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arXiv:quant-ph/0306166 · Physical Review Letters

Unconventional Geometric Quantum Computation

Shi-Liang Zhu, Ziang Wang

Published Oct 30, 2003• 271 citations
arXivHTMLPDF

Abstract

We propose a new class of unconventional geometric gates involving nonzero dynamic phases, and elucidate that geometric quantum computation can be implemented by using these gates. Comparing with the conventional geometric gate operation, in which the dynamic phase shift must be removed or avoided, the gates proposed here may be operated more simply. We illustrate in detail that unconventional nontrivial two-qubit geometric gates with built-in fault-tolerant geometric features can be implemented in real physical systems.

DOI
10.1103/physrevlett.91.187902

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We propose a new class of unconventional geometric gates involving nonzero dynamic phases, and elucidate that geometric quantum computation can be implemented by using these gates. Comparing with the conventional geometric gate operation, in which the dynamic phase shift must be removed or avoided, the gates proposed here may be operated more simply. We illustrate in detail that unconventional nontrivial two-qubit geometric gates with built-in fault-tolerant geometric features can be implemented in real physical systems.
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.
Abstract Unconventional nonadiabatic geometric quantum computation not only possesses the geometric robustness of conventional ones but also avoids the need to remove the dynamical phase. Dynamical decoupling is a promising method to protect unconventional nonadiabatic geometric gates against decoherence. Here, we propose a protocol to implement unconventional nonadiabatic geometric quantum computation protected by dynamical decoupling. By using three physical qubits to encode a logical qubit and choosing a system Hamiltonian that commutes with the decoupling group, a universal set of dynamical-decoupling-protected unconventional nonadiabatic geometric gates can be realized. Our work maintains the robustness of unconventional nonadiabatic geometric gates while protecting quantum gates against both collective and independent decoherence.
ABSTRACT Geometric quantum gates play crucial role in quantum computation, and the diffident types of geometric gates are constructed, including conventional and unconventional geometric gates. The physical natures of the conventional geometric gates with zero dynamical phase are investigated widely. In this paper, we analyze the physical natures of the unconventional geometric gates, and present the unconventional geometric gate is related with the Levi–Civita connection. We establish its physical constraint, and demonstrate the results by employing two‐level system.
Quantum computation has demonstrated advantages over classical computation for special hard problems, where a set of universal quantum gates is essential. Geometric phases, which have built-in resilience to local noise, have been used to construct quantum gates with excellent performance. However, this advantage has been smeared in previous schemes. Here, we propose a state-independent nonadiabatic geometric quantum-gate scheme that is able to realize a more fully geometric gate than previous approaches, allowing for the cancelation of dynamical phases accumulated by an arbitrary state. Numerical simulations demonstrate that our scheme has significantly stronger gate robustness than the previous geometric and dynamical ones. Meanwhile, we give a detailed physical implementation of our scheme with the Rydberg atom system based on the Rydberg blockade effect, specifically for multiqubit control-phase gates, which exceeds the fault-tolerance threshold of multiqubit quantum gates within the considered error range. Therefore, our scheme provides a promising way for fault-tolerant quantum computation in atomic systems.
A different way to realize nonadiabatic geometric quantum computation is proposed by varying parameters in the Hamiltonian for nuclear-magnetic resonance, where the dynamical and geometric phases are implemented separately without the usual operational process. Therefore the phase accumulated in the geometric gate is a pure geometric phase for any input state. In comparison with the conventional geometric gates by rotating operations, our approach simplifies experimental implementations making them robust to certain experimental errors. In contrast to the unconventional geometric gates, our approach distinguishes the total and geometric phases and offers a wide choice of the relations between the dynamical and geometric phases.