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physica status solidi (b)

Holonomic Quantum Computation with Josephson Networks

Jens Siewert, Lara Faoro, Rosario Fazio

Published Oct 1, 2002• 2 citations
Publisher

Abstract

It is well known that it is possible to build quantum gates based entirely on the action of geometric phases. This approach is called holonomic quantum computation. Here we show that it is possible to generate and detect non-Abelian geometric phases in the charge dynamics of a network of mesoscopic Josephson junctions. The scheme can be used to realize a universal set of quantum gates.

DOI
10.1002/1521-3951(200210)233:3<490::aid-pssb490>3.0.co;2-j

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In 1984 Berry pointed out that abelian gauge structures exist naturally in slowly varying quantum system. Since then, a great deal of interest both theoretically and experimentally has been devoted to the interpretation, generalization and detection of geometric phases. In particular, relevant for the present work, it has been shown that a non-Abelian gauge structure emerges if a set of quantum states remains degenerate as the Hamiltonian varies. Non-Abelian phases have been originally investigated in Nuclear Quadrupole Resonance. More recently, they have found a renewed interest within the field of Quantum Computation, since it has been proved that quantum gates can be also implemented by geometrical means using both Abelian and non-Abelian phases. In this novel geometric approach, known as Holonomic Quantum Computation (HQC), information is encoded in a degenerate eigenspace of a parametric family of Hamiltonians and the computational network of unitary quantum gates is realized by driving adiabatically the Hamiltonian parameters along loops in a control manifold. By properly designing such loops, the non trivial curvature of the geometry of the control manifold gives rise to unitary transformations, i.e. holonomies, that implement desired quantum gates. Until now only one scheme has been proposed to realize HQC in trapped ions by means of pulsed coherent excitations. On the other hand, it would be of great interest to have an implementation for solid-state systems, in view of their scalability and integrability. In particular, the impressing development of nanotechnologies with the consequent control of artificial two-level system in superconducting devices stimulates a more detailed investigation of geometric interference in mesoscopic systems. Here we show that non-Abelian phases can appear in the quantum charge dynamics of Josephson networks. Besides their possible detection, we shall see how they may be employed both in adiabatic charge pumping and as an implementation of HQC with Josephson junctions. 2. Josephson network for non abelian phases

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It is well known that it is possible to build quantum gates based entirely on the action of geometric phases. This approach is called holonomic quantum computation. Here we show that it is possible to generate and detect non-Abelian geometric phases in the charge dynamics of a network of mesoscopic Josephson junctions. The scheme can be used to realize a universal set of quantum gates.
In 1984 Berry pointed out that abelian gauge structures exist naturally in slowly varying quantum system. Since then, a great deal of interest both theoretically and experimentally has been devoted to the interpretation, generalization and detection of geometric phases. In particular, relevant for the present work, it has been shown that a non-Abelian gauge structure emerges if a set of quantum states remains degenerate as the Hamiltonian varies. Non-Abelian phases have been originally investigated in Nuclear Quadrupole Resonance. More recently, they have found a renewed interest within the field of Quantum Computation, since it has been proved that quantum gates can be also implemented by geometrical means using both Abelian and non-Abelian phases. In this novel geometric approach, known as Holonomic Quantum Computation (HQC), information is encoded in a degenerate eigenspace of a parametric family of Hamiltonians and the computational network of unitary quantum gates is realized by driving adiabatically the Hamiltonian parameters along loops in a control manifold. By properly designing such loops, the non trivial curvature of the geometry of the control manifold gives rise to unitary transformations, i.e. holonomies, that implement desired quantum gates. Until now only one scheme has been proposed to realize HQC in trapped ions by means of pulsed coherent excitations. On the other hand, it would be of great interest to have an implementation for solid-state systems, in view of their scalability and integrability. In particular, the impressing development of nanotechnologies with the consequent control of artificial two-level system in superconducting devices stimulates a more detailed investigation of geometric interference in mesoscopic systems. Here we show that non-Abelian phases can appear in the quantum charge dynamics of Josephson networks. Besides their possible detection, we shall see how they may be employed both in adiabatic charge pumping and as an implementation of HQC with Josephson junctions. 2. Josephson network for non abelian phases
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 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.