Partitioning qubits in hypergraph product codes to implement logical gates

Abstract

The promise of high-rate low-density parity check (LDPC) codes to substantially reduce the overhead of fault-tolerant quantum computation depends on constructing efficient, fault-tolerant implementations of logical gates on such codes. Transversal gates are the simplest type of fault-tolerant gate, but the potential of transversal gates on LDPC codes has hitherto been largely neglected. We investigate the transversal gates that can be implemented in hypergraph product codes, a class of LDPC codes. Our analysis is aided by the construction of a symplectic canonical basis for the logical operators of hypergraph product codes, a result that may be of independent interest. We show that in these codes transversal gates can implement Hadamard (up to logical SWAP gates) and control-Z on all logical qubits. Moreover, we show that sequences of transversal operations, interleaved with error correction, allow implementation of entangling gates between arbitrary pairs of logical qubits in the same code block. We thereby demonstrate that transversal gates can be used as the basis for universal quantum computing on LDPC codes, when supplemented with state injection.

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• Transversal Diagonal Logical Operators for Stabiliser Codes
• Distance-preserving stabilizer measurements in hypergraph product codes
• CSS code surgery as a universal construction
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• Parallel Logical Measurements via Quantum Code Surgery
• Targeted Clifford logical gates for hypergraph product codes
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• Hierarchical Quantum Error Correction with Hypergraph Product Code and Rotated Surface Code
• Color code with a logical control-$S$ gate using transversal $T$ rotations
• Transversal architecture for megaquop-scale quantum simulation with neutral atoms