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Hamiltonian Simulation by Qubitization

Author(s)
Low, Guang Hao; Chuang, Isaac L.
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Creative Commons Attribution 4.0 International license https://creativecommons.org/licenses/by/4.0/
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Abstract
We present the problem of approximating the time-evolution operator e-iHt to error ϵ, where the Hamiltonian H = ((G|⊗I)U (|Gi⊗I) is the projection of a unitary oracle U onto the state |Gi created by another unitary oracle. Our algorithm solves this with a query complexity O ( t + log(1/ϵ) ) to both oracles that is optimal with respect to all parameters in both the asymptotic and non-asymptotic regime, and also with low overhead, using at most two additional ancilla qubits. This approach to Hamiltonian simulation subsumes important prior art considering Hamiltonians which are d-sparse or a linear combination of unitaries, leading to significant improvements in space and gate complexity, such as a quadratic speed-up for precision simulations. It also motivates useful new instances, such as where H is a density matrix. A key technical result is 'qubitization', which uses the controlled version of these oracles to embed any H in an invariant SU(2) subspace. A large class of operator functions of H can then be computed with optimal query complexity, of which e-iHt is a special case.
Date issued
2019-07
URI
https://hdl.handle.net/1721.1/129822
Department
Massachusetts Institute of Technology. Department of Physics; Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science; Massachusetts Institute of Technology. Research Laboratory of Electronics
Journal
Quantum
Publisher
Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften
Citation
Low, Guang Hao and Isaac L. Chuang. "Hamiltonian Simulation by Qubitization." Quantum 3 (July 2019): 163 © Verein zur Forderung des Open Access Publizierens in den Quantenwissenschaften
Version: Final published version
ISSN
2521-327X

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