Magic state distillation with low overhead

TL;DR

This paper introduces a new family of error-detecting stabilizer codes enabling low-overhead magic state distillation with poly-logarithmic scaling, significantly improving efficiency over previous protocols.

Contribution

The authors develop triorthogonal matrices to construct stabilizer codes with transversal T-gates, reducing distillation overhead and introducing a numerical method for generating these matrices.

Findings

Achieved two-fold reduction in distillation overhead for high-precision magic states.

Developed a polynomial-time method for generating triorthogonal matrices.

Demonstrated scalable distillation protocols with poly-logarithmic overhead.

Abstract

We propose a new family of error detecting stabilizer codes with an encoding rate 1/3 that permit a transversal implementation of the pi/8-rotation $T$ on all logical qubits. The new codes are used to construct protocols for distilling high-quality `magic' states $T|+>$ by Clifford group gates and Pauli measurements. The distillation overhead has a poly-logarithmic scaling as a function of the output accuracy, where the degree of the polynomial is $\log_2{3}\approx 1.6$. To construct the desired family of codes, we introduce the notion of a triorthogonal matrix --- a binary matrix in which any pair and any triple of rows have even overlap. Any triorthogonal matrix gives rise to a stabilizer code with a transversal $T$-gate on all logical qubits, possibly augmented by Clifford gates. A powerful numerical method for generating triorthogonal matrices is proposed. Our techniques lead to a two-fold overhead reduction for distilling magic states with output accuracy $10^{-12}$ compared with the best previously known protocol.