Geometric analysis of minimum-time trajectories for a two-level quantum system

Romano, Raffaele

doi:10.1103/physreva.90.062302

Cited by 17 publications

(50 citation statements)

References 15 publications

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“…where s j , with = j x y z , , , are the Pauli matrices. We remark that the reachable set of a single qubit subject to two independent control fields was recently analyzed in great detail [42,43] can be calculated exactly [30] using the Euler angle decomposition above. This calculation shows that the minimum time is determined by Ω.…”

Section: Single Qubit Casementioning

confidence: 99%

The roles of drift and control field constraints upon quantum control speed limits

et al. 2017

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In this work we derive a lower bound for the minimum time required to implement a target unitary transformation through a classical time-dependent field in a closed quantum system. The bound depends on the target gate, the strength of the internal Hamiltonian and the highest permitted control field amplitude. These findings reveal some properties of the reachable set of operations, explicitly analyzed for a single qubit. Moreover, for fully controllable systems, we identify a lower bound for the time at which all unitary gates become reachable. We use numerical gate optimization in order to study the tightness of the obtained bounds. It is shown that in the single qubit case our analytical findings describe the relationship between the highest control field amplitude and the minimum evolution time remarkably well. Finally, we discuss both challenges and ways forward for obtaining tighter bounds for higher dimensional systems, offering a discussion about the mathematical form and the physical meaning of the bound.

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Section: Single Qubit Casementioning

confidence: 99%

The roles of drift and control field constraints upon quantum control speed limits

et al. 2017

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“…This is indeed the case since the Lie algebra generated by the Hamiltonians C z , B x , B y and B z in (12) is, in the considered representation, the Lie algebra of block diagonal 4 × 4 matrices with arbitrary 2 × 2 blocks in su (2). The set of reachable operators is the associated Lie group of block diagonal 4 × 4 matrices with 2 × 2 blocks in SU (2).…”

Section: The Lie Algebraic Structure Of the Problemmentioning

confidence: 99%

“…This group is compact, semisimple, and isomorphic to SU(2) ⊕ SU(2). The associated Lie algebra, isomorphic to su(2) ⊕ su (2), is given by l = b ⊕ c, where…”

Section: The Lie Algebraic Structure Of the Problemmentioning

confidence: 99%

“…The second term in (2) represents an Ising interaction between the two systems; we assume ω 0 = 0, otherwise the problem reduces to that of controlling an isolated two-level system with bounded control, which has been solved in previous work [1,2]. Without loss of generality we can take ω 0 > 0, as we shall shortly see.…”

Section: Introduction a Statement Of The Problemmentioning

confidence: 99%

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Minimum time control of a pair of two-level quantum systems with opposite drifts

Romano¹,

D’Alessandro²

2016

J. Phys. A: Math. Theor.

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In this paper we solve two equivalent time optimal control problems. On one hand, we design the control field to implement in minimum time the SWAP (or equivalent) operator on a two-level system, assuming that it interacts with an additional, uncontrollable, two-level system. On the other hand, we synthesize the SWAP operator simultaneously, in minimum time, on a pair of twolevel systems subject to opposite drifts. We assume that it is possible to perform three independent control actions, and that the total control strength is bounded. These controls either affect the dynamics of the target system, under the first perspective, or, simultaneously, the dynamics of both systems, in the second view. We obtain our results by using techniques of geometric control theory on Lie groups. In particular, we apply the Pontryagin Maximum Principle, and provide a complete characterization of singular and non-singular extremals. Our analysis shows that the problem can be formulated as the motion of a material point in a central force, a well known system in classical mechanics. Although we focus on obtaining the SWAP operator, many of the ideas and techniques developed in this work apply to the time optimal implementation of an arbitrary unitary operator.

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“…The resonance phenomenon happens to play a key role in problems where one seeks to drive an initial source state towards a final target state. For instance, typical problems of this kind can be found when controlling population transfer in quantum systems [3][4][5] or when searching for a target state [6,7]. In the framework of quantum control theory, it is commonly believed that an off-resonant driving scheme cannot help achieving population transfer with high fidelity between two quantum states.…”

Section: Introductionmentioning

confidence: 99%

Information Geometric Perspective on Off-Resonance Effects in Driven Two-Level Quantum Systems

2020

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We present an information geometric analysis of off-resonance effects on classes of exactly solvable generalized semi-classical Rabi systems. Specifically, we consider population transfer performed by four distinct off-resonant driving schemes specified by su (2; C) time-dependent Hamiltonian models. For each scheme, we study the consequences of a departure from the on-resonance condition in terms of both geodesic paths and geodesic speeds on the corresponding manifold of transition probability vectors. In particular, we analyze the robustness of each driving scheme against off-resonance effects. Moreover, we report on a possible tradeoff between speed and robustness in the driving schemes being investigated. Finally, we discuss the emergence of a different relative ranking in terms of performance among the various driving schemes when transitioning from on-resonant to off-resonant scenarios.

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Geometric analysis of minimum-time trajectories for a two-level quantum system

Cited by 17 publications

References 15 publications

The roles of drift and control field constraints upon quantum control speed limits

The roles of drift and control field constraints upon quantum control speed limits

Minimum time control of a pair of two-level quantum systems with opposite drifts

Information Geometric Perspective on Off-Resonance Effects in Driven Two-Level Quantum Systems

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