Partitioning qubits in hypergraph product codes to implement logical gates

Armanda O. Quintavalle1,2, Paul Webster3, and Michael Vasmer4,5

1Department of Physics & Astronomy, University of Sheffield, Sheffield, S3 7RH, United Kingdom
2Dahlem Center for Complex Quantum Systems, Freie Universität Berlin, 14195 Berlin, Germany
3Centre for Engineered Quantum Systems, School of Physics, The University of Sydney, Sydney, NSW 2006, Australia
4Perimeter Institute for Theoretical Physics, Waterloo, ON N2L 2Y5, Canada
5Institute for Quantum Computing, University of Waterloo, Waterloo, ON N2L 3G1, Canada

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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.

Error correction codes would be of no use without a method for dynamically manipulating the information they store. While the literature offers several techniques for performing gates on codes with just one logical qubit, there are far fewer solutions available for codes that encode multiple logical qubits.
In this paper, we introduce an approach to implement logical encoded gates within hypergraph product codes, even for codes with multiple logical qubits. Our method extends the concept of transversal gates and relies on partitioning the physical qubits within the code in alignment with its logical structure. After demonstrating the fault tolerance of our method, we showcase its application in realizing certain Clifford gates for hypergraph product codes that adhere to specific symmetry constraints.

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