Cartan decomposition of SU(2n) and control of spin systems

Khaneja, Navin; Glaser, Steffen J.

doi:10.1016/s0301-0104(01)00318-4

Cited by 197 publications

(276 citation statements)

References 26 publications

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“…In the two-qubit demonstration of our variational methods, we have obtained a general solution for the optimal Hamiltonian without attempting to match it to a prescribed NMR experiment, which was a main concern in [1]. Our formalism also naturally allows for the treatment of the more general and physical situation in which one-qubit local controls require a nonzero time cost.…”

Section: Summary and Discussionmentioning

confidence: 99%

“…It also provides a physical ground to describe the complexity of quantum algorithms, whereas gate complexity should be regarded as a more abstract concept in which physics is implicit. Works relevant to the former subject can be found, e.g., in [1] and [2], which discuss the time optimal generation of unitary operations for a small number of qubits using a Cartan decomposition scheme and assuming that one-qubit operations can be performed arbitrarily fast. An adiabatic solution to the optimal control problem in holonomic quantum computation was given in [6], while Schulte-Herbrüggen et al [3] numerically obtained improved upper bounds on the time complexity of certain quantum gates.…”

Section: Introductionmentioning

confidence: 99%

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Time-optimal unitary operations

et al. 2007

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Extending our previous work on time optimal quantum state evolution [A. Carlini, A. Hosoya, T. Koike and Y. Okudaira, Phys. Rev. Lett. 96, 060503 (2006)], we formulate a variational principle for finding the time optimal realization of a target unitary operation, when the available Hamiltonians are subject to certain constraints dictated either by experimental or by theoretical conditions. Since the time optimal unitary evolutions do not depend on the input quantum state this is of more direct relevance to quantum computation. We explicitly illustrate our method by considering the case of a two-qubit system self-interacting via an anisotropic Heisenberg Hamiltonian and by deriving the time optimal unitary evolution for three examples of target quantum gates, namely the swap of qubits, the quantum Fourier transform and the entangler gate. We also briefly discuss the case in which certain unitary operations take negligible time.

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Section: Summary and Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Time-optimal unitary operations

et al. 2007

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“…e.g. [38,32,34,42]). The task of computing the Cartan factors for a specific unitary matrix is greatly facilitated by the existence of Cartan involutions.…”

Section: Theorem 1: For Any Two Maximal Abelian Subalgebras H and Hmentioning

confidence: 99%

“…We must now define a Cartan subalgebra h contained in m. Here Nakajima et al make a different choice of h to that used by Khaneja and Glaser in [38] and [39]. Recall that all maximal Abelian subalgebras share an adjoint orbit, namely m itself, and that one may, as a result, switch between them with relative ease.…”

Section: The Qsd From Cartan Involutionsmentioning

confidence: 99%

Constructive quantum Shannon decomposition from Cartan involutions

Drury

Love

2008

J. Phys. A: Math. Theor.

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Abstract. The work presented here extends upon the best known universal quantum circuit, the Quantum Shannon Decomposition proposed in [Vivek V. Shende, Stephen S. Bullock and Igor Markov, Synthesis of Quantum Logic Circuits, IEEE : 1000-1010 (2006)]. We obtain the basis of the circuit's design in a pair of Cartan decompositions. This insight gives a simple constructive algorithm for obtaining the Quantum Shannon Decomposition of a given unitary matrix in terms of the corresponding Cartan involutions.

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“…For instance, Alvarez and Gómez [1] showed that the quantum state in Grover's algorithm [2], known as the optimal quantum search algorithm [3], actually follows a geodesic curve derived from the Fubini-Study metric in the projective space. Khaneja et al [4] and Zhang et al [5], using a Cartan decomposition scheme for unitary operations, discussed the time optimal way to realize a two-qubit universal unitary gate under the condition that one-qubit operations can be performed in an arbitrarily short time. On the other hand, Tanimura et al [6] gave an adiabatic solution to the optimal control problem in holonomic quantum computation, in which a desired unitary gate is generated as the holonomy corresponding to the minimal length loop in the space of control parameters for the Hamiltonian.…”

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confidence: 99%

Time-Optimal Quantum Evolution

et al. 2006

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We present a general framework for finding the time-optimal evolution and the optimal Hamiltonian for a quantum system with a given set of initial and final states. Our formulation is based on the variational principle and is analogous to that for the brachistochrone in classical mechanics. We reduce the problem to a formal equation for the Hamiltonian which depends on certain constraint functions specifying the range of available Hamiltonians. For some simple examples of the constraints, we explicitly find the optimal solutions.PACS numbers: 03.67.Lx, 03.65.Ca, 02.30.Xx, 02.30.Yy In quantum mechanics one can change a given state to another by applying a suitable Hamiltonian on the system. In many situations, e.g. quantum computation, it is desirable to know the pathway in the shortest time.In this paper we consider the problem of finding the time-optimal path for the evolution of a pure quantum state and the optimal driving Hamiltonian. Recently, a growing number of works related to this topic have appeared. For instance, Alvarez and Gómez [1] showed that the quantum state in Grover's algorithm [2], known as the optimal quantum search algorithm [3], actually follows a geodesic curve derived from the Fubini-Study metric in the projective space. Khaneja et al. [4] and Zhang et al.[5], using a Cartan decomposition scheme for unitary operations, discussed the time optimal way to realize a two-qubit universal unitary gate under the condition that one-qubit operations can be performed in an arbitrarily short time. On the other hand, Tanimura et al. [6] gave an adiabatic solution to the optimal control problem in holonomic quantum computation, in which a desired unitary gate is generated as the holonomy corresponding to the minimal length loop in the space of control parameters for the Hamiltonian. Schulte-Herbrüggen et al.[7] exploited the differential geometry of the projective unitary group to give the tightest known upper bounds on the actual time complexity of some basic modules of quantum algorithms. More recently, Nielsen [8] introduced a lower bound on the size of the quantum circuit necessary to realize a given unitary operator based on the geodesic distance, with a suitable metric, between the unitary and the identity operators. However, a general method for generating the time optimal Hamiltonian evolution of quantum states was not known until now.Here we are going to study this problem by exploiting the analogy with the so-called brachistochrone problem in classical mechanics and the elementary properties * Electronic address: carlini@th.phys.titech.ac.jp † Electronic address: ahosoya@th.phys.titech.ac.jp ‡ Electronic address: koike@phys.keio.ac.jp § Electronic address: okudaira@th.phys.titech.ac.jp of quantum mechanics. In ordinary quantum mechanics the initial state and the Hamiltonian of a physical system are given and one has to find the final state using the Schrödinger equation. In our work we generalize this framework so as to optimize a certain cost functional with respect to the Hamiltonian a...

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Cartan decomposition of SU(2n) and control of spin systems

Cited by 197 publications

References 26 publications

Time-optimal unitary operations

Time-optimal unitary operations

Constructive quantum Shannon decomposition from Cartan involutions

Time-Optimal Quantum Evolution

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