Universal single-qubit non-adiabatic holonomic quantum gates in optomechanical system

# Universal single-qubit non-adiabatic holonomic quantum gates in optomechanical system

## Abstract

The non-adiabatic holonomic quantum computation with the advantages of fast and robustness attracts widespread attention in recent years. Here, we propose the first scheme for realizing universal single-qubit gates based on an optomechanical system working with the non-adiabatic geometric phases. Our quantum gates are robust to the control errors and the parameter fluctuations, and have unique functions to achieve the quantum state transfer and entanglement generation between cavities. We discuss the corresponding experimental parameters and give some simulations. Our scheme may have the practical applications in quantum computation and quantum information processing.

###### pacs:
03.67.Lx, 03.67.Pp, 32.80.Qk, 37.90.+j

## I Introduction

Quantum geometric phases (1); (2); (3) are very important resource for quantum computation. They have unique advantage of robustness in quantum computation due to their global geometric property in evolution process and thus attrack much attention from both theoretical and experimental aspects (4); (5); (6); (7); (8); (9); (10); (11); (12). One of the important contributions in this field comes from Zanardi and Rasetti (4), who proposed the adiabatic holonomic quantum computation (AHQC) by using the geometric phases. It is showed that AHQC can be used to implement the high-fidelity quantum gates because of its robustness to small random perturbations of the path in parameter space and experimental imperfection (13). Following the idea of above AHQC, several AHQC schemes based on different physical systems like trapped ions (14), superconducting qubits (15), and semiconductor quantum dots (16), etc., were developed. However, an adiabatic process may bring in more decoherence due to a long evolution time, while the decoherence will result in the decrease of fidelity. To solved this problem, the non-adiabatic holonomic quantum computation (NHQC), such as the early non-adiabatic geometric phase shift gate with NMR (17), universal non-adiabatic geometric quantum gates (18) and, subsequently, more theoretical shemes (19); (20); (21); (22); (23); (24); (25); (26); (27); (28) and the experimental realizations (29); (30); (31); (32); (33); (34); (35); (36) of the NHQC were proposed. The investigations have confirmed the features of the built-in noise-resilience and less decoherence of the NHQC.

An optomechanical system, where light and mechanical motion are coupled by radiation pressure, is an important platform to realize, in the systems ranging from quantum to classical ones, the quantum effects in the content of quantum optics (37) and quantum information processing (38); (39). The fundamental study in this field includes cooling of the mechanical resonator to its ground state (40); (41); (42), strong coupling between the cavities and the mechanical resonator (44); (43) and optomechanically induced transparency (45); (46); (47), etc. The relevant application study concerns with quantum state operation (48); (49); (50); (51) and the quantum gate operation (38); (39); (52); (53).

In this paper, we propose the first scheme to achieve a set of universal single-qubit non-adiabatic holonomic quantum gates (SQNAHQGs) based on an optomechanical system working with the non-adiabatic geometric phases. This optomechanical system is composed of two optical cavities coupling to an mechanical oscillator, and the universal SQNAHQGs include noncommute Not gate, phase gate and Hadamard gate, obtained in the computational basis of the single excited state of the optomechanical system after a cyclic evolution of the system is finished. With these universal single-qubit gates, we can also achieve the quantum state transfer and the entanglement generation between two cavity-modes. Our scheme is of all the good properties of the NHQC based on a quantum system, such as the built-in noise-resilience, faster operation, less decoherence and non-requirement for the resource and time to remove the dynamical phases. It provides a prototype of quantum gates realized in the space of the mechanical motion degree of freedom, which has the promising application in quantum computation and quantum information processing.

This paper is organized as follows: In Sec. II, we give the review description of the optomechanical system. In Sec. III, we show how to realize the universal single-qubit gates in the optomechanics by using the non-adiabatic geometric phases. In Sec. IV, we give some numerical simulations and discussions. A summary is given in Sec. V.

