## Description

Qubit stabilizer code used to correct faults in Clifford circuits, i.e., circuits up made of Clifford gates and Pauli measurements. The code utilizes redundancy in the measurement outcomes of a circuit to correct circuit faults.

The set of measurement outcomes of a circuit can be made into a classical binary linear code called the outcome code [3; Corr. 2]. The spacetime circuit code is defined such that its error syndromes can be used to obtain the parity checks of the outcome code.

More technically, given an \([m,k]\) outcome code associated with an \(n\)-qubit circuit of depth \(\Delta\) with \(m\) measurements and \(2^k\) outcomes, the corresponding spacetime circuit code is an \([[ n (\Delta + 1), n (\Delta + 1) - (m - k) ]]\) code [3; Thm. 2].

The spacetime circuit code is the stabilizer code corresponding to the subsystem codes of earlier works [1,2], which dealt with specific families of Clifford circuits. The general case was developed in Ref. [3].

## Decoding

## Parents

## Cousins

- Qubit stabilizer code — Spacetime circuit codes are useful for constructing fault-tolerant syndrome extraction circuits for qubit stabilizer codes.
- Linear binary code — The set of measurement outcomes of a Clifford circuit can be made into a classical binary linear code. Error syndromes of the spacetime circuit code can be used to obtain the parity checks of the outcome code.
- Kitaev surface code — Stabilizer generators of a spacetime code are called detectors in Refs. [3,4].
- Floquet code — Spacetime circuit codes are useful for constructing fault-tolerant encoding and syndrome extraction circuits for Floquet codes.
- Subsystem qubit stabilizer code — Spacetime circuit codes can be upgraded to subsystem codes by gauging a subgroup of the logical Pauli group which causes trivial faults in the corresponding Clifford circuit.

## References

- [1]
- D. Bacon et al., “Sparse Quantum Codes From Quantum Circuits”, IEEE Transactions on Information Theory 63, 2464 (2017) DOI
- [2]
- D. Gottesman, “Opportunities and Challenges in Fault-Tolerant Quantum Computation”, (2022) arXiv:2210.15844
- [3]
- N. Delfosse and A. Paetznick, “Spacetime codes of Clifford circuits”, (2023) arXiv:2304.05943
- [4]
- C. Gidney, “Stim: a fast stabilizer circuit simulator”, Quantum 5, 497 (2021) arXiv:2103.02202 DOI

## Page edit log

- Victor V. Albert (2023-05-11) — most recent

## Cite as:

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