Efficient Magic State Distillation by Zero-Level Distillation

Abstract

Magic state distillation (MSD) is an essential element for universal fault-tolerant quantum computing, which distills a high-fidelity magic state from noisy magic states using ideal (error-corrected) Clifford operations. For ideal Clifford operations, it needs to be performed on the logical qubits and hence incurs a large spatiotemporal overhead, which is one of the major bottlenecks for the realization of fault-tolerant quantum computers (FTQCs). Here we propose zero-level distillation, which prepares a high-fidelity logical magic state at the physical level, namely , using physical qubits and nearest-neighbor two-qubit gates on a square lattice. We develop a zero-level distillation circuit and show that distillation can be made even more efficient than the conventional sophisticated approaches with logical level distillations. The key idea involves the Knill -type distillation using the Steane code and its careful mapping to the square-lattice architecture with error detection. The distilled magic state on the Steane-code state is then teleported or converted to surface codes. We numerically find that the error rate of the logical magic state scales as approximately <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mn>100</a:mn> <a:mo>×</a:mo> <a:msup> <a:mi>p</a:mi> <a:mn>2</a:mn> </a:msup> </a:math> in terms of the physical error rate <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mi>p</c:mi> </c:math> . For example, with a physical error rate of <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>p</e:mi> <e:mo>=</e:mo> <e:msup> <e:mn>10</e:mn> <e:mrow> <e:mo>−</e:mo> <e:mn>4</e:mn> </e:mrow> </e:msup> </e:math> ( <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:msup> <g:mn>10</g:mn> <g:mrow> <g:mo>−</g:mo> <g:mn>3</g:mn> </g:mrow> </g:msup> </g:math> ), the logical error rate is reduced to <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:msub> <i:mi>p</i:mi> <i:mi>L</i:mi> </i:msub> <i:mo>=</i:mo> <i:msup> <i:mn>10</i:mn> <i:mrow> <i:mo>−</i:mo> <i:mn>6</i:mn> </i:mrow> </i:msup> </i:math> ( <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:msup> <k:mn>10</k:mn> <k:mrow> <k:mo>−</k:mo> <k:mn>4</k:mn> </k:mrow> </k:msup> </k:math> ), resulting in an improvement of 2 (1) orders of magnitude. This contributes to reducing both space and time overhead for early FTQC as well as full-fledged FTQC combined with conventional multilevel distillation protocols.

Topics & Concepts