dorsal/arxiv
View SchemaThe geometry of quantum computation
| Authors | Mark R. Dowling, Michael A. Nielsen |
|---|---|
| Categories | |
| ArXiv ID | quant-ph/0701004 |
| URL | https://arxiv.org/abs/quant-ph/0701004 |
Abstract
Determining the quantum circuit complexity of a unitary operation is closely related to the problem of finding minimal length paths in a particular curved geometry [Nielsen et al, Science 311, 1133-1135 (2006)]. This paper investigates many of the basic geometric objects associated to this space, including the Levi-Civita connection, the geodesic equation, the curvature, and the Jacobi equation. We show that the optimal Hamiltonian evolution for synthesis of a desired unitary necessarily obeys a simple universal geodesic equation. As a consequence, once the initial value of the Hamiltonian is set, subsequent changes to the Hamiltonian are completely determined by the geodesic equation. We develop many analytic solutions to the geodesic equation, and a set of invariants that completely determine the geodesics. We investigate the problem of finding minimal geodesics through a desired unitary, U, and develop a procedure which allows us to deform the (known) geodesics of a simple and well understood metric to the geodesics of the metric of interest in quantum computation. This deformation procedure is illustrated using some three-qubit numerical examples. We study the computational complexity of evaluating distances on Riemmanian manifolds, and show that no efficient classical algorithm for this problem exists, subject to the assumption that good pseudorandom generators exist. Finally, we develop a canonical extension procedure for unitary operations which allows ancilla qubits to be incorporated into the geometric approach to quantum computing.
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"abstract": "Determining the quantum circuit complexity of a unitary operation is closely\nrelated to the problem of finding minimal length paths in a particular curved\ngeometry [Nielsen et al, Science 311, 1133-1135 (2006)]. This paper\ninvestigates many of the basic geometric objects associated to this space,\nincluding the Levi-Civita connection, the geodesic equation, the curvature, and\nthe Jacobi equation. We show that the optimal Hamiltonian evolution for\nsynthesis of a desired unitary necessarily obeys a simple universal geodesic\nequation. As a consequence, once the initial value of the Hamiltonian is set,\nsubsequent changes to the Hamiltonian are completely determined by the geodesic\nequation. We develop many analytic solutions to the geodesic equation, and a\nset of invariants that completely determine the geodesics. We investigate the\nproblem of finding minimal geodesics through a desired unitary, U, and develop\na procedure which allows us to deform the (known) geodesics of a simple and\nwell understood metric to the geodesics of the metric of interest in quantum\ncomputation. This deformation procedure is illustrated using some three-qubit\nnumerical examples. We study the computational complexity of evaluating\ndistances on Riemmanian manifolds, and show that no efficient classical\nalgorithm for this problem exists, subject to the assumption that good\npseudorandom generators exist. Finally, we develop a canonical extension\nprocedure for unitary operations which allows ancilla qubits to be incorporated\ninto the geometric approach to quantum computing.",
"arxiv_id": "quant-ph/0701004",
"authors": [
"Mark R. Dowling",
"Michael A. Nielsen"
],
"categories": [
"quant-ph"
],
"title": "The geometry of quantum computation",
"url": "https://arxiv.org/abs/quant-ph/0701004"
},
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