## Description

A code constructed in a multi-partite quantum system, i.e., a physical space consisting of a tensor product of \(n > 1\) identical factors called subsystems, parties, or bodies. The subsystems include qubits, modular qudits, Galois qudits, oscillators, or more general groups. For finite dimensional codes, the dimension of the underlying subsystem is denoted by \(q\) and is sometimes called the local dimension.

While codewords \(c\) of block codes are elements of \(\Sigma^n\) for some alphabet \(\Sigma\), quantum states of block quantum codes are \(L^2\)-normalizable functions on said alphabet. Put differently, the configuration space of the canonical (a.k.a. computational) basis states \(|c\rangle\) of an \(n\)-body quantum system is the classical \(n\)-coordinate alphabet \(\Sigma^n\). The table below lists the most common alphabets used in block quantum codes, along with names of the corresponding subsystems.

alphabet \(\Sigma\) | subsystem |
---|---|

\(\mathbb{Z}_{2}=GF(2)\) | qubit |

\(GF(q)\) | Galois qudit |

\(\mathbb{Z}_{q}\) | modular qudit |

\(\mathbb{R}\) | bosonic mode |

\(G\) | group-valued qudit |

\(\mathcal{C}\) | category-valued qudit |

## Protection

Block codes protect from erasures or, more generally, errors acting on a few of the \(n\) subsystems. A block code with distance \(d\) detects errors acting on up to \(d-1\) subsystems, and corrects erasure errors on up to \(d-1\) subsystems. The subsystems that are erased are known to the receiver, and erasures of subsystems at unknown locations are called deletion errors [1–4]. More general forms of noise are caused by insertion errors [1–4], where subsystems are inserted into the block, and synchronization errors (a.k.a. misalignment) [5], where the code block is misplaced in a larger block by one or more locations.

The weight of an operator on a tensor-product Hilbert space is the number of subsystems on which the operator acts non-trivially. For example, an operator acting on two subsystem is called a weight-two operator or a two-body operator.

General noise models for block codes include stochastic noise, in which every possible error is assigned a probability. In the case of local stochastic noise, the probability decreases rapidly (typically, exponentially) with the number of subsystems that an error acts on. On the other hand, the adversarial noise model consists of errors acting on at most a fixed number of subsystems. Errors acting on subsystems in a geometrically local region are called burst errors [6,7].

### Bounds on code parameters

Bounds on finite dimensional block code performance include the quantum Singleton bound, quantum Hamming bound, quantum GV bound, various quantum linear programming (LP) bounds [8,9] (see the book [10]), and other bounds [11,12]. A code whose parameters attain the quantum Hamming bound (quantum Singleton bound) is called a perfect quantum code (a quantum MDS code). We are often interested in how parameters of particular infinite block quanutm code families scale with increasing block length \(n\), necessitating the use of asymptotic notation.

Quantum GV bound: The quantum GV bound [13] (see also Refs. [14–17]) for Galois qudits states that a pure \([[n,k,d]]_q\) Galois-qudit stabilizer code exists if \begin{align} \frac{q^{n-k+2}-1}{q^{2}-1}>\sum_{j=1}^{d-1}(q^{2}-1)^{j-1}\binom{n}{j}~. \tag*{(1)}\end{align} The quantum GV bound gives rise to the asymptotic quantum GV bound (i.e., quantum GV bound in the \(n\to\infty\) limit), expressed in terms of the maximum achievable rate \(R\) and relative distance \(\delta\), \begin{align} R\geq 1-\delta\log_q(q+1) - h_{q}(\delta)~, \tag*{(2)}\end{align} where \(h_q\) is the \(q\)-ary entropy function.

## Transversal Gates

Eastin-Knill theorem: Transversal gatesare logical gates on block codes that can be realized as tensor products of unitary operations acting on subsets of subsystems whose size is independent of \(n\). For subsets of size one, gates are sometimes called strongly transversal the single-subsystem unitaries are identical and weakly transversal otherwise. A universal gate set for a finite-dimensional block quantum code cannot be transversal for any code that detects single-block errors due to the Eastin-Knill theorem [18].

