Partitioning qubits in hypergraph product codes to implement logical gates

TL;DR

This paper explores the capabilities of transversal gates in hypergraph product LDPC codes, demonstrating their potential for universal fault-tolerant quantum computation with efficient logical gate implementation.

Contribution

It introduces a symplectic basis for logical operators in hypergraph product codes and shows how transversal gates can implement a universal set of logical operations.

Findings

Transversal gates can implement Hadamard and control-Z on all logical qubits.

Sequences of transversal gates enable entangling logical operations.

Transversal gates, combined with error correction, support universal quantum computing on LDPC codes.

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.