Non-Adiabatic Universal Holonomic Quantum Gates Based on Abelian Holonomies

Abstract

We implement a non-adiabatic universal set of holonomic quantum gates based on abelian holonomies using dynamical invariants, by Lie-algebraic methods. Unlike previous implementations, presented scheme does not rely on secondary methods such as double-loop or spin-echo and avoids associated experimental difficulties. It turns out that such gates exist purely in the non-adiabatic regime for these systems.

Geometry and symmetry of quantum and classical-quantum variational principles

“…Fubini-Study geodesics on P C n are used in various situations of quantum mechanics. Their applications in quantum search algorithms [33,45], time-optimal control problems [10,11,16] and holonomic quantum computation [22,31,40] emphasizes their importance and makes this example especially interesting. In the general case, geodesics are optimal curves in the sense that they minimize the distance between two quantum states.…”

“…Here, we recall iρ ψ 0 = Ad * U (iρ ψ ), from the definition ρ ψ := ψψ † = U ρ ψ 0 U −1 . Therefore, because of the symmetry property (22) possessed by any Lagrangian of the type (2), the corresponding Euler-Poincaré equations ( 6) conserve J 1 (U, δℓ/δξ). More particularly, one shows that any quantum system with an arbitrary Lagrangian of the type (21) produces dynamics on the zero-level set of J 1 .…”

“…Therefore, because of the symmetry property (22) possessed by any phase-invariant Lagrangian of the type (2), the corresponding Euler-Poincaré quantum dynamics (8) conserves the momentum map J 2 and lies on the kernel of J 1 .…”

“…After Kibble's investigation [27] of the geometric properties of quantum state spaces, geometric formulations of quantum dynamics have been attracting much attention over the last decades [3,4,6,7,8,13,14,15,18,21,34,39,26,42]. In turn, geometric quantum dynamics has opened several modern perspectives: for example, Fubini-Study geodesics have been introduced in Grover's quantum search algorithms [33,45] and in time-optimal quantum control [10,11,16], while the holonomy features arising from the quantum geometric phase [1,7] and its non-Abelian extensions [2,14] have been proposed in quantum computation algorithms [22,31,40].…”

Doubly geometric quantum control

“…This is made challenging by the ubiquitous noise from the environment and unavoidable control imperfections, which can substantially reduce the control fidelity. Holonomic quantum computation [1], where the gates are based on geometric phases [2][3][4][5][6][7][8], is one approach to boosting gate fidelities in the presence of noise. Using geometric rather than dynamical phases to implement quantum gates can mitigate the effect of noise that leaves holonomy loops in the control space unperturbed.…”