Environment-Assisted Holonomic Quantum Maps

Abstract

Holonomic quantum computation uses non-Abelian geometric phases to realize error resilient quantum gates. Nonadiabatic holonomic gates are particularly suitable to avoid unwanted decoherence effects, as they can be performed at high speed. By letting the computational system interact with a structured environment, we show that the scope of error resilience of nonadiabatic holonomic gates can be widened to include systematic parameter errors. Our scheme maintains the geometric properties of the evolution and results in an environmentassisted holonomic quantum map that can mimic the effect of a holonomic gate. We demonstrate that the sensitivity to systematic errors can be reduced in a proof-of-concept spin-bath model.Quantum holonomies are non-Abelian (non-commuting) unitary operators that only depend on paths in state space of a quantum system. The non-commuting property makes them useful for implementing quantum gates that manipulate quantum information by purely geometric means. Holonomic quantum computation (HQC) [1] is a network of holonomic gates that unifies geometric characteristics of quantum systems and information processing, as well as is conjectured to be robust to errors in experimental control parameters [2].Nonadiabatic HQC has recently been proposed [3] and experimentally implemented [4-9] as a tool to realize quantum gates based upon nonadiabatic non-Abelian geometric phases [10]. The basic setup for nonadiabatic HQC in [3] is a threelevel Λ configuration, where two simultaneous resonant laser pulses drive transitions between the qubit levels and an auxiliary state level. This scheme has been generalized to offresonant pulses [11,12]. The off-resonant setup uses two simultaneous laser pulses with the same variable detuning, which enhances the flexibility of the holonomic scheme. For experimental realization of off-resonant nonadiabatic holonomic gates, see Refs. [13][14][15][16].The nonadiabatic version of HQC avoids the drawback of the long run time associated with adiabatic holonomies [17], on which the original holonomic schemes are based [1,18]. Nonadiabatic holonomic gates are therefore particularly suitable to avoid unwanted decoherence effects [19]. The resilience to decoherence errors can be further improved by combining nonadiabatic HQC with decoherence-free subspaces [20-23] and subsystems [24], as well as dynamical decoupling [25-27]. On the other hand, it has been pointed out [28] that the original version of nonadiabatic HQC has no particular advantage compared to standard dynamical schemes in the presence of systematic errors in experimental parameters. To deal with this, we here show that the sensitivity to systematic parameter errors can be reduced by letting the system interact with a structured environment. Our approach is inspired by earlier findings [29][30][31] that transport efficiency in quantum systems can be enhanced in such environments.We modify the off-resonant non-adiabatic holonomic scheme by coupling the auxiliary state to a finite thermal bath, the latter playing t...

Experimental Realization of Nonadiabatic Holonomic Single-Qubit Quantum Gates with Two Dark Paths in a Trapped Ion

“…To resolve this dilemma, holonomic quantum computation based on nonadiabatic evolutions has been proposed [15,16], which was theoretically expanded [17,18,19,20,21,22,23,24,25] and experimentally demonstrated [26,27,28,29,30,31,32,33,34,35,36,37] in various three-level physical systems. Unfortunately, the noise-resilience feature of geometric phases is smeared in this type of implementation [38,39,40,41].…”

Noncyclic nonadiabatic holonomic quantum gates via shortcuts to adiabaticity

“…Recently, nonadiabatic holonomic quantum computation (NHQC), based on the non-Abelian geometric phase, has been proposed for high-fidelity quantum operation [11,12], and then significant theoretical [13][14][15][16][17][18][19][20] and experimental progresses [21][22][23][24][25][26][27][28][29][30][31] have been made. However, the above NHQC implementations are sensitive to the systematic error [32,33] caused by external control imperfections, which can lead to infidelity of the implemented gate.…”

Dynamical-decoupling-protected nonadiabatic holonomic quantum computation

“…By using this approach, one can easily find a Hamiltonian making the quantum system evolve along a desired path so that nonadiabatic holonomic gates can be realized with an economical evolution time. Up to now, a lot of works both in theories [23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38] and experiments [39][40][41][42][43][44][45][46][47][48][49][50][51][52] have contributed to nonadiabatic holonomic quantum computation.…”