April 12, 2021

# Holonomic Quantum Computation

By Antonio Hernández G.

One of the main challenges of quantum computation is to overcome the degradation of qubits due to decoherence. Thus much effort has been devoted to the implementation of fault-tolerant quantum gates. Holonomic quantum computation is a promising tool for achieving this goal.

The idea of holonomy goes back to Michael Berry’s discovery of the geometric phase [1]. An elementary account of the geometric phase can be found in [2]. It is worth noting that geometric phases appear naturally in classical mechanics as well; as an example, see a contribution of ours in the context of vortex dynamics [4].

In the context of quantum systems, the geometric phase appears as an evolution history. Say that we have a quantum state subjected to the influence of a slowly changing environment, and that the state of the environment is characterized by $$\lambda \in M$$, where $$M$$ is some parameter space of dimension greater than one. Since the environment changes, $$\lambda$$ moves along a curve $$\lambda: [t_0, t_1] \longrightarrow M$$. We assume for now that this movement is slow –an adiabatic evolution as it is called. As a consequence of a changing environment, the wave function $$| \psi(t) \rangle$$ of our quantum state also evolves in time. Now, let us further assume that the parameter $$\lambda$$ describes a closed loop in parameter space, that is to say, $$\lambda(t_0) = \lambda(t_1)$$. Before the 1980’s, it was understood that $$| \psi (t _1) \rangle = e ^{ i \alpha } | \psi (t _0) \rangle$$, where $$\alpha$$ is a dynamic phase that results from the time evolution of the wave function. M. Berry’s celebrated contribution was the realization that there is an additional phase factor $$\Phi$$, so that $| \psi (t _1) \rangle = e ^{ i \alpha } e ^{ i \Phi } | \psi (t _0) \rangle \,,$ and that this phase only depends on the geometry of the loop traversed by $$\lambda$$. It turns out that this geometric phase is an instance of the differential geometric concept of parallel transport. Experimental evidence of the presence of the geometric phase $$\Phi$$ can be obtained through interference phenomena.

A renewed interest in the geometric phase in quantum physics has been recently motivated by its use in implementing universal quantum computation. In the seminal paper [6], Zanardi and Rasetti proposed the following model of holonomic quantum computation. Suppose that a parameter dependent Hamiltonian $$H(\lambda(t))$$ is controlled by the time dependent parameters represented by $$\lambda(t)$$. As the parameters evolve in time, the eigenvalues and eigenspaces of $$H(\lambda(t))$$ change (although the eigenspaces never change dimension due to the adiabatic assumption). An initial quantum state $$| \psi (t _0) \rangle$$ is prepared so that it is in an eigenstate of $$H(\lambda(t_0))$$. By the adiabatic theorem, the state $$| \psi (t) \rangle$$ remains an eigenstate of the instantaneous Hamiltonian $$H(\lambda(t))$$. Thus, after the loop $$\lambda(t)$$ finishes its journey, a geometric phase is obtained. To achieve universal quantum computation, the eigenspace associated to $$H(\lambda(t_0))$$ is assumed to be degenerate. In this circumstance, the geometric phase gets associated to a unitary transformation, and through this a quantum gate can be implemented.

Due to its geometric nature, holonomic quantum gates are insensitive to small fluctuations of their control parameters and thus lend well to fault-tolerance.

Several experimental realizations of holonomic quantum computation have been achieved. Further generalizations have appeared since the original proposal presented in [6]. The adiabatic condition has been dropped to make the implementation less susceptible to decoherence. The first experimental demonstration of non-adiabatic quantum computation was reported in [3]. More recently, researchers at the Yokohama National University have experimentally demonstrated fault-tolerant universal non-adiabatic holonomic quantum gates under zero magnetic field at room temperature [5]. The field remains very active.

## References

1. Berry, M. (). Quantal phase factors accompanying adiabatic changes. Proceedings of the Royal Society of London. a. Mathematical and Physical Sciences, 392(1802), 45–57.
2. Berry, M. (). The Geometric Phase. Scientific American, 259(6), 46–52.
3. Feng, G., Xu, G., & Long, G. (). Experimental Realization of Nonadiabatic Holonomic Quantum Computation. Physical Review Letters, 110(19), 190501.
4. Hernández-Garduño, A., & Shashikanth, B. N. (). Reconstruction phases in the planar three- and four-vortex problems. Nonlinearity, 31(3), 783–814.
5. Nagata, K., Kuramitani, K., Sekiguchi, Y., & Kosaka, H. (). Universal holonomic quantum gates over geometric spin qubits with polarised microwaves. Nature Communications, 9(1), 1–10.
6. Zanardi, P., & Rasetti, M. (). Holonomic quantum computation. Physics Letters A, 264(2-3), 94–99.