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Fault-tolerant holonomic computation on quantum error-correcting codes

Ognyan Oreshkov, University of Southern California

(Session 9 : Saturday from 16:15-16:45)

Collaborators: Ognyan Oreshkov, Todd Brun, Daniel Lidar, and Paolo Zanardi

Abstract. Holonomic quantum computation is a method of computation that uses non-abelian generalizations of the Berry phase. Due to its geometric nature, this approach is robust against various types of errors in the control parameters driving the evolution. In this study, we propose a scheme for fault-tolerant holonomic computation on stabilizer codes, which combines the virtues of error correction with those of the geometric approach. The scheme implements single-qubit operations on different qubits in the code by adiabatically varying Hamiltonians that are elements of the stabilizer, or in the case of subsystem codes---operators that act on the noisy subsystem. Two-qubit operations between qubits from different blocks require Hamiltonians whose weights are higher by one. Thus for certain codes, like the 9-qubit Shor code or its subsystem versions, it is possible to realize universal fault-tolerant computation using Hamiltonians of weight two and three, which is the optimal Hamiltonian weight for holonomic computation on a system of qubits. We also study the regime in which the adiabaticity condition becomes compatible with the fault-tolerance condition for fast gates on the time scale of the noise. Both conditions can be satisfied for a sufficiently large Hamiltonian strength, or equivalently, for a sufficiently low noise rate. This requires only a constant overhead of resources compared to those needed for fault-tolerant dynamical computation.