Conceptual aspects of geometric quantum computation

Sjöqvist, Erik; Mousolou, Vahid Azimi; Canali, Carlo M.

doi:10.1007/s11128-016-1381-1

Cited by 23 publications

(21 citation statements)

References 43 publications

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“…Due to their intrinsic tolerance to local fluctuations [9,10], geometric phases offer an attractive route for implementing high-fidelity quantum logic. This approach, termed holonomic quantum computation (HQC) [3,[11][12][13][14][15], employs the cyclic evolution of quantum states and derives its resilience from the global geometric structure of the evolution in Hilbert space. Arising both for adiabatic [16] and non-adiabatic [17] cycles, geometric phases can be either Abelian (phase shifts) or non-Abelian (matrix transformations) [18] by acting on a single state or a subspace of states, respectively.…”

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confidence: 99%

Holonomic Quantum Control by Coherent Optical Excitation in Diamond

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Although geometric phases in quantum evolution were historically overlooked, their active control now stimulates strategies for constructing robust quantum technologies. Here, we demonstrate arbitrary single-qubit holonomic gates from a single cycle of non-adiabatic evolution, eliminating the need to concatenate two separate cycles. Our method varies the amplitude, phase, and detuning of a two-tone optical field to control the non-Abelian geometric phase acquired by a nitrogen-vacancy center in diamond over a coherent excitation cycle. We demonstrate the enhanced robustness of detuned gates to excited-state decoherence and provide insights for optimizing fast holonomic control in dissipative quantum systems.Besides its central role in the understanding of contemporary physics [1,2], the quantum geometric phase is gaining recognition as a powerful resource for practical applications using quantum systems [3][4][5]. The manipulation of nanoscale systems has progressed rapidly towards realizing quantum-enhanced information processing and sensing, but also revealed the necessity for new methods to combat noise and decoherence [6][7][8]. Due to their intrinsic tolerance to local fluctuations [9,10], geometric phases offer an attractive route for implementing high-fidelity quantum logic. This approach, termed holonomic quantum computation (HQC) [3,[11][12][13][14][15], employs the cyclic evolution of quantum states and derives its resilience from the global geometric structure of the evolution in Hilbert space. Arising both for adiabatic [16] and non-adiabatic [17] cycles, geometric phases can be either Abelian (phase shifts) or non-Abelian (matrix transformations) [18] by acting on a single state or a subspace of states, respectively.Recently, non-Abelian, non-adiabatic holonomic gates using three-level dynamics [19] were proposed to match the computational universality of earlier adiabatic schemes [3,[11][12][13], but also eliminate the restriction of slow evolution. By reducing the run-time of holonomic gates, and thus their exposure to decoherence, this advance enabled experimental demonstration of HQC in superconducting qubits [20], nuclear spin ensembles in liquid [21], and nitrogen-vacancy (NV) centers in diamond [22,23]. However, these initial demonstrations were limited to fixed rotation angles about arbitrary axes, and thus required two non-adiabatic loops of evolution, from two iterations of experimental control, to execute an arbitrary gate [20][21][22][23]. Alternatively, variable angle rotations from a single non-adiabatic loop can be achieved using Abelian geometric phases [14,24] or hyperbolic secant pulses [25][26][27], but these approaches are complicated by a concomitant dynamic phase. To address these shortcomings, non-Abelian, non-adiabatic single-loop schemes * awsch@uchicago.edu [28,29] were designed to allow purely geometric, arbitrary angle rotations about arbitrary axes with a single experimental iteration.In this Letter, we realize single-loop, single-qubit holonomic gates by implementing th...

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Holonomic Quantum Control by Coherent Optical Excitation in Diamond

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“…We conclude this section with two remarks on the entangling nature of the two-qubit gate U (C 0 ) in Eq. (13). First, from the above analysis and illustrations in Figs.…”

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confidence: 98%

“…Namely, quantum gates based upon quantum holonomies have some built-in fault-tolerant features and stability, which can be employed to achieve robust quantum computation. Among geometric gates, nonadiabatic non-Abelian geometric gates may have some additional advantages, such as they are exact in a sense that there is no adiabatic approximation, there is no need for slow manipulation and in fact there is more freedom in the gates operation times [13]. Therefore, one may expect that nonadiabatic geometric gates can be made robust to wider class of noises compare to their adiabatic counterpart.…”

