Block quantum code 

Description

A code constructed in a multi-partite quantum system, i.e., a physical space consisting of a tensor product of \(n > 1\) identical subsystems, e.g., qubits, modular qudits, Galois qudits, or oscillators.

Protection

Block codes protect from 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.

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.

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 [1].

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References

[1]
B. Eastin and E. Knill, “Restrictions on Transversal Encoded Quantum Gate Sets”, Physical Review Letters 102, (2009) arXiv:0811.4262 DOI
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Zoo Code ID: block_quantum

Cite as:
“Block quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/block_quantum
BibTeX:
@incollection{eczoo_block_quantum, title={Block quantum code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/block_quantum} }
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“Block quantum code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/block_quantum

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/properties/block/block_quantum.yml.