Low-Overhead Magic State Distillation with Color Codes

Fault-tolerant implementation of non-Clifford gates is a major challenge for achieving universal fault-tolerant quantum computing with quantum error-correcting codes. Magic state distillation is the most well-studied method for this but requires significant resources. Hence, it is crucial to tailor and optimize magic state distillation for specific codes from both logical- and physical-level perspectives. In this work, we perform such optimization for two-dimensional color codes, which are promising due to their higher encoding rates compared to surface codes, transversal implementation of Clifford gates, and efficient lattice surgery. We propose two carefully designed distillation schemes based on the 15-to-1 distillation circuit and lattice surgery, differing in their methods for handling faulty rotations. Our first scheme employs faulty <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>T</a:mi></a:math> measurement, achieving infidelities of <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>O</c:mi><c:mo stretchy="false">(</c:mo><c:msup><c:mi>p</c:mi><c:mn>3</c:mn></c:msup><c:mo stretchy="false">)</c:mo></c:math> for physical noise strength <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"><g:mi>p</g:mi></g:math>. To achieve lower infidelities, our second scheme integrates distillation with “cultivation” (a distillation-free approach to fault tolerantly prepare magic states through transversal Clifford measurements). Our second scheme achieves significantly lower infidelities (e.g., approximately <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"><i:mn>2</i:mn><i:mo>×</i:mo><i:msup><i:mn>10</i:mn><i:mrow><i:mo>−</i:mo><i:mn>16</i:mn></i:mrow></i:msup></i:math> at <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"><k:mi>p</k:mi><k:mo>=</k:mo><k:msup><k:mn>10</k:mn><k:mrow><k:mo>−</k:mo><k:mn>3</k:mn></k:mrow></k:msup></k:math>), surpassing the capabilities of both cultivation and single-level distillation. Notably, to reach a given target infidelity, our schemes require approximately 2 orders of magnitude fewer resources than the previous best magic-state-distillation schemes for color codes.