Spatial overhead reduction for 2D hypergraph product codes

TL;DR

This paper presents methods to reduce the physical qubit overhead of 2D hypergraph product quantum codes while preserving key properties, enabling more efficient fault-tolerant quantum computation.

Contribution

It introduces techniques for qubit reduction in hypergraph product codes that maintain code parameters and fault-tolerance features, with practical examples and simulation results.

Findings

Reduced codes maintain code dimension and minimum distance.

Simulation shows reduced codes have similar error thresholds with fewer qubits.

Overhead reduction is compatible with logical gates and measurement gadgets.

Abstract

The hypergraph product creates a quantum stabilizer code from two input classical linear codes; a paradigmatic example being the surface code as a hypergraph product of two classical repetition codes. Many properties of the hypergraph product code can be inherited from those of the classical codes such as the code dimension, minimum distance and certain fault-tolerant gadgets. We investigate ways to reduce the number of physical qubits in hypergraph product codes while maintaining some of their useful properties for fault tolerance. We show that the code dimension, canonical logical basis, and minimum distances of the hypergraph product code are preserved through this reduction. We also provide distance-preserving syndrome measurement schedules as well as examples of reduced hypergraph product codes with parameter improvements such as $[\![610,64,6]\!] \rightarrow [\![441,64,6]\!]$ and $[\![1225,49,11]\!] \rightarrow [\![931,49,11]\!]$. In memory simulations with circuit-level depolarizing noise, we observe that the reduced codes can have similar subthreshold performance as their unreduced versions, but using fewer physical qubits. Finally, we show how overhead reduction can be compatible with homomorphic measurement gadgets, fold-transversal gates and automorphisms, which extends the savings to logical computation.