Section: Robustnessmentioning

confidence: 99%

“…Recently, this approach has been generalized [10] based on the nonadiabatic non-Abelian geometric phase [11]. The generalized approach allows us to combine the necessary components for realization of quantum processors, i.e., robustness, universality and speed [10,12,13]. It has been shown that the idea of nonadiabatic holonomic quantum computation can be incorporated with decoherence free subspaces [14][15][16][17][18][19], noiseless subsystems [20], and dynamical decoupling [21].…”

Section: Introductionmentioning

confidence: 99%

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Electric nonadiabatic geometric entangling gates on spin qubits

Mousolou¹

2017

Phys. Rev. A

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Producing and maintaining entanglement reside at the heart of the optimal construction of quantum operations and are fundamental issues in the realization of universal quantum computation. We here introduce a setup of spin qubits that allows for geometric implementation of entangling gates between the register qubits with any arbitrary entangling power. We show this by demonstrating a circuit through a spin chain, which performs universal nonadiabatic holonomic two-qubit entanglers. The proposed gates are all electric and geometric, which would help to realize fast and robust entangling gates on spin qubits. This family of entangling gates contains gates that are as efficient as the CNOT gate in quantum algorithms. We examine the robustness of the circuit to some extent.

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“…[2]). As shown in [10], the non-Abelian geometric gates of Sjöqvist et al can be interpreted as the Abelian geometric gates of Zhu and Wang.…”

Section: Geometric Quantum Computationmentioning

confidence: 99%

Observable-Geometric Phases and Quantum Computation

Chen

2020

Int J Theor Phys

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This paper presents an alternative approach to geometric phases from the observable point of view. Precisely, we introduce the notion of observable-geometric phases, which is defined as a sequence of phases associated with a complete set of eigenstates of the observable. The observablegeometric phases are shown to be connected with the quantum geometry of the observable space evolving according to the Heisenberg equation. It is shown that the observable-geometric phases can be used to realize a universal set of quantum gates in quantum computation. IntroductionThe geometric phase can be used in quantum computing by implementing each of quantum gates in terms of geometric phases only [12]. This scheme for computing is called geometric quantum computation. Geometric quantum computation involves adiabatic or nonadiabatic, as well as Abelian or non-Abelian characteristics of the underlying quantum evolution for the quantum state. It is shown in [14,15] that the non-adiabatic Abelian geometric phases can be used to realize a universal set of quantum gates and thus are sufficient for universal all-geometric quantum computation. Furthermore, the non-adiabatic non-Abelian geometric phases are shown in [11] to achieve universality as well. We refer to [10] for a up-date review and background.Here, we present an alternative approach to the geometric phase from the observable point of view. Precisely, we introduce the concept of observable-geometric phases, which is defined as a sequence of phases associated with a complete set of eigenstates of the observable. The observable-geometric phases are shown to be connected with the geometry of the observable space evolving according to the Heisenberg equation. In contrast to that classical (Riemannian) geometry of the state space is suited to the usual geometric phases, the geometry suited for the observable-geometric phases is a quantum geometric structure over the observable space induced by a complete measurement. This quantum-geometric structure of the observable space will be described in details in Section 5.The paper is organized as follows. In Section 2, we will define the notion of the observable-geometric phase using the propagator of a time-dependent Hamiltonian, which is shown to be independent of of the choice of the Hamiltonian as long as the Heisenberg equations involving the Hamiltonian. Two examples are included in which qubit systems subject to an orientated magnetic field, a rotating magnetic field, and a rotating plus a constant magnetic field. Subsequently, in Section 3, we show that the observable-geometric phases can be used to realize a universal set of quantum gates in quantum computation. The quantum gate based on the observable-geometric phases can be interpreted as the Abelian geometric gates of Zhu and Wang [14,15]; indeed the two schemes lead to the same gates. However, the ZW gates are related with the classical geometry of the state space of the system, while ours are connected with the quantum-geometric structure over the observable space. We wil...

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Conceptual aspects of geometric quantum computation

Cited by 23 publications

References 43 publications

Holonomic Quantum Control by Coherent Optical Excitation in Diamond

Holonomic Quantum Control by Coherent Optical Excitation in Diamond

Electric nonadiabatic geometric entangling gates on spin qubits

Observable-Geometric Phases and Quantum Computation

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