majorana moving
• Computing Defects Associated to Bounded Domain Wall Structures: The $\mathrm{Vec}(\mathbb{Z}/p\mathbb{Z})$ Case
• Logical multi-qubit entanglement with dual-rail superconducting qubits
• Single-Shot Decoding of Linear Rate LDPC Quantum Codes with High Performance
• Direct Characterization of Quantum Dynamics with Noisy Ancilla
• Local Probabilistic Decoding of a Quantum Code
• A Practical Framework for Quantum Error Mitigation
• Memory effects in quantum dynamics modelled by quantum renewal processes
• Efficient fault-tolerant code switching via one-way transversal CNOT gates
• Probabilistic state synthesis based on optimal convex approximation
• Protecting information via probabilistic cellular automata
• Quantum smoothed particle hydrodynamics algorithm inspired by quantum walks
• Superconducting Quantum Simulation for Many-Body Physics beyond Equilibrium
• Scalable fast benchmarking for individual quantum gates with local twirling
• General superposition states associated to the rotational and inversion symmetries in the phase space
• Lightcone Bounds for Quantum Circuit Mapping via Uncomplexity
• Lindbladians with multiple steady states: theory and applications
• Distributed Encoding and Decoding of Quantum Information over Networks
• Quantum delegated and federated learning via quantum homomorphic encryption
• Edge mode locality in perturbed symmetry protected topological order
• Robustness of longitudinal transmon readout to ionization
• Robust Topological Degeneracy of Classical Theories
• Plane-selective manipulations of nuclear spin qubits in a three-dimensional optical tweezer array
• Entangled Cloning of Stabilizer Codes and Free Fermions
• Geometric structure and transversal logic of quantum Reed-Muller codes
• Fault-tolerant simulation of Lattice Gauge Theories with gauge covariant codes
• Constant Depth Code Deformations in the Parity Architecture
• Designing quantum error correction codes for practical spin qudit
• Surpassing spectator qubits with photonic modes and continuous measurement for Heisenberg-limited noise mitigation
• Low-overhead quantum error correction codes with a cyclic topology
• Modular Architectures and Entanglement Schemes for Error-Corrected Distributed Quantum Computation
• A new twist on the Majorana surface code: Bosonic and fermionic defects for fault-tolerant quantum computation
• Measurement-Induced Heating of Trapped Ions
• A Universal Circuit Set Using the $S_3$ Quantum Double
• Foliated order parameter in a fracton phase transition
• Probabilistic unitary synthesis with optimal accuracy
• Learning from physics experiments, with quantum computers: Applications in muon spectroscopy
• Weak-Measurement-Induced Heating in Bose-Einstein Condensates
• Quantum Information Processing with Molecular Nanomagnets: an introduction
• Fidelity-based distance bounds for $N$-qubit approximate quantum error correction
• Microscopic theory on magnetic-field-tuned sweet spot of exchange interactions in multielectron quantum-dot systems
• Generalized quantum data-syndrome codes and belief propagation decoding for phenomenological noise
• Noise-aware variational eigensolvers: a dissipative route for lattice gauge theories
• Scalable noncontextuality inequalities and certification of multiqubit quantum systems
• Bridging three-dimensional coupled-wire models and cellular topological states: Solvable models for topological and fracton orders
• Braiding defects in topological stabiliser codes of any dimension cannot be universal
• Normalizer Circuits and Quantum Computation
• Probing the non-Abelian fusion of a pair of Majorana zero modes
• Methods for Classically Simulating Noisy Networked Quantum Architectures
• Generalized Number-Phase Lattice Encoding of a Bosonic Mode for Quantum Error Correction
• Pauli component erasing quantum channels
• Rescaling decoder for 2D topological quantum color codes on 4.8.8 lattices
• Demonstration of two-dimensional connectivity for a scalable error-corrected ion-trap quantum processor architecture
• Gauging Defects in Quantum Spin Systems
• Statistical mechanical mapping and maximum-likelihood thresholds for the surface code under generic single-qubit coherent errors
• Atomic interactions for qubit-error compensations