## Notes

## Parent

## Children

- Oscillator-into-oscillator code
- Amplitude-damping (AD) code
- Covariant block quantum code — Covariant codes for \(n>1\) are block quantum codes.
- Dynamically-generated QECC
- Quantum maximum-distance-separable (MDS) code
- Single-shot code
- Small-distance block quantum code
- Quasi-cyclic quantum code
- Tensor-network code
- Topological code — Topological codes are block codes because an infinite family of tensor-product Hilbert spaces is required to formally define a phase of matter.
- Quantum locally testable code (QLTC)
- Quantum low-weight check (QLWC) code
- Quantum locally recoverable code (QLRC)
- Modular-qudit code
- Galois-qudit code

## Cousins

- Monolithic quantum code — Block quantum codes for \(n=1\) are monolithic codes.
- Block code

## References

- [1]
- M. Hagiwara and A. Nakayama, “A Four-Qubits Code that is a Quantum Deletion Error-Correcting Code with the Optimal Length”, (2020) arXiv:2001.08405
- [2]
- A. Nakayama and M. Hagiwara, “Single Quantum Deletion Error-Correcting Codes”, (2020) arXiv:2004.00814
- [3]
- Y. Ouyang, “Permutation-invariant quantum coding for quantum deletion channels”, 2021 IEEE International Symposium on Information Theory (ISIT) (2021) arXiv:2102.02494 DOI
- [4]
- T. Shibayama and M. Hagiwara, “Permutation-Invariant Quantum Codes for Deletion Errors”, 2021 IEEE International Symposium on Information Theory (ISIT) (2021) arXiv:2102.03015 DOI
- [5]
- Y. Fujiwara, “Block synchronization for quantum information”, Physical Review A 87, (2013) arXiv:1206.0260 DOI
- [6]
- S. Kawabata, “Quantum Interleaver: Quantum Error Correction for Burst Error”, Journal of the Physical Society of Japan 69, 3540 (2000) DOI
- [7]
- F. Vatan, V. P. Roychowdhury, and M. P. Anantram, “Spatially correlated qubit errors and burst-correcting quantum codes”, IEEE Transactions on Information Theory 45, 1703 (1999) DOI
- [8]
- E. M. Rains, “Quantum shadow enumerators”, (1997) arXiv:quant-ph/9611001
- [9]
- A. Ashikhmin and S. Litsyn, “Upper Bounds on the Size of Quantum Codes”, (1997) arXiv:quant-ph/9709049
- [10]
- D. Gottesman. Surviving as a quantum computer in a classical world (2024) URL
- [11]
- Keqin Feng, San Ling, and Chaoping Xing, “Asymptotic bounds on quantum codes from algebraic geometry codes”, IEEE Transactions on Information Theory 52, 986 (2006) DOI
- [12]
- G. Chiribella et al., “Quantum error correction with degenerate codes for correlated noise”, Physical Review A 83, (2011) arXiv:1007.3655 DOI
- [13]
- K. Feng and Z. Ma, “A Finite Gilbert–Varshamov Bound for Pure Stabilizer Quantum Codes”, IEEE Transactions on Information Theory 50, 3323 (2004) DOI
- [14]
- A. Ekert and C. Macchiavello, “Error Correction in Quantum Communication”, (1996) arXiv:quant-ph/9602022
- [15]
- A. Ashikhmin et al., “Quantum Error Detection II: Bounds”, (1999) arXiv:quant-ph/9906131
- [16]
- A. Ashikhmin and E. Knill, “Nonbinary quantum stabilizer codes”, IEEE Transactions on Information Theory 47, 3065 (2001) DOI
- [17]
- Y. Ma, “The asymptotic probability distribution of the relative distance of additive quantum codes”, Journal of Mathematical Analysis and Applications 340, 550 (2008) DOI
- [18]
- B. Eastin and E. Knill, “Restrictions on Transversal Encoded Quantum Gate Sets”, Physical Review Letters 102, (2009) arXiv:0811.4262 DOI
- [19]
- M. Grassl, “Searching for linear codes with large minimum distance”, Discovering Mathematics with Magma 287 DOI
- [20]
- M. F. Ezerman et al., “Characterization of Nearly Self-Orthogonal Quasi-Twisted Codes and Related Quantum Codes”, (2024) arXiv:2405.15057

## Page edit log

- Victor V. Albert (2023-02-14) — most recent

## Cite as:

“Block quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/block_quantum