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arXiv:1412.2848 · Scientific Reports

Fast non-Abelian geometric gates via transitionless quantum driving

Jiang Zhang, Thi Ha Kyaw, D. M. Tong, Erik Sjöqvist, L. C. Kwek

Published Dec 21, 2015• 103 citations
arXivHTMLPDF

Abstract

A practical quantum computer must be capable of performing high fidelity quantum gates on a set of quantum bits (qubits). In the presence of noise, the realization of such gates poses daunting challenges. Geometric phases, which possess intrinsic noise-tolerant features, hold the promise for performing robust quantum computation. In particular, quantum holonomies, i.e., non-Abelian geometric phases, naturally lead to universal quantum computation due to their non-commutativity. Although quantum gates based on adiabatic holonomies have already been proposed, the slow evolution eventually compromises qubit coherence and computational power. Here, we propose a general approach to speed up an implementation of adiabatic holonomic gates by using transitionless driving techniques and show how such a universal set of fast geometric quantum gates in a superconducting circuit architecture can be obtained in an all-geometric approach. Compared with standard non-adiabatic holonomic quantum computation, the holonomies obtained in our approach tends asymptotically to those of the adiabatic approach in the long run-time limit and thus might open up a new horizon for realizing a practical quantum computer.

DOI
10.1038/srep18414

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A practical quantum computer must be capable of performing high fidelity quantum gates on a set of quantum bits (qubits). In the presence of noise, the realization of such gates poses daunting challenges. Geometric phases, which possess intrinsic noise-tolerant features, hold the promise for performing robust quantum computation. In particular, quantum holonomies, i.e., non-Abelian geometric phases, naturally lead to universal quantum computation due to their non-commutativity. Although quantum gates based on adiabatic holonomies have already been proposed, the slow evolution eventually compromises qubit coherence and computational power. Here, we propose a general approach to speed up an implementation of adiabatic holonomic gates by using transitionless driving techniques and show how such a universal set of fast geometric quantum gates in a superconducting circuit architecture can be obtained in an all-geometric approach. Compared with standard non-adiabatic holonomic quantum computation, the holonomies obtained in our approach tends asymptotically to those of the adiabatic approach in the long run-time limit and thus might open up a new horizon for realizing a practical quantum computer.
Geometric phases and holonomies are a promising resource for the realization of high-fidelity quantum operations in noisy devices, due to their intrinsic fault-tolerance against parametric noise. However, for a long time their practical use in quantum computing was limited to proof of principle demonstrations. This was partly due to the need for adiabatic time evolution or the requirement of complex, high-dimensional state spaces and a large number of driving field parameters to achieve universal quantum gates employing holonomies. In 2016 Liang et al. proposed universal, superadiabatic, geometric quantum gates exploiting transitionless quantum driving, thereby offering fast and universal quantum gate performance on a simple two-level system. Here, we report on the experimental implementation of a set of non-commuting single-qubit superadiabatic, geometric quantum gates on the electron spin of the nitrogen-vacancy center in diamond under ambient conditions. This provides a promising and powerful tool for large-scale quantum computing under realistic, noisy experimental conditions.
Abstract Geometric phases and holonomies are a promising resource for the realization of high-fidelity quantum operations in noisy devices, due to their intrinsic fault-tolerance against parametric noise. However, for a long time their practical use in quantum computing was limited to proof of principle demonstrations. This was partly due to the need for adiabatic time evolution or the requirement of complex, high-dimensional state spaces and a large number of driving field parameters to achieve universal quantum gates employing holonomies. In 2016 Liang et al. proposed universal, superadiabatic, geometric quantum gates exploiting transitionless quantum driving, thereby offering fast and universal quantum gate performance on a simple two-level system. Here, we report on the experimental implementation of a set of non-commuting single-qubit superadiabatic, geometric quantum gates on the electron spin of the nitrogen-vacancy center in diamond under ambient conditions. This provides a promising and powerful tool for large-scale quantum computing under realistic, noisy experimental conditions.
Geometric quantum phases enable the construction geometric quantum gates, whose action only depend on the evolution path in state space. Geometric gates may offer increased robustness to control parameter noise. In this report, the conditions under which non-adiabatic geometric and holonomic quantum gates preserve their geometric character under sequential composition are analysed. Grover’s search algorithm requiring non-commuting operations is used as a representative example. The basic concepts of qubits, quantum gates and quantum circuits are explained. The concept of geometric phases is introduced and derivations of Abelian and non-Abelian non-adiabatic geometric phases are provided (Aharonov-Anandan and Anandan phase respectively). Construction of geometric and holonomic universal quantum gate sets with the aforementioned geometric phases are briefly described.Explicit calculation of two non-commuting Abelian and non-adiabatic geometric time evolutions show that the successively combined operation is not geometric. A proof of successive combinations of non-Abelian and non-adiabatic geometric time evolutions preserving their non-adiabatic Abelian and geometric properties is given. The report concludes that it is in general not possible to perform quantum circuits completely geometrically with a geometric universal quantum gate set. However, performing quantum computations with gates from a holonomic universal gate set is geometric and holonomic throughout the computation.
High-fidelity quantum gates are essential for large-scale quantum computation. However, any quantum manipulation will inevitably affected by noises, systematic errors and decoherence effects, which lead to infidelity of a target quantum task. Therefore, implementing high-fidelity, robust and fast quantum gates is highly desired. Here, we propose a fast and robust scheme to construct high-fidelity holonomic quantum gates for universal quantum computation based on resonant interaction of three-level quantum systems via shortcuts to adiabaticity. In our proposal, the target Hamiltonian to induce noncyclic non-Abelian geometric phases can be inversely engineered with less evolution time and demanding experimentally, leading to high-fidelity quantum gates in a simple setup. Besides, our scheme is readily realizable in physical system currently pursued for implementation of quantum computation. Therefore, our proposal represents a promising way towards fault-tolerant geometric quantum computation.
Geometrical phases acquired in cyclic quantum evolutions can be utilized as natural resources for constructing robust logic gates for quantum information processing, which was originally proposed through adiabatic changes of the driving Hamiltonian. Practically, much efforts have been made to shorten the construction times of geometrical gates, known as non-adiabatic holonomic quantum computation (NHQC). However, previous NHQC gates require the driving Hamiltonian to satisfy a set of rather restrictive conditions, reducing the robustness of the resulting geometric gates against control errors. Here we show that non-adiabatic geometric gates can be constructed in an alternative way where the driving Hamiltonian can exhibit a high degree of flexibility for retaining the robustness of the geometric phases. Consequently, this approach makes it possible to incorporate most of the existing optimal control methods, such as dynamical decoupling, composite pulses, and shortcut to adiabaticity, into the construction of geometrical gates, to further address different types of errors. Furthermore, this extensible approach of geometric quantum computation can be applied to various physical platform such as superconducting qubits and nitrogen-vacancy (NV) centers. Specifically, we performed numerical simulation to show how the noise robustness in the recent experimental implementations [Phys. Rev. Lett. 119, 140503 (2017)] and [Nat. Photonics 11, 309 (2017)] of NHQC with NV centers can be significantly improved by our optimized NHQC approach. These results cover a large class of new techniques combing the noise robustness of both geometric phase and optimal control theory.