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.
Quantum holonomies are investigated in different contexts. A geometric phase is proposed for decomposition dependent evolution, where each component of a given decomposition of a mixed state evolves independently. It is shown that this geometric phase only depends on the path traversed in the space of decompositions. A holonomy is associated to general paths of subspaces of a Hilbert space, both discrete and continuous. This opens up the possibility of constructing quantum holonomic gates in the open path setting. In the discrete case it is shown that it is possible to associate two distinct holonomies to a given path. Interferometric setups for measuring both holonomies are provided. It is further shown that there are cases when the holonomy is only partially defined. This has no counterpart in the Abelian setting. An operational interpretation of amplitudes of density operators is provided. This allows for a direct interferometric realization of Uhlmann's parallelity condition, and the possibility of measuring the Uhlmann holonomy for sequences of density operators. Off-diagonal geometric phases are generalized to the non-Abelian case. These off-diagonal holonomies are undefined for cyclic evolution, but must contain members of non-zero rank if all standard holonomies are undefined. Experimental setups for measuring the off-diagonal holonomies are proposed. The concept of nodal free geometric phases is introduced. These are constructed from gauge invariant quantities, but do not share the nodal point structure of geometric phases and off-diagonal geometric phases. An interferometric setup for measuring nodal free geometric phases is provided, and it is shown that these phases could be useful in geometric quantum computation. A holonomy associated to a sequence of quantum maps is introduced. It is shown that this holonomy is related to the Uhlmann holonomy. Explicit examples are provided to illustrate the general idea.
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.
In the context of two-particle interferometry, we construct a parallel transport condition that is based on the maximization of coincidence intensity with respect to local unitary operations on one of the subsystems. The dependence on correlation is investigated and it is found that the holonomy group is generally non-Abelian, but Abelian for uncorrelated systems. It is found that our framework contains the Lévay geometric phase [2004 {\it J. Phys. A: Math. Gen.} {\bf 37} 1821] in the case of two-qubit systems undergoing local SU(2) evolutions.
Implementing quantum gates as non-Abelian holonomies, a class of topologically protected unitary operators, is a particularly promising paradigm for the design of intrinsically stable quantum computers [1]. In contrast to dynamic phases, the geometric phase accumulated by a quantum system propagating through a Hilbert space
Hdepends exclusively on its path. In general, geometric phases can exhibit arbitrary dimensionality. Wilczek and Zee introduced the idea of multi-dimensional, non-Abelian geometric phases - so called holonomies [2]. Anandan later dropped the requirement of adiabaticity to create holonomies, that are truly time-independent [3]. Non-adiabatic holonomies rely on a subspace
Hgeoof the Hilbert-space that is spanned by states
{∣Φk⟩}kthat fulfill
(Φk∣H^∣Φj⟩=0, where
H^is the system's Hamiltonian. Restricting the propagation to
Hgeoensures parallel transport and, thus, a purely geometric phase (see Fig. 1a) [4], [5]. Quantum optics constitutes a particularly versatile platform for quantum information processing, and in particular for the construction of non-adiabatic holonomic quantum computers: In addition to integration and miniaturization provided by the platform, the bosonic nature of photons also conveniently allows for multiple excitations of the same mode, readily expanding
Hgeoand enabling the synthesis of holonomies from higher symmetry groups
U(N)as larger and more capable computational units [6], [7].