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Universal quantum computation and quantum error correction using discrete holonomies

Holonomic quantum computation exploits a quantum state's non-trivial, matrix-valued geometric phase (holonomy) to perform fault-tolerant computation. Holonomies arising from systems where the Hamiltonian traces a continuous path through parameter space have been well-researched. Discrete holonomies, on the other hand, where the state jumps from point to point in state space, have had little prior investigation. Using a sequence of incomplete projective measurements of the spin operator, we build an explicit approach to universal quantum computation. We show that quantum error correction codes integrate naturally in our scheme, providing a model for measurement-based quantum computation that combines the passive error resilience of holonomic quantum computation and active error correction techniques. In the limit of dense measurements we recover known continuous-path holonomies.