Non-adiabatic holonomic quantum computation (NHQC) has been developed to shorten the construction times of geometric quantum gates. However, previous NHQC gates require the driving Hamiltonian to satisfy a set of rather restrictive conditions, reducing the robustness of the resulting geometric gates against control errors. Here we show that non-adiabatic geometric gates can be constructed in an extensible way, called NHQC+, for maintaining both flexibility and robustness. Consequently, this approach makes it possible to incorporate most of the existing optimal control methods, such as dynamical decoupling, composite pulses, and shortcut to adiabaticity, into the construction of single-looped geometric gates. Furthermore, this extensible approach of geometric quantum computation can be applied to various physical platform such as superconducting qubits and nitrogen-vacancy centers. Specifically, we performed numerical simulation to show how the noise robustness in the recent experimental implementations [Phys. Rev. Lett. 119, 140503 (2017)] and [Nat. Photonics 11, 309 (2017)] can be significantly improved by our NHQC+ approach. These results cover a large class of new techniques combing the noise robustness of both geometric phase and optimal control theory.

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

Geometric phases are well known to be noise resilient in quantum evolutions and operations. Holonomic quantum gates provide us with a robust way towards universal quantum computation, as these quantum gates are actually induced by non-Abelian geometric phases. Here we propose and elaborate how to efficiently implement universal nonadiabatic holonomic quantum gates on simpler superconducting circuits, with a single transmon serving as a qubit. In our proposal, an arbitrary single-qubit holonomic gate can be realized in a single-loop scenario by varying the amplitudes and phase difference of two microwave fields resonantly coupled to a transmon, while nontrivial two-qubit holonomic gates may be generated with a transmission-line resonator being simultaneously coupled to the two target transmons in an effective resonant way. Moreover, our scenario may readily be scaled up to a two-dimensional lattice configuration, which is able to support large scalable quantum computation, paving the way for practically implementing universal nonadiabatic holonomic quantum computation with superconducting circuits.

Nonadiabatic geometric quantum computation (NGQC) has been developed to realize fast and robust geometric gate. However, the conventional NGQC is that all of the gates are performed with exactly the same amount of time, whether the geometric rotation angle is large or small, due to the limitation of cyclic condition. Here, we propose an unconventional scheme, called nonadiabatic noncyclic geometric quantum computation (NNGQC), that arbitrary single-and two-qubit geometric gate can be constructed via noncyclic non-Abelian geometric phase. Consequently, this scheme makes it possible to accelerate the implemented geometric gates against the effects from the environmental decoherence. Furthermore, this extensible scheme can be applied in various quantum platforms, such as superconducting qubit and Rydberg atoms. Specifically, for single-qubit gate, we make simulations with practical parameters in neutral atom system to show the robustness of NNGQC and also compare with NGQC using the recent experimental parameters to show that the NNGQC can significantly suppress the decoherence error. In addition, we also demonstrate that nontrivial two-qubit geometric gate can be realized via unconventional Rydberg blockade regime within current experimental technologies. Therefore, our scheme provides a promising way for fast and robust neutral-atom-based quantum computation.

Holonomic quantum computation is a quantum computation strategy that promises some built-in noiseresilience features. Here, we propose a scheme for nonadiabatic holonomic quantum computation with nitrogenvacancy center electron spins, which are characterized by fast quantum gates and long qubit coherence times. By varying the detuning, amplitudes, and phase difference of lasers applied to a nitrogen-vacancy center, one can directly realize an arbitrary single-qubit holonomic gate on the spin. Meanwhile, with the help of cavityassisted interactions, a nontrivial two-qubit holonomic quantum gate can also be induced. The distinct merit of this scheme is that all the quantum gates are obtained via an all-optical geometric manipulation of the solid-state spins. Therefore, our scheme opens the possibility for robust quantum computation using solid-state spins in an all-optical way.

Adiabatic process has found many important applications in modern physics, the distinct merit of which is that it does not need accurate control over the timing of the process. However, it is a slow process, which limits the application in quantum computation, due to the limited coherent times of typical quantum systems. Here, we propose a scheme to implement quantum state conversion in opto-electro-mechanical systems via shortcut to adiabaticity, where the process can be greatly speeded up while the precise timing control is still not necessary. In our scheme, only by modifying the coupling strength, we can achieve fast quantum state conversion with high fidelity, where the adiabatic condition does not need to be met. In addition, the population of the unwanted intermediate state can be further suppressed. Therefore, our protocol presents an important step towards practical state conversion between optical and microwave photons, and thus may find many important applications in hybrid quantum information processing.

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