Validity of the rotating-wave approximation in nonadiabatic holonomic quantum computation

Abstract

We examine the validity of the rotating wave approximation (RWA) in non-adiabatic holonomic single-qubit gates [New J. Phys. 14, 103035 (2012)]. We demonstrate that the adoption of RWA may lead to a sharp decline in fidelity for rapid gate implementation and small energy separation between the excited and computational states. The validity of the RWA in the recent experimental realization [Nature (London) 496, 482 (2013)] of non-adiabatic holonomic quantum computation for a superconducting qubit is examined.PACS numbers: 03.65. Vf, 03.67.Lx, 85.25.Cp Holonomic quantum computation (HQC) is the idea of using non-Abelian geometric phases to implement robust quantum gates [1]. By using adiabatic holonomies, HQC becomes tolerant to errors caused by fluctuations of the slowly changing control parameters. On the other hand, dissipation may have detrimental effects on the gates, leading to the need of performing the gate operations as fast as possible by using non-adiabatic holonomies. Nonadiabatic strategies have been shown [2] to be effective to minimize this error source. However, a shortening of the run time may in turn lead to other errors that can lower the gate fidelity and therefore put a limitation on the speed of holonomic gate operations. Here, we examine how the validity of the rotating wave approximation (RWA) depends on the run time and energy structure of the three level Λ setting used to implement non-adiabatic non-Abelian geometric gates first proposed in Ref. [3] and experimentally demonstrated in Refs. [4,5].The speed of quantum gate operations is generally limited by unwanted effects that become more pronounced when the run time is decreased. One such effect is related to the quasi-monochromatic approximation [6] that breaks down for short pulses, causing population of energy levels outside the computational subspace [7,8]. Another speed-limiting feature is the RWA, which is expected to break down when the run time of the gate becomes too short. This leads to a situation where the Rabi flopping is accompanied by faster fidelity reducing oscillations [9]; an effect that can be suppressed by embedding the qubit in an off-resonant Λ system [10]. We quantify the validity of the RWA by computing the fidelity of the ideal RWA-based non-adiabatic holonomic single-qubit gate operations with respect to numerical solutions of the exact Schrödinger equation.The Λ system consists of states |0 and |1 coupled to the auxiliary excited state |e via a pair of oscillating electric field pulses E j (t) = j g j (t) cos(ω j t), j = 0, 1, g j (t) being envelope functions describing the pulse shape and duration. The polarization j (t) is chosen so as to allow for the j ↔ e transition only. The Hamiltonian of the system readsĤ(t) =Ĥ 0 +μ · [E 0 (t) + E 1 (t)], wherê H 0 = −f e0 |0 0| − f e1 |1 1| is the bare Hamiltonian (by putting the energy of the excited state to zero) andμ is the electric dipole operator. By tuning the oscillation frequencies ω j on resonance with the bare transition frequencies f ej , the Hamiltonia...

Experimental realization of single-shot nonadiabatic holonomic gates in nuclear spins

“…Fig. 4 illustrates some example experimental NMR spectra observed at 13 C, in which (a) is the spectrum of the thermal equilibrium state; (b) is the spectrum of input state where ρ in = σ x and 1 H is in state |1 1|; (c), (d), (e) and (f) are the spectra of output states after applying…”

“…13 C labeled chloroform sample is used as our two-qubit quantum information processor, where 13 C acts as the work qubit and 1 H acts as the ancillary qubit. At first, we prepare the NMR system in |1 1| ⊗ ρ in with a cat-state method; Then, at the stage of implementing the quantum circuits of Fig.…”

“…At the quantum state preparation stage, the ancillary qubit 1 H is prepared in state |1 1| and the work qubit 13 C is prepared in ρ in . Writing ρ in in form of deviation matrix [39], here we can get a set of ρ in :…”

“…Although NHQC was just proposed recently, it gains increasing attentions because of its combining speed and universality, where lots of theoretical schemes [1,2,[12][13][14][15][16][17][18][19][20][21][22][23][24][25] are prompted and meanwhile NHQC has been realized experimentally in NMR [26], superconducting quantum devices [27] and diamond nitrogen-vacancy centers [28,29] recently.…”

Numerical simulations of the quantum systems dynamics in the path integral approach

“…Thus, different approximations are used. For example, there are two-or three-level quantum system models [7] and rotating wave approximation [8]. For high-intensity laser field the perturbation theory runs into problems.…”