A unified framework for magic state distillation and multi-qubit gate-synthesis with reduced resource cost

TL;DR

This paper introduces a unified 'synthillation' framework that combines magic state distillation and multiqubit gate synthesis, significantly reducing resource costs in fault-tolerant quantum computing.

Contribution

It presents a novel approach that merges distillation and synthesis into one step, achieving quadratic error suppression and eliminating the need for separate distillation rounds.

Findings

Synthillation reduces resource overheads for quantum circuits.

The framework achieves quadratic error suppression with the same T-state count as gate synthesis.

Efficient algorithms for multiqubit unitary synthesis, including optimal controlled-unitary synthesis.

Abstract

The standard approach to fault-tolerant quantum computation is to store information in a quantum error correction code, such as the surface code, and process information using a strategy that can be summarized as distill-then-synthesize. In the distill step, one performs several rounds of distillation to create high-fidelity logical qubits in a magic state. Each such magic state provides one good T gate. In the synthesize step, one seeks the optimal decomposition of an algorithm into a sequence of many T gates interleaved with Clifford gates. This gate-synthesis problem is well understood for multiqubit gates that do not use any Hadamards. We present an in-depth analysis of a unified framework that realises one round of distillation and multiqubit gate synthesis in a single step. We call these synthillation protocols, and show they lead to a large reduction in resource overheads. This is because synthillation can implement a general class of circuits using the same number of T-states as gate synthesis, yet with the benefit of quadratic error suppression. This general class includes all circuits primarily dominated by control-control-Z gates, such as adders and modular exponentiation routines used in Shor's algorithm. Therefore, synthillation removes the need for a costly round of magic state distillation. We also present several additional results on the multiqubit gate-synthesis problem. We provide an efficient algorithm for synthesizing unitaries with the same worst-case resource scaling as optimal solutions. For the special case of synthesizing controlled-unitaries, our techniques are not just efficient but exactly optimal. We observe that the gate-synthesis cost, measured by T-count, is often strictly subadditive. Numerous explicit applications of our techniques are also presented.