Non-adiabatic holonomic quantum computation

Sjöqvist, Erik; Tong, D. M.; Andersson, Mauritz; Hessmo, Björn; Johansson, Markus; Singh, Kuldip

doi:10.1088/1367-2630/14/10/103035

Cited by 376 publications

(495 citation statements)

References 21 publications

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“…However, they are difficult to realize in experiment because of the long evolution time needed to fulfill the adiabatic condition. Instead, Sjöqvist et al 15 have proposed a scheme based on non-adiabatic non-abelian holonomies 14 which combines universality and speed and can thus be implemented in experiments.…”

mentioning

confidence: 99%

Experimental realization of non-Abelian non-adiabatic geometric gates

Abdumalikov

Fink

Júlíusson

et al. 2013

Nature

354

301

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The geometric aspects of quantum mechanics are underlined most prominently by the concept of geometric phases, which are acquired whenever a quantum system evolves along a closed path in Hilbert space. The geometric phase is determined only by the shape of this path [1][2][3][4] and is -in its simplest form -a real number. However, if the system contains degenerate energy levels, matrix-valued geometric phases, termed non-abelian holonomies, can emerge 5 . They play an important role for the creation of synthetic gauge fields in cold atomic gases 6 and the description of non-abelian anyon statistics 7 . Moreover, it has been proposed to exploit non-abelian holonomic gates for robust quantum computation [8][9][10] . In contrast to abelian geometric phases 11 , nonabelian ones have been observed only in nuclear quadrupole resonance experiments with a large number of spins and without fully characterizing the geometric process and its non-commutative nature 12,13 . Here, we realize non-abelian holonomic quantum operations 14,15 on a single superconducting artificial three-level atom 16 by applying a well controlled two-tone microwave drive. Using quantum process tomography, we determine fidelities of the resulting non-commuting gates exceeding 95%. We show that a sequence of two paths in Hilbert space traversed in different order yields inequivalent transformations, which is an evidence for the non-abelian character of the implemented holonomic quantum gates. In combination with two-qubit operations, they form a universal set of gates for holonomic quantum computation.A cyclic evolution of a non-degenerate quantum system is in general accompanied by a phase change of its wave function. The acquired abelian phase can be divided into two parts: The dynamical phase which is proportional to the evolution time and the energy of the system, and the geometric phase which depends only on the path of the system in Hilbert space. This characteristic feature leads to a resilience of the geometric phase to certain fluctuations during the evolution 17-19 , a property which has attracted particular attention in the field of quantum information processing 20 . However, universal quantum computation cannot be based on simple phase gates, which modify only the relative phase of a superposition state, unless they act on specific basis states 21 . Furthermore, geometric operations acting on degenerate subspaces have * abdumalikov@phys.ethz.ch † Now at Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125, USA been proposed for holonomic quantum computation fully based on geometric concepts 8 . In this scheme, quantum bits are encoded in a doubly degenerate eigenspace of the system hamiltonian h( λ). The parameters λ are varied to induce a cyclic evolution of the system. When the system returns back to its initial state, it can acquire not only a simple geometric phase factor, but also undergoes a path-dependent unitary transformation, a non-abelian holonomy, which causes a transition bet...

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mentioning

confidence: 99%

Experimental realization of non-Abelian non-adiabatic geometric gates

Abdumalikov

Fink

Júlíusson

et al. 2013

Nature

354

301

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“…See, for example, the Hamiltonians in Refs. [3,14]. Although both physical systems and controlling methods are different in these researches, the Hamiltonians can be uniformly written as H(t) = Ω jk e iφ jk | jk a| + Ω lm e iφ lm |lm a| + h.c.,…”

Section: B Two-qubit Gatementioning

confidence: 99%

Composite nonadiabatic holonomic quantum computation

Zhao

Xing

et al. 2017

Phys. Rev. A

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Nonadiabatic holonomic quantum computation has robust feature in suppressing control errors because of its holonomic feature. However, this kind of robust feature is challenged since the usual way of realizing nonadiabatic holonomic gates introduces errors due to systematic errors in the control parameters. To resolve this problem, we here propose a composite scheme to realize nonadiabatic holonomic gates. Our scheme can suppress systematic errors while preserving holonomic robustness. It is particularly useful when the evolution period is shorter than the coherence time. We further show that our composite scheme can be protected by decoherence-free subspaces. In this case, the strengthened robust feature of our composite gates and the coherence stabilization virtue of decoherence-free subspaces are combined.

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“…Although adiabatic GQC has geometric robustness, the evolution time associated with adiabatic requirement is usually longer than the coherence time and thus the practical computation is seriously collapsed. To overcome this problem, nonadiabatic GQC based on nonadiabatic and Abelian geometric phases [8] was proposed [9,10], and more interestingly nonadiabatic holonomic quantum computation, which is based on nonadiabatic and non-Abelian geometric phase [11], was recently found [12,13]. In realizing both adiabatic and nonadiabatic geometric gates, dynamical phases are usually removed and this kind of schemes is known as conventional GQC.…”

Section: Introductionmentioning

confidence: 99%

Nonadiabatic geometric quantum computation in decoherence-free subspaces based on unconventional geometric phases

Zhao

Tong

2016

Phys. Rev. A

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Nonadiabatic geometric quantum computation in decoherence-free subspaces has received increasing attention due to the merits of its high-speed implementation and robustness against both control errors and decoherence. However, all the previous schemes in this direction have been based on the conventional geometric phases, of which the dynamical phases need to be removed. In this paper, we put forward a scheme of nonadiabatic geometric quantum computation in decoherence-free subspaces based on unconventional geometric phases, of which the dynamical phases do not need to be removed. Specifically, by using three physical qubits undergoing collective dephasing to encode one logical qubit, we realize a universal set of geometric gates nonadiabatically and unconventionally. Our scheme not only maintains all the merits of nonadiabatic geometric quantum computation in decoherence-free subspaces, but also avoids the additional operations required in the conventional schemes to cancel the dynamical phases.

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Non-adiabatic holonomic quantum computation

Cited by 376 publications

References 21 publications

Experimental realization of non-Abelian non-adiabatic geometric gates

Experimental realization of non-Abelian non-adiabatic geometric gates

Composite nonadiabatic holonomic quantum computation

Nonadiabatic geometric quantum computation in decoherence-free subspaces based on unconventional geometric phases

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