Error-Correcting Code Table

A quantum code specifies a protected space or subsystem and an error model; a fault-tolerant implementation additionally specifies preparation, syndrome extraction, decoding, and logical operations. This table compares mathematical constructions rather than experimental records. The definitions and correction condition are developed in Why Quantum Error Correction Is Possible and Stabilizer Formalism.

Qubit code parameters

For a qubit stabilizer code, $[[n,k,d]]$ denotes n physical data qubits, k encoded qubits, and minimum weight d of an undetectable nontrivial logical Pauli. Ideal recovery corrects arbitrary errors on at most $\lfloor(d-1)/2\rfloor$ unknown qubits, or up to $d-1$ known erased positions. These statements assume ideal syndrome information and do not count extraction faults. Subsystem notation $[[n,k,r,d]]$ adds r gauge qubits; distance refers to protected logical information under dressed logical errors.

Code or family	Parameters, checks, and protection
Three-qubit repetition	$[[3,1,1]]$ for unrestricted Pauli noise; bit-flip version checks ZZI and IZZ Corrects one X error in its declared model; a single Z is already a logical error. Phase-flip version exchanges X and Z.
Five-qubit code	$[[5,1,3]]$; generators XZZXI, IXZZX, XIXZZ, ZXIXZ Corrects an arbitrary single-qubit error; a finite syndrome table suffices for ideal recovery.
Shor code	$[[9,1,3]]$; Z-pair checks within triples and X checks across paired triples Nested repetition protects both Pauli sectors; noisy check extraction still needs fault-tolerant design.
Steane code	$[[7,1,3]]$; CSS checks from the classical Hamming construction Separate X and Z syndrome processing; transversal Clifford operations with the appropriate logical conventions.
CSS codes	Binary checks satisfying $H_XH_Z^T=0$; the number of independent checks determines k X errors trigger Z checks and vice versa; joint decoding may exploit correlations between sectors.
Rotated planar surface code	Standard square patch $[[d^2,1,d]]$; local X- and Z-type checks Repeated noisy extraction requires spacetime decoding; data count excludes syndrome ancillas.
Color codes	Trivalent, face-colorable lattices with paired X/Z face checks; parameters depend on patch Some planar constructions support transversal Clifford gates; check weights and decoder differ from surface codes.
Bacon–Shor subsystem code	Square family $[[m^2,1,(m-1)^2,m]]$; local two-qubit gauge operators Gauge outcomes infer stabilizer syndromes; noncommuting gauges cannot all be assigned simultaneous eigenvalues.
Quantum LDPC codes	Families with bounded check weight and bounded qubit degree Sparse checks do not imply geometric locality, a universal decoder, or low circuit overhead.

For CSS codes, the number of encoded qubits is

$$k=n-\operatorname{rank}H_X-\operatorname{rank}H_Z.$$

The five-qubit generators are one cyclic convention; permutation and local Clifford changes produce equivalent presentations. Gottesman provides the stabilizer language; Bacon constructs the lattice gauge-subsystem family developed in Subsystem Codes. Roffe provides an accessible comparison between code properties and syndrome circuits. For asymptotic context, Panteleev and Kalachev establish that qLDPC families can have both $k=\Theta(n)$ and $d=\Theta(n)$; that existence result alone is not an implementation performance guarantee.

Bosonic encodings

An oscillator is not a finite collection of physical qubits. Its physical requirements include energy, accessible levels, control nonlinearities, ancillas, and measurement. A qubit distance label should not be imposed on a bosonic code without specifying the error basis.

Family	Encoding and error model
Cat codes	Coherent-state superpositions; parity and phase-space structure Protection depends on the selected encoding, loss monitoring, and engineered stabilization; cat size alone is not distance.
Binomial codes	Finite superpositions of Fock states with selected number moments Constructed to correct a specified finite set of loss, gain, and dephasing errors to the stated order.
GKP codes	Phase-space lattice encoding with displacement syndromes Ideal combs correct bounded shifts; finite peak width and envelope create residual logical error.

See Bosonic Codes for the error-algebra comparison and Bosonic-Encoded Computation Models for how an encoding participates in a computation.

Logical gates and recovery

No row specifies a universal fault-tolerant gate set by itself. Transversal gates depend on the code; lattice surgery, code deformation, gauge fixing, or injected resources can add operations. Use Fault-Tolerant Gates and Magic-State Distillation to check how faults propagate through the resulting circuit.

Likewise, a decoder is an algorithm with a noise model and latency budget. Lookup tables serve small codes; graph matching is useful for suitable surface-code noise; belief propagation and related methods apply to selected sparse-code constructions. Decoding the most likely individual error need not be the same as choosing the most likely logical equivalence class. See Decoders.

All listed families have established mathematical definitions. Claims of break-even memory, suppression with increasing distance, or application-scale utility require separate experimental evidence. Those claims belong with Logical Benchmarking and the dated fault-tolerance research record.

A distance check

For the bit-flip repetition code, $\lvert0_L\rangle=\lvert000\rangle$ and $\lvert1_L\rangle=\lvert111\rangle$. Each single X produces a distinct pair of Z-check signs, so ideal recovery can identify it. But $Z_1$ commutes with both checks and changes the relative phase of logical codewords. Its weight is one; therefore the full quantum distance is one despite the code’s ability to correct one error of the selected X type.

References

• Bacon, Dave, “Operator quantum error-correcting subsystems for self-correcting quantum memories,” Physical Review A 73, 012340 (2006), arXiv:quant-ph/0506023 — lattice subsystem construction and gauge recovery.
• Gottesman, Daniel, Stabilizer Codes and Quantum Error Correction, Ph.D. thesis, California Institute of Technology, 1997, arXiv:quant-ph/9705052 — stabilizer codes, distance, and fault-tolerant operations.
• Gottesman, Daniel, Alexei Kitaev, and John Preskill, “Encoding a qubit in an oscillator,” Physical Review A 64, 012310 (2001), doi:10.1103/PhysRevA.64.012310 — oscillator displacement codes.
• Panteleev, Pavel, and Gleb Kalachev, “Asymptotically good quantum and locally testable classical LDPC codes,” Proceedings of STOC 2022, 375–388 (2022), arXiv:2111.03654 — existence of asymptotically good qLDPC families.
• Roffe, Joschka, “Quantum error correction: an introductory guide,” Contemporary Physics 60, 226–245 (2019), arXiv:1907.11157 — code examples and implementation distinctions.