• A Substrate Scheduler for Compiling Arbitrary Fault-tolerant Graph States
• Verification of a resetting protocol for an uncontrolled superconducting qubit
• Reliable Quantum Memories with Unreliable Components
• Quantum error detection in qubit-resonator star architecture
• Time-delayed quantum feedback and incomplete decoherence suppression with no-knowledge measurement
• Comparing planar quantum computing platforms at the quantum speed limit
• Advancing Quantum Otto Engine Performance via Additional Magnetic Field and Effective Negative Temperature
• Electric-magnetic duality and ${\mathbb{Z}}_2$ symmetry enriched Abelian lattice gauge theory
• The domain wall color code
• Noise-adapted Quantum Error Correction for Non-Markovian Noise
• ArsoNISQ: Analyzing Quantum Algorithms on Near-Term Architectures
• Exact results on finite size corrections for surface codes tailored to biased noise
• Error correctable efficient quantum homomorphic encryption using Calderbank-Shor-Steane codes
• Effectiveness of the syndrome extraction circuit with flag qubits on IBM quantum hardware
• Fundamental thresholds for computational and erasure errors via the coherent information
• Towards robust variational quantum simulation of Lindblad dynamics via stochastic Magnus expansion
• Theory of the Simultaneous Transient Dispersive Readout of Multiple Spin Qubits
• Mitigating decoherence in molecular spin qudits
• Noisy Probabilistic Error Cancellation and Generalized Physical Implementability
• Fight or Flight: Cosmic Ray-Induced Phonons and the Quantum Surface Code
• Quantum simulation of Pauli channels and dynamical maps: algorithm and implementation
• Bounds on Autonomous Quantum Error Correction
• Hamiltonian Switching Control of Noisy Bipartite Qubit Systems
• Decoding across transversal Clifford gates in the surface code
• Introduction to Quantum Error Correction with Stabilizer Codes
• Improved performance of the Bacon-Shor code with Steane's syndrome extraction method
• Methodology for bus layout for topological quantum error correcting codes
• Thresholds for post-selected quantum error correction from statistical mechanics
• Stabilizing Non-Abelian Topological Order against Heralded Noise via Local Lindbladian Dynamics
• Coherent Information Phase Transition in a Noisy Quantum Circuit
• Distance-four quantum codes with combined postselection and error correction
• Long-range Ising interactions mediated by $\lambda {\varphi }^4$ fields: probing the renormalisation of sound in crystals of trapped ions
• Quantum Computing with sine-Gordon Qubits
• Magic Mirror on the Wall, How to Benchmark Quantum Error Correction Codes, Overall ?
• Magic State Injection with Erasure Qubits
• High cooperativity coupling to nuclear spins on a circuit QED architecture
• Almost Linear Decoder for Optimal Geometrically Local Quantum Codes
• Error threshold in active steering protocols for few-qubit systems
• Transversal CNOT gate with multi-cycle error correction
• Magic State Injection on IBM Quantum Processors Above the Distillation Threshold
• Viewing protected superconducting qubits through the lens of the cat qubit
• Color codes with twists: construction and universal gate set implementation
• Practical quantum error correction with the XZZX code and Kerr-cat qubits
• Contextuality of Quantum Error-Correcting Codes
• Revealing correlated noise with single-qubit operations
• Distributed Realization of Color Codes for Quantum Error Correction
• Consequences of unitary evolution of coupled qubit-resonator systems for stabilizing circuits in surface codes
• Lindblad-like quantum tomography for non-Markovian quantum dynamical maps
• Robustness of Magic in the quantum Ising chain via Quantum Monte Carlo tomography
• Low-density parity-check representation of fault-tolerant quantum circuits
• Lattice-enabled detection of spin-dependent three-body interactions
• Revisiting Nishimori multicriticality through the lens of information measures
• Shift symmetries and duality web in gauge theories
• Mode multiplexing for scalable cavity-enhanced operations in neutral-atom arrays
• Transport and fusion of Majorana zero modes in the presence of nonadiabatic transitions