## Ii Basic model for an optomechanical system

The optomechanical system under consideration is shown by Fig. 1, where the two cavity modes coupled to each other by radiation pressure force via a mechanical oscillator, and also are driven respectively by a laser in the red sideband resonant with mechanical mode. After the linearization procedure, the Hamiltonian of this optomechanical system in the interaction picture is given by() (49); (50)

 ^H1=∑i=1,2δi^a†i^ai+Gi^ai^b†m+H.c., (1)

where () () and () are the annihilation (creation) operators for the th cavity of frequency and the mechanical oscillator of frequency , respectively. with the detuning between the laser and the cavity mode . is the effective coupling strength which depends on the single-photon coupling strength and the intracavity photon number .

In this paper, we choose . We assume that , , and represent the single excited states on cavities 1, 2, and the mechanical oscillator, respectively. We makes and as the qubit basis states and as the ancillary qubit to construct a single-qubit state subspace . In this single-excitation subspace, the Hamiltonian (1) can be rewritten as

 ^H2=G0(t)[sinθ2eiφ|e⟩⟨g1|−cosθ2|e⟩⟨g2|+H.c.], (2)

where . The Rabi frequencies and satisfy the ratio and , respectively. Therefore, the Hamiltonian (2) can be expressed in the matrix form

 ^H3=G0(t)⎡⎢ ⎢ ⎢⎣0sinθ2e−iφ0sinθ2eiφ0−cosθ20−cosθ20⎤⎥ ⎥ ⎥⎦, (3)

where , , and are shown as , , and , respectively. The instantaneous eigenvectors of the Hamiltonian (3) are given by

 |E0⟩ = cosθ2|g1⟩+sinθ2eiφ|g2⟩, |E+⟩ = sinθ2e−iφ|g1⟩−cosθ2|g2⟩+|e⟩, |E−⟩ = sinθ2e−iφ|g1⟩−cosθ2|g2⟩−|e⟩, (4)

and the corresponding eigenvalues are , , and , respectively. In the dressed state representation, we can get the bright state and the dark state . The bright state couples to the excited state and the dark state decouples from the state .

## Iii Universal single-qubit non-adiabatic holonomic quantum gates in an optomechanical system

To implement the single-qubit gates based on the non-adiabatic geometric dynamics in an optomechanical system, two conditions (19) should be satisfied. First, one should make with to ensure the states undergo a cyclic evolution. Second, one should make the parallel-transport condition with to keep the zero dynamical phases. In this way, the total evolution phases are the purely geometric phases. The bright and the dark states evolve as

 |ψ1(t)⟩ = ^U1(t)|d⟩=|d⟩, |ψ2(t)⟩ = eiα(t)^U1(t)|b⟩ (5) = eiα(t)[cosα(t)|b⟩−isinα(t)|e⟩],

where the evolution operator is . The inserted factor is used to ensure the cyclic evolution in the projective Hilbert space. According to Eq.(III), one can derive that the accumulated purely geometric phases during the evolution process of the dark state and the bright state are and , respectively. As shown in Fig. 2, the dark state keeps unchange in the evolution process under the driving of the Hamiltonian with the basis and the bright state evolves along the longitude with the basis }.

Changing the dark-bright basis into the subspace spanned by , one makes a transformation of coordinates with the form

 |ξ1(t)⟩ = sinθ2eiφ|ψ2(t)⟩+cosθ2|d⟩, |ξ2(t)⟩ = −cosθ2|ψ2(t)⟩+sinθ2e−iφ|d⟩. (6)

The above computational states satisfy to ensure the cyclic evolution after the above transformations. The non-adiabatic holonomic dynamics can be described by , where is the time-ordering operator (19). The matrix is given by

 ^A=˙α(t)⎡⎣−sin2θ2   e−iφsinθ2cosθ2eiφsinθ2cosθ2   −cos2θ2⎤⎦. (7)

Therefore, one can obtain the evolution operator with

 ^U(θ,φ)=[cosθsinθe−iφsinθeiφ−cosθ], (8)

where and are the corresponding parameter values in the Bloch sphere. By changing the different values of coupling strength, i.e, and , one can get the NOT gate, rotation gate, and Hadamard gate with , , and , respectively (31). One can realize a phase gate by the combination of and

 [0e−iπ4eiπ40][0110]=[e−iπ400eiπ4]. (9)

And the phase-flip gate can be given with and

 ^U(0,0)=[100−1]. (10)

With these gates, one can obtain a set of universal single-qubit gates which are based on the subspace spanned by . Besides, the NOT gate and Hadamard gate can be applied in the optomechanics.

