Superadiabatic holonomic quantum computation in cavity QED

Abstract

Superadiabatic control for four-level systemWe consider a tripod configuration in the scheme. The Hamiltonian is written as Eq.1 of the maintext. In the adiabatic basis, the Hamiltonian can be reduced to three-level form when ϕ is kept constant. The reduced Hamiltonian in the bases {|+ , |d 1 , |− } is Eq. (3) of the maintext, i.e.,where the termθ corresponds to the nonadiabatic couplings.In order to correct the nonadiabatic errors, there are two approaches. Here, we modify the pulses via dressed states methods [1]. We look for a correction Hamiltonian H C (t) such that the superadiabatic Hamiltonian H ′ (t) = H(t) + H C (t) governs a perfect state transfer. We choose the general formwhere the unitary operator is defined, in the basis of {(sin ϕ |0 + cos ϕ |1 ), |e , |2 }, aswhere the modified pulses areSimilar with the treatments in Ref.[1], we define a new basis of dressed states by the action of a time-dependent unitary operator V (t) on the time-independent eigenstates |±, d 1 .In our model, without loss of generality, we takes V (t) = e iµ(t)Mx , with a Euler angle µ(t), moving in the frame defined by V in which the Hamiltonian takes the formFor a perfect state transfer in new dressed states,the Hamiltonian H new (t) should be diagonalized. In other words, H C (t)has to be designed such that the unwanted off-diagonal elements in H new (t) equal zero. In order to satisfy Eq. (6) the control parameters have to be chosen asThe additional control Hamiltonian and dressed state basis must be chosen so that the dressed medium states coincide with the medium states at initial time and final time. Because of this condition, µ has to satisfy µ(t i ) = µ(t f ) = 0(2π).The simplest nontrivial choice of the dressed states basis is the superadiabatic basis, for whichThis choice will be referred to as superadiabatic transitionless driving (SATD). To reduce the intermediate-level occupancy, we can further modify the SATD by choosing the suit parameter as [1]Taking Eq. (8) into Eq. (5) we can get the modified diving field strength with SATD as Eq. (4) of the maintext. Similarly, taking Eq. (8) into Eq. (5), we can obtained similar result with modified superadiabatic transitionless driving (MSA). We plot the modified pulses for realizing NOT gate with (a) and (d), Hardarmd gate with (b) and (e), and two-qubit control phase gate with (c) and (f) by the SATD and MSA in the Fig. 1, and these modified pules can achieve high fidelity quantum gate. Effective atom-cavity couplingOur scheme relies on the Raman excitation of two threelevel atom by a driving field of frequency ω Rj and a quantized cavity mode of frequency ω 0 in a lambda configuration. The field drives dispersively the transition from level 6 2 S 1/2 , |1 = |F = 4, m = 3 to level 6 2 P 1/2 , |e = |F = 4, m = 3 , with coupling strength Ω R and detuning ∆ = ω e1 − ω R ≫ |Ω Rj |. The cavity mode couples level 6 2 S 1/2 ,|0 = |F = 3, m = 2 to level 6 2 P 1/2

Nonadiabatic noncyclic geometric quantum computation in Rydberg atoms

“…Geometric quantum logic gates [22,23] based on adiabatic or nonadiabatic geometric phase [24][25][26][27], which depends only on the global properties of the evolution paths, provides us the possibility for robust quantum computation [28][29][30][31][32][33][34]. In contrast to the earlier adiabatic-process-based geometric quantum computation [35][36][37][38], nonadiabatic geometric quantum computation (NGQC) and nonadiabatic holonomic quantum computation (NHQC) based on Abelian [39][40][41][42][43][44][45][46] and non-Abelian geometirc phases [47][48][49][50][51][52][53][54][55][56] in two-and threelevel system, respectively, can intrinsically protect against environment-induced decoherence, since the the construction times of geometric quantum gates is reduced. The nonadiabatic geometric gates of NGQC and NHQC have been experimentally demonstrated in many systems including superconducting qubit [57][58][59][60][61], NMR [62][63][64]…”

Error‐Insensitive 3D Entanglement Gate with Optimal Control of Geometric Evolution

“…In order to shorten the adiabatic evolution time, many schemes were proposed to expedite the implementation of AGQC. [ 44–50 ] Zhang et al. [ 44 ] expedited an implementation of adiabatic geometric gates by using transitionless quantum driving.…”

Accelerated adiabatic quantum gates: Optimizing speed versus robustness

“…We also note that while Ref. [30] straightforwardly applied the dressed state technique of Ref. [24] to accelerate an adiabatic gate, they did not consider the potential difficulties associated with this procedure (stemming from STA-induced modification of phases).…”