Transversal Gates for Highly Asymmetric qLDPC Codes

TL;DR

This paper demonstrates the existence of transversal phase gates in highly asymmetric qLDPC codes, challenging prior beliefs and providing new constructions with potential for fault-tolerant quantum computing.

Contribution

It introduces novel constructions of qLDPC codes with transversal phase gates and analyzes their properties, including asymmetry and limitations of distance rebalancing techniques.

Findings

Existence of qLDPC codes with transversal phase gates and linearly growing logical qubits.

Distance against bit flip errors grows linearly, while phase flip error distance remains limited.

Rebalancing techniques only preserve transversality when increasing asymmetry.

Abstract

Transversal gates are the ideal gates in a fault-tolerant scenario; relatively easy to implement, and minimally error propagating. Their availability will maximise fault tolerant thresholds, enabling universal quantum computation in a wider range of noisy hardware. Transversal gates in quantum low density parity check (qLDPC) codes are largely unstudied, with the early results of Burton & Browne suggesting that transversal non-Clifford gates may be impossible. In this paper, we contradict this expectation with constructions for both hypergraph product codes and balanced product codes, although these first examples have weak properties. We find qLDPC codes with transversal phase gates that have a number of logical qubits that grows linearly with $n$, the number of physical qubits. The distance is highly asymmetric; while the distance against bit flip errors also grows (almost) linearly in $n$, the distance against phase flip errors is limited to $O(1)$. Moreover, existing distance rebalancing techniques are one-sided; we show that they only preserve the transversality when rebalancing to increase the asymmetry, not decrease it. We also collate a toolbox of techniques that identify a single transversal gate from which we can individuate a large range of different transversal gates. This is critical when addressing the question of what a transversal phase gate truly means when there are many logical qubits in the system.