Optimal control, geometry, and quantum computing

Abstract

We prove upper and lower bounds relating the quantum gate complexity of a unitary operation, U, to the optimal control cost associated to the synthesis of U. These bounds apply for any optimal control problem, and can be used to show that the quantum gate complexity is essentially equivalent to the optimal control cost for a wide range of problems, including time-optimal control and finding minimal distances on certain Riemannian, subriemannian, and Finslerian manifolds. These results generalize the results of Nielsen, Dowling, Gu, and Doherty, Science 311, 1133-1135 (2006), which showed that the gate complexity can be related to distances on a Riemannian manifold

7 Pages Added Full Names to Authors

References in corpus (6)

• Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
• Time Optimal Control in Spin Systems
• Quantum Computation as Geometry
• Universality in Quantum Computation
• Quantum Brachistochrone
• Optimal Control-Based Efficient Synthesis of Building Blocks of Quantum Algorithms Seen in Perspective from Network Complexity towards Time Complexity

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• Gate complexity using Dynamic Programming
• Geometry of a two-spin quantum state in evolution
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• Probing the geometry of two-qubit state space by evolution
• Quantum simulation of time-dependent Hamiltonians via commutator-free quasi-Magnus operators
• Numerical Solution of the Dynamic Programming Equation for the Optimal Control of Quantum Spin Systems
• Coherence and Imaginarity as Resources in Quantum Circuit Complexity
• Complexity in One- and Two-Qubit Systems
• Wasserstein Complexity of Quantum Circuits
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• Geometry of quantum state manifolds generated by the Lie algebra operators
• Monte Carlo approach for finding optimally controlled quantum gates with differential geometry
• Resource-Dependent Complexity of Quantum Channels
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• Krylov Spread Complexity of Quantum-Walks
• Probing mean values and correlations of high-spin systems on a quantum computer
• Geometry of quantum complexity
• Geometric methods in quantum information and entanglement variational principle
• Toward a physically motivated notion of Gaussian complexity geometry
• Circuit Complexity for Coherent-Thermal States in Bosonic String Theory
• Measuring distance between quantum states on a quantum computer
• An explicit family of unitaries with exponentially minimal length Pauli geodesics
• Holographic Interpretation of Relative State Complexity
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• Geometric quantum control and the random Schrödinger equation
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