Now, we use the NOT gate and the Hadamard gate to accomplish the quantum state transfer and the generation of the entanglement in the optomechanical system, respectively. For the quantum state transfer, with and , one can obtain a NOT gate given by

 U(π2,0)=[0110]. (11)

If the initial quantum state is chosen with , one can accomplish the quantum state transfer between two cavities under the driving of the NOT gate described by

 |g2⟩=U(π2,0)|g1⟩. (12)

In this paper, we choose the coupling strengths with MHz, MHz, and MHz to perform the quantum state transfer. We calculate the variation of population and fidelities in Fig. 3. The fidelity is defined with , where represents the ideal final state. For the quantum state transfer, . represents a reduced density matrix, where the mechanical oscillator degree of freedom has been removed by tracing. One can find that when the time is , the complete population inversion indicates the system achieves the quantum state transfer successfully and the system satisfies the cyclic evolution very well.

Also, one can generate a discrete variable entangled state between two cavities by choosing and to construct a Hadamard gate. The process is given by

 |ψideal⟩|0⟩b=U(π4,0)|g1⟩. (13)

Here, the Hadamard gate is given by

 U(π4,0)=1√2[111−1]. (14)

With the parameters MHz, MHz, and MHz, one performs the process of entanglement generation in Fig. 4. When the evolution time is satisfied, the fidelity arrives , and the state becomes .

## Iv simulations and fidelities

To evaluate the performance of the SQNAHQGs, we calculate the fidelities of the NOT gate and the Hadamard gate. With dissipation, the dynamics of process can be calculated by the master equation with the Lindblad form given by

 d^ρdt=i[^ρ,^H1]+κ1^L[^a1]^ρ+κ2^L[^a2]^ρ+γm^D[^bm]^ρ, (15)

where , and represent the mechanical damping rate, the decay rates of the cavities 1 and 2, respectively. . , where is the thermal phonon number of the environment. is the density operator and is the Hamiltonian of the optomechanical system.

Here, we choose the parameters and with the range MHz and MHz, respectively. . The frequencies of the cavities 1, 2, and the mechanical oscillator are THz, THz, and MHz, respectively. For the NOT gate, the influence induced by different and  on the fidelity of the quantum state transfer is shown in Fig. 5, and the maximum and minimum fidelities are 0.96 and 0.56, respectively. The fidelity is inversely proportional to and . For the Hadamard gate, the maximum and the minimum fidelities become, accordingly, 0.97 and 0.65. In the both cases, the fidelity tends to decrease monotonously with respect to and . The higher the damping of the mechanical oscillator or the cavity mode is, the lower the fidelity we can obtain will be. Therefore, the preparing of the high quality optomechanical system is helpful for implementing universal single-qubit holonomic gates with a high fidelity.

## V summary

In summary, we have proposed a prototype of the universal single-qubit quantum gates based on an optomechanical system working with the non-adiabatic geometric phases. We have shown its typical application by changing the different coupling strengths to get various noncommute quantum gates, such as Not gates, phase gates and Hadamard gates, and apply these gates for achieving quantum state transfer and the two-cavity mode entanglement generation in the optomechanical system. The result has shown that the quantum gates can have high fidelity against the negative influence of their dissipative environment. Our scheme is of all the good properties of the NHQC based on a quantum system and can be extended into other hybrid optomechanical quantum systems.

## Acknowledgment

This work is supported by the National Natural Science Foundation of China under Grants No. 11654003, No. 11174040, No. 61675028, No. 11474026, and No. 11674033, the Fundamental Research Funds for the Central Universities under Grant No. 2015KJJCA01 and No. 2017TZ01, and the National High Technology Research and Development Program of China under Grant No. 2013AA122902.

### Footnotes

1. The first two authors contributed equally to this work.
2. Corresponding author: yanggj@bnu.edu.cn

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