Subsystem surface codes with three-qubit check operators

TL;DR

This paper introduces a simplified subsystem surface code that uses only three-qubit measurements, maintains high error thresholds, and is topologically equivalent to the standard surface code, enhancing fault-tolerant quantum computing.

Contribution

The paper presents a new subsystem surface code with three-qubit check operators, providing an exactly solvable Hamiltonian and comparable topological properties to the standard surface code.

Findings

Error threshold ~0.6% with circuit-based errors

Error threshold ~0.97% with direct three-qubit parity measurements

Code preserves topological order and efficient decoding methods

Abstract

We propose a simplified version of the Kitaev's surface code in which error correction requires only three-qubit parity measurements for Pauli operators XXX and ZZZ. The new code belongs to the class of subsystem stabilizer codes. It inherits many favorable properties of the standard surface code such as encoding of multiple logical qubits on a planar lattice with punctured holes, efficient decoding by either minimum-weight matching or renormalization group methods, and high error threshold. The new subsystem surface code (SSC) gives rise to an exactly solvable Hamiltonian with 3-qubit interactions, topologically ordered ground state, and a constant energy gap. We construct a local unitary transformation mapping the SSC Hamiltonian to the one of the ordinary surface code thus showing that the two Hamiltonians belong to the same topological class. We describe error correction protocols for the SSC and determine its error thresholds under several natural error models. In particular, we show that the SSC has error threshold approximately 0.6% for the standard circuit-based error model studied in the literature. We also consider a model in which three-qubit parity operators can be measured directly. We show that the SSC has error threshold approximately 0.97% in this setting.