Description
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 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 [1,2].
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 [3].
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
- Quantum Lego code
- Planar-perfect-tensor 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]
- S. Kawabata, “Quantum Interleaver: Quantum Error Correction for Burst Error”, Journal of the Physical Society of Japan 69, 3540 (2000) DOI
- [2]
- 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
- [3]
- B. Eastin and E. Knill, “Restrictions on Transversal Encoded Quantum Gate Sets”, Physical Review Letters 102, (2009) arXiv:0811.4262 DOI
- [4]
- M. Grassl, “Searching for linear codes with large minimum distance”, Discovering Mathematics with Magma 287 DOI
- [5]
- 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