Spacetime circuit code[1–3]

Description

Qubit stabilizer code constructed from a Clifford circuit, i.e., a circuit made up of Clifford gates and Pauli measurements, in order to detect and correct circuit faults. 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. A more general construction includes circuits with intermediate and multi-qubit measurements [3].

Many features of the spacetime circuit formalism can be understood through ZX calculus [4]. Two circuits are fault-equivalent if all undetectable faults on one circuit have a corresponding fault on the other [5].