• Do we have a quantum computer? Expert perspectives on current status and future prospects
• Building Bridges in Quantum Information Science Education: Expert Insights to Guide Framework Development for Interdisciplinary Teaching and Evolution of Common Language
• Generating magnon Bell states via parity measurement
• Exact Entanglement Dynamics of Two Spins in Finite Baths
• Purity-Assisted Zero-Noise Extrapolation for Quantum Error Mitigation
• An Energy Estimation Benchmark for Quantum Computers
• Approximate quantum error correction, covariance symmetry, and their relation
• Realizing Quantum Algorithms on Real Quantum Computing Devices
• Mixed-State Quantum Spin Liquids and Dynamical Anyon Condensations in Kitaev Lindbladians
• Minimising surface-code failures using a color-code decoder
• Universal set of Dynamically Protected Gates for Bipartite Qubit Networks II: Soft Pulse Implementation of the [[5,1,3]] Quantum Error Correcting Code
• Band twisting and resilience to disorder in long-range topological superconductors
• Optimizing the Frequency of Quantum Error Correction using the [[7,1,3]] Steane Code
• Hybrid spin-phonon architecture for scalable solid-state quantum nodes
• Geometrical Approach to Logical Qubit Fidelities of Neutral Atom CSS Codes
• Distinguishing Graph States by the Properties of Their Marginals
• Information conservation relations for weak measurement and its reversal
• Generalized Parity Measurements and Efficient Large Multi-component Cat State Preparation with Quantum Signal Processing
• Expedited Noise Spectroscopy of Transmon Qubits
• Sparse Non-Markovian Noise Modeling of Transmon-Based Multi-Qubit Operations
• Dissipation-assisted quantum correlations in coupled qubits
• Signaling versus distinguishing different superpositions of same pure quantum state
• Speeding up adiabatic holonomic quantum gates via $\pi$-pulse modulation
• Universal quantum computation and quantum error correction with ultracold atomic mixtures
• Stabilization of symmetry-protected long-range entanglement in stochastic quantum circuits
• Chiral-cat-state generation via the Sagnac-Fizeau effect
• Experimental measurement and a physical interpretation of quantum shadow enumerators
• Circuit-Noise-Resilient Virtual Distillation
• Hybrid quantum gap estimation algorithm using a filtered time series
• Distributing Quantum Computations, Shot-wise
• Partial randomized benchmarking
• Stable quantum memories with limited measurement
• Diabatic quantum annealing for training energy-based generative models
• Break-even point of the quantum repetition code
• Intrinsic Heralding and Optimal Decoders for Non-Abelian Topological Order
• Smallest quantum codes for amplitude damping noise
• Quantum Computation and Communication in Bosonic Systems
• Optimal Scheduling of Graph States via Path Decompositions
• Optimal demonstration of Autler Townes splitting
• Weight-based measure of quantum memory as a universal and operational benchmark
• Revisiting thermodynamics in computation and information theory
• Q-Pandora Unboxed: Characterizing Noise Resilience of Quantum Error Correction Codes
• Thermal stability of two-dimensional Topological Color Code
• Heralded High-Dimensional Photon-Photon Quantum Gate
• The Parity Flow Formalism: Tracking Quantum Information Throughout Computation
• Layer codes as partially self-correcting quantum memories
• Arbitrary high-fidelity binomial codes from multiphoton spin-boson interactions
• Ancilla-Assisted Protection of Information: Application to Atom-Cavity Systems
• Measurement-only circuit of perturbed toric code on triangular lattice: Topological entanglement, 1-form symmetry and logical qubits
• Quantum computation with logical gates between hot systems
• Self-attention U-Net decoder for toric codes
• Experimental Quantum Channel Purification
• Dephasing superchannels
• Symmetry boosts quantum computer performance
• Streaming Belief Propagation on Mixed-Alphabet Tanner Graphs for Practical Quantum Memory
• Snowflake: A Distributed Streaming Decoder
• Managing Classical Processing Requirements for Quantum Error Correction
• Quantum Thermodynamics Allows Quantum Measurement Almost Without Collapse
• Quantum-Inspired Weight-Constrained Neural Network: Reducing Variable Numbers by 100x Compared to Standard Neural Networks
