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Physica Scripta

Adiabatic holonomic quantum gates for a single qubit

Vladimir S. Malinovsky, S. Rudin

Published Apr 1, 2014• 2 citations
Publisher

Abstract

A universal set of single qubit holonomic quantum gates using the geometric phase that the qubit wave function acquires after a cyclic evolution is discussed. The proposed scheme utilizes ultrafast linearly chirped pulses and provides a possibility to substantially suppress transient population of the ancillary state in a generic three-level system. That provides a possibility to reduce the decoherence effect and achieve a higher fidelity of the quantum gates.

DOI
10.1088/0031-8949/2014/t160/014029

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A universal set of single qubit holonomic quantum gates using the geometric phase that the qubit wave function acquires after a cyclic evolution is discussed. The proposed scheme utilizes ultrafast linearly chirped pulses and provides a possibility to substantially suppress transient population of the ancillary state in a generic three-level system. That provides a possibility to reduce the decoherence effect and achieve a higher fidelity of the quantum gates.
Abstract Using the analytic expression of the evolution operator for the electron spin in a quantum dot, we propose a scheme to design a universal set of single‐qubit gates based solely on the geometric phase that the qubit state acquires after a cyclic evolution in the parameter space. The scheme is using ultrafast linearly chirped pulses providing adiabatic excitation of the qubit states and the geometric phase is fully controlled by the relative phase between pulses. © 2012 Wiley Periodicals, Inc.
Holonomic quantum computation offers a promising route to robust quantum gates, but decoherence remains a central obstacle in realistic implementations. Here we develop a nonadiabatic holonomic scheme for a driven three-level system in the no-jump regime described by an effective non-Hermitian Hamiltonian. Within a biorthogonal framework, tailored complex pulses enforce exact closure of the computational-subspace evolution at the final time despite the underlying nonunitary dynamics, enabling arbitrary single-qubit holonomic gates without requiring cyclic evolution in its orthogonal complement. In contrast to existing non-Hermitian treatments, which either neglect the overall exponential prefactor or, in adiabatic settings, include dissipation only on the auxiliary excited level, our scheme incorporates decay and dephasing of all bare eigenstates directly into the pulse design, so that dissipation does not reduce the no-jump gate fidelity.
Nonadiabatic holonomic quantum computation provides the means to perform fast and robust quantum gates by utilizing the resilience of non-Abelian geometric phases to fluctuations of the path in state space. While the original scheme [E. Sj\"oqvist et al., New J. Phys. 14, 103035 (2012)] needs two loops in the Grassmann manifold (i.e., the space of computational subspaces of the full state space) to generate an arbitrary holonomic one-qubit gate, we propose single-loop one-qubit gates that constitute an efficient universal set of holonomic gates when combined with an entangling holonomic two-qubit gate. Our one-qubit gate is realized by dividing the loop into path segments, each of which is generated by a $\mathrm{\ensuremath{\Lambda}}$-type Hamiltonian. We demonstrate that two path segments are sufficient to realize arbitrary single-loop holonomic one-qubit gates. We describe how our scheme can be implemented experimentally in a generic atomic system exhibiting a three-level $\mathrm{\ensuremath{\Lambda}}$-coupling structure by utilizing carefully chosen laser pulses.
Holonomic quantum gates offer a promising approach for achieving high fidelity and robustness in quantum computation by leveraging geometric phases to suppress the influence of noise. However, existing schemes often suffer from limitations such as complex segmented evolution paths and an inability to implement certain gates, including those with a geometric phase of 𝜋𝜋. In this work, we propose a circular-path holonomic quantum gate scheme with multiple degrees of freedom, which enables the realization of any single-qubit gate, including gates with a geometric phase of 𝜋𝜋. The scheme allows for the optimization of pulse performance, effectively mitigating control errors in both the addressing frequency and Rabi frequency. Applied to a Rydberg-atoms system, numerical simulations demonstrate high fidelity and robustness against control errors, with fidelities exceeding 99.97% under various conditions. This work addresses the limitations of previous schemes and provides a flexible framework for implementing robust quantum gates in three-level quantum systems.