paper

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

arXiv:1606.01904 · doi:10.1103/PhysRevA.95.022316

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.

Authors' final copy (Accepted to Phys Rev A). 23 pages. This is one of a pair of companion papers. The more concise 5 page paper is entitled "Unifying gate-synthesis and magic state distillation". Fixed typo in discussion of Lemma 3. Thanks to Anqi Gong for spotting some (postpublication) errors in case 6 of Table I and Example IV.3, both of which are fixed in this version

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