• Universal quantum computation via scalable measurement-free error correction
• Relaxation times do not capture logical qubit dynamics
• Decoding Quantum Error Correction Codes with Local Variation
• Universal Dephasing Noise Injection via Schrodinger Wave Autoregressive Moving Average Models
• Affect of Simulated Atmospheric Turbulence on Three Entangled Photons
• Fully Parallelized BP Decoding for Quantum LDPC Codes Can Outperform BP-OSD
• Collective dynamics in holographic fractonic solids
• Decoding Correlated Errors in Quantum LDPC Codes
• Sampling (noisy) quantum circuits through randomized rounding
• From Bell Products to Greenberger-Horne-Zeilinger states: Quantum Memories via emergent Hamiltonians
• Quantum Error Correction with Superpositions of Squeezed Fock States
• Benchmarking fault-tolerant quantum computing hardware via QLOPS
• Efficient Characterization of Coherent and Correlated Low-Degree Noise in Layers of Gates
• Structural encoding with classical codes for computational-basis bit-flip correction in the early fault-tolerant regime
• Reduced State Embedding for Error Correction in Quantum Cryptography
• Fault-tolerant and secure long-distance quantum communication via uncorrectable-error-injection
• Dense packing of the surface code: code deformation procedures and hook-error-avoiding gate scheduling
• A digital quantum simulator in the presence of a bath
• General many-body entanglement swapping protocol: opportunities for distributed quantum computing
• Fault-Tolerant Non-Clifford GKP Gates using Polynomial Phase Gates and On-Demand Noise Biasing
• Error Resilience of Fracton Codes and Near Saturation of Code-Capacity Threshold in Three Dimensions
• Pinball: A Cryogenic Predecoder for Surface Code Decoding Under Circuit-Level Noise
• Linear-optical protocols for mitigating and suppressing noise in bosonic systems
• Non-normality and dissipation in Markovian quantum dynamics: Implications for quantum simulation
• Overflow-Safe Polylog-Time Parallel Minimum-Weight Perfect Matching Decoder: Toward Experimental Demonstration
• Comparison of spin-qubit architectures for quantum error-correcting codes
• Efficient decoding of stabilizer code by single-qubit local operations and classical communication
• Nonlinear quantum error correction
• Realization of Surface Code Quantum Memory on Systems with Always-On Interactions
• Q-Embroidery: A Study on Weaving Quantum Error Correction into the Fabric of Quantum Classifiers
• Certification of multi-qubit quantum systems with temporal inequalities
• The abelian state hidden subgroup problem: Learning stabilizer groups and beyond
• Information storage and transmission under Markovian noise
• Automated discovery and optimization of autonomous quantum error correction codes for a general open quantum system
• Robust Error Accumulation Suppression for Quantum Circuits
• Feedforward suppression of readout-induced faults in quantum error correction
• Influence of errors on the transport of quantum information through distant quantum dot spin qubits
• Geometric and holonomic quantum computation
• Emergent Kitaev materials in synthetic Fermi-Hubbard bilayers
• Biased-Noise Thresholds of Zero-Rate Holographic Codes with Tensor-Network Decoding
• Noise-resilient and resource-efficient hybrid algorithm for robust quantum gap estimation
• Global estimates of errors in quantum computation by the Feynman-Vernon formalism
• Entanglement Protection of Classically Driven Qubits in a Lossy Cavity
• Characterize noise correlation and enhance coherence via qubit motion
• Universal fault tolerant quantum computation in 2D without getting tied in knots
• Quantum Algorithms for State Preparation and Data Classification based on Stabilizer Codes
• Near-Term Fermionic Simulation with Subspace Noise Tailored Quantum Error Mitigation
• qSIEVE: Efficient qLDPC Memory via Systolic Movement in Atom Arrays
• Capacity of Summation over a Symmetric Quantum Erasure MAC with Partially Replicated Inputs
• A Useful Metric for the NISQ Era: Qubit Error Probability and Its Role in Zero Noise Extrapolation
• Approximate Quantum Error Correction with 1D Log-Depth Circuits
• Logical operations with a dynamical qubit in Floquet-Bacon-Shor code