Constant-Overhead Magic State Distillation
Magic state distillation is a crucial yet resource-intensive process in fault-tolerant quantum com- putation. The protocol’s overhead, defined as the number of input magic states required per output magic state with an error rate below ε, typically grows as O(logγ (1/ε)) as ε → 0. Achieving smaller overheads, i.e., smaller exponents γ, is highly desirable; however, all existing protocols require poly- logarithmically growing overheads with some γ > 0, and identifying the smallest achievable exponent γ for distilling magic states of qubits has remained challenging. To address this issue, we develop magic state distillation protocols for qubits with efficient, polynomial-time decoding that achieve an O(1) overhead, meaning the optimal exponent γ = 0; this improves over the previous best of γ ≈ 0.678 due to Hastings and Haah. In our construction, we employ algebraic geometry codes to explicitly present asymptotically good quantum codes for 210-dimensional qudits that support transversally implementable logical gates in the third level of the Clifford hierarchy. The use of asymptotically good codes with non-vanishing rate and relative distance leads to the constant over- head. These codes can be realised by representing each 210-dimensional qudit as a set of 10 qubits, using stabiliser operations on qubits. The 10-qubit magic states distilled with these codes can be converted to and from conventional magic states for the controlled-controlled-Z (CCZ) and T gates on qubits with only a constant overhead loss, making it possible to achieve constant-overhead distil- lation of such standard magic states for qubits. These results resolve the fundamental open problem in quantum information theory concerning the construction of magic state distillation protocols with the optimal exponent.
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