## 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, which correspond to Pauli errors of the code.

The structure of the Clifford circuit yields correlations between the circuit's possible measurement outcomes. The set of outcomes 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 backpropagated to obtain the parity checks of the outcome code. In other words, both codes have the same set of parity check outcomes.

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].

Many features of the spacetime circuit formalism can be understood through ZX calculus [4].

## 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. [3,5].
- Subsystem qubit stabilizer code — Spacetime circuit codes can be upgraded to subsystem codes by gauging out a subgroup of the logical Pauli group which causes trivial faults in the corresponding Clifford circuit.
- Quantum LDPC (QLDPC) code — General spacetime circuit codes can be sparsified to yield QLDPC spacetime circuit codes [3].
- Low-density parity-check (LDPC) code — There is an equivalence between Clifford circuits and LDPC codes with bit-check symmetry [6].
- Subsystem spacetime circuit code — Spacetime circuit codes can yield subsystem spacetime circuit codes by gauging out a subgroup of the logical Pauli group which causes trivial faults in the corresponding Clifford circuit. This construction is used to show the existance of geometrically local subsystem codes that nearly saturate the subsystem BT bound [1].

## References

- [1]
- D. Bacon et al., “Sparse Quantum Codes From Quantum Circuits”, IEEE Transactions on Information Theory 63, 2464 (2017) arXiv:1411.3334 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]
- J. C. M. de la Fuente et al., “The XYZ ruby code: Making a case for a three-colored graphical calculus for quantum error correction in spacetime”, (2024) arXiv:2407.08566
- [5]
- C. Gidney, “Stim: a fast stabilizer circuit simulator”, Quantum 5, 497 (2021) arXiv:2103.02202 DOI
- [6]
- Y. Li, “Low-density parity-check representation of fault-tolerant quantum circuits”, (2024) arXiv:2403.10268

## 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