Efficient Magic State Distillation by Zero-Level Distillation

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.