Exact solutions of the isoholonomic problem and the optimal control problem in holonomic quantum computation

Tanimura, Shinji; Nakahara, Mikio; Hayashi, Daisuke

doi:10.1063/1.1835545

Cited by 26 publications

(32 citation statements)

References 23 publications

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“…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. The present authors [7] discussed the quantum brachistochrone for state evolution, i.e.…”

Section: Introductionmentioning

confidence: 99%

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: Introductionmentioning

confidence: 99%

Time-optimal unitary operations

et al. 2007

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“…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…”

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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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“…The resultant geometric quantum gate is Eq. (19). The values of θ(= χ 2 − χ 1 ) and Θ(= (χ 2 + χ 1 )/2) are given in Table I.…”

Section: E Implementation In Liquid-state Nmrmentioning

confidence: 99%

“…It is necessary to employ different processes between the loops 1 and 2 to prevent the cancellation of the geometric phase associated with the two loops. The matrix representation (19) implies that U echo contains three parameters Γ, θ, and Θ. Due to the limitation in the control parameters, it may be difficult to choose them independently in a standard liquid-state NMR.…”

Section: Cancellation Of Dynamical Phasesmentioning

confidence: 99%

“…A promising way to achieve this is to employ geometric phases (or, more generally, nonAbelian holonomies) [1,2], because geometric phases are expected to be robust against noise and decoherence under a proper condition [3,4]. A large number of studies for applying their potential robustness to quantum computing have been done, e,g., phase-shift gates with Berry phases [5], nonadiabatic geometric quantum gates [6,7,8,9,10,11,12,13], holonomic quantum computing [14,15,16,17,18,19,20,21], quantum gates with noncyclic geometric phases [22], and so on.…”

Section: Introductionmentioning

confidence: 99%

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Geometric quantum gates in liquid-state NMR based on a cancellation of dynamical phases

et al. 2009

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A proposal for applying nonadiabatic geometric phases to quantum computing, called double-loop method [S.-L. Zhu and Z. D. Wang, Phys. Rev. A 67, 022319 (2003)], is demonstrated in a liquid state nuclear magnetic resonance quantum computer. Using a spin-echo technique, the original method is modified so that quantum gates are implemented in a standard high-precision nuclear magnetic resonance system for chemical analysis. We show that a dynamical phase is successfully eliminated and a one-qubit quantum gate is realized, although the gate fidelity is not high.

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Exact solutions of the isoholonomic problem and the optimal control problem in holonomic quantum computation

Cited by 26 publications

References 23 publications

Time-optimal unitary operations

Time-optimal unitary operations

Time-Optimal Quantum Evolution

Geometric quantum gates in liquid-state NMR based on a cancellation of dynamical phases

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