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Center for Nonlinear Studies |
Fault-tolerant quantum computers will compute by applying a sequence of elementary unitary operations, or gates, to an error-protected subspace. While algorithms are typically expressed over arbitrary local gates, there is unfortunately no known theory that can correct errors for a continuous set of quantum gates. However, theory does support the fault-tolerant construction of various finite gate sets, which, in some cases, generate circuits that can approximate arbitrary gates to any desired precision. In this talk, I will present joint work with Kliuchnikov, Bocharov and Roetteler on a framework for approximating arbitrary qubit unitaries over a very general but natural class of gate sets. These gate sets are derived from the theory of integral quaternions over number fields and generate S-arithmetic subgroups of SU(2). In this framework, the complexity of a unitary is algebraically encoded in the length of a corresponding quaternion. The algorithm achieves epsilon-approximations with circuits of length O(log(1/epsilon)), which is optimal up to constant factors.
Host: Rolando Somma