Universal Weakly Fault-Tolerant Quantum Computation via Code Switching in the [[8,3,2]] Code

Abstract

Code-switching offers a route to universal, fault-tolerant quantum computation by circumventing the limitation implied by the Eastin-Knill theorem against a universal transversal gate set within a single quantum code. Here, we present a fault-tolerant code-switching protocol between two versions of the $[[8, 3, 2]]$ code. One version supports weakly fault-tolerant single-qubit Clifford gates, while the other supports a logical $\overline{\mathrm{CCZ}}$ gate via transversal $T/T^\dagger$ together with logical $\overline{\mathrm{CZ}}$, $\overline{\mathrm{CNOT}}$, and $\overline{\mathrm{SWAP}}$ gates. Because both codes have distance 2, the protocol operates in a postselected, error-detecting regime: single faults lead to detectable outcomes, and accepted runs exhibit quadratic suppression of logical error rates. This yields a universal scheme for postselected fault-tolerant computation. We validate the protocol numerically through simulations of state preparation, code switching, and a three-logical-qubit implementation of Grover's search.

Figures (16)

• Figure 1: Summary of this work. The two versions of the $[[8,3,2]]$ code, each of which can be geometrically represented by a cube. The three red edges of the $[[8,3,2]]_1$ code represent the fixed weight-2 gauge $G^X$ operators, and the three blue faces of the $[[8,3,2]]_2$ code represent the fixed weight-4 gauge $G^Z$ operators.
• Figure 2: Obtaining the two versions of the $[[8,3,2]]$ code by fixing the gauge operators of the $[8,3,3,2]$ subsystem code. Combining with the transversal $X^{\otimes 8}$ and $Z^{\otimes 8}$ stabilizers of the $[[8,3,3,2]]$ subsystem code, fixing the gauge operators give the different stabilizers that define the two stabilizer $[[8,3,2]]$ codes.
• Figure 3: The logical circuit for the single-pair inter-block $\overline{\mathrm{CNOT}}$ from logical qubit $\overline{1}$ of block $1$ to logical qubit $\overline{1}$ of block $2$hangleiter2025fault. If the two logical $\overline{\mathrm{SWAP}}$ gates are omitted, the circuit instead implements $\overline{\mathrm{CNOT}}(\ket{1,\overline{1}},\ket{2,\overline{2}})$; the SWAPs simply swaps the target from $\ket{2,\overline{2}}$ to $\ket{2,\overline{1}}$. Variants of this construction are given in \ref{['app:C']}.
• Figure 4: (a) Flag-based measurement of a weight-4 $Z$-type gauge operator used in switching from Version 1 to Version 2. The upper four wires are the data qubits in the support of the measured operator, the fifth wire is the ancilla measured in the $Z$ basis to decide whether $G^X$ corrections are necessary, and the last wire is a flag ancilla measured in the $X$ basis. A triggered flag identifies single-fault events that can propagate to weight-2 $Z$ errors on the data qubits. (b) Fault-tolerant measurement of a weight-2 $X$-type gauge operator used in switching from Version 2 to Version 1. The upper two wires are the data qubits in the support of the measured operator, the third wire is the ancilla measured in the $X$ basis to decide whether $G^Z$ corrections are necessary. Because the measured operator has weight 2, no flag ancilla is required.
• Figure 5: Fault-tolerant GHZ state encoding circuit chao2018quantum. The black part is the encoding circuit, and the red part checks for error propagation.