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

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

Shan-Shan Chen, Hao Zhang111The first two authors contributed equally to this work., Xue-Ke Song, Fu-Guo Deng, Ahmed Alsaedi, Tasawar Hayat, Haibo Wang, and Guo-Jian Yang222Corresponding author: yanggj@bnu.edu.cn Department of Physics, Applied Optics Beijing Area Major Laboratory, Beijing Normal University, Beijing 100875, China
NAAM Research Group, Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia
Department of Mathematics, Quaid-I-Azam University, Islamabad 44000, Pakistan
July 17, 2019

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.

03.67.Lx, 03.67.Pp, 32.80.Qk, 37.90.+j

I Introduction

Quantum geometric phases Pancharatnam1956 (); SimonPRL1983 (); Berry1984 () 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 ZanardiPRA1999 (); VedralPRL2000 (); TongPRL2004 (); YiPRL2004 (); YiPRA2007 (); YiPRA2009 (); TondPRA2009 (); YiPRA2010 (); SongNJP2016 (). One of the important contributions in this field comes from Zanardi and Rasetti ZanardiPRA1999 (), 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 PachosMPB2001 (). Following the idea of above AHQC, several AHQC schemes based on different physical systems like trapped ions DuanScience2001 (), superconducting qubits Faoro2003 (), and semiconductor quantum dots Solinas2003 (), 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 WangPRL2001 (), universal non-adiabatic geometric quantum gates ZhuPRL2002 () and, subsequently, more theoretical shemes SjoqvistNJP2012 (); XU2012PRL (); LongSR2014 (); LongPRA2014 (); zhouOe2015 (); XuePRA2015 (); XuPRA2015 (); TongPRA2016 (); XuPRA2017 (); XuePRApp2017 () and the experimental realizations AbdumalikovNature2013 (); FengPRL2013 (); DuanNature2014 (); ArroyoCamejoNatcommu2014 (); AwschalomNatPhoton2016 (); PRLAwschalom2017 (); NatphtonicsKosaka2017 (); SciChinaLong2017 () 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 MAspelmeyerRMP2014 () and quantum information processing LudwigPRL2012 (); StannigelPRL2012 (). The fundamental study in this field includes cooling of the mechanical resonator to its ground state Schliesser2009 (); Liu201301 (); Peterson2016 (), strong coupling between the cavities and the mechanical resonator Groblacher2009 (); Thompson2008 () and optomechanically induced transparency Weis2010 (); Safavi2011 (); LongOe2015 (), etc. The relevant application study concerns with quantum state operation FiorePRL2011 (); WangPRL2012 (); LTianPRL2012 (); Palomaki2013 () and the quantum gate operation LudwigPRL2012 (); StannigelPRL2012 (); ZhangJPB2015 (); AsjadOE2015 ().

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.

Figure 1: Schematic diagram of the optomechanical system composed of the two cavities and a mechanical oscillator. The mechanical oscillator is realized by a membrane fixed in middle. , , and represent the frequencies of cavity 1, cavity 2, and the mechanical oscillator, respectively.

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() WangPRL2012 (); LTianPRL2012 ()


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


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


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


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 .

Figure 2: Two geometric dynamics process pictures for the non-adiabatic quantum state transfer process. (a) Bloch sphere for the evolution of the dark state vector; (b) Bloch sphere for the evolution of the bright state vector. Under the cyclic evolution and the parallel-transport condition, the dark state and the bright state acquire the geometric phases of 0 and , respectively.

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 SjoqvistNJP2012 () 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


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


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 SjoqvistNJP2012 (). The matrix is given by


Therefore, one can obtain the evolution operator with


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 DuanNature2014 (). One can realize a phase gate by the combination of and


And the phase-flip gate can be given with and


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


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


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


Here, the Hadamard gate is given by


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 .

Figure 3: The simulation of the populations and the fidelity of the NOT gate. , , and represent the qubit-state populations of cavities 1, 2, and the phonon, respectively. is the fidelity of the NOT gate. When the initial state is , the ideal final state can be achieved when the evolution time meets the condition , which means the population of cavity 1 can be transferred completely to that of cavity 2.
Figure 4: The simulation of the populations and the fidelity of the Hadamard gate. , , and represent the qubit-state populations of cavities 1, 2, and phonon, respectively. is the fidelity of the Hadamard gate. With the initial state as , we realize the quantum entanglement between the two cavities at the time .

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


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.

Figure 5: The fidelity of the NOT gate vs the decays of the cavities () and the dissipation of the mechanical oscillator . Here we choose MHz, MHz. and with the range MHz and MHz, respectively.
Figure 6: The fidelity of the Hadamard gate vs the decays of the cavities () and the dissipation of the mechanical resonator . Here we choose MHz, MHz. and with the range MHz and MHz, respectively.

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.


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.


  • (1) S. Pancharatnam, Generalized theory of interference and its applications. Proc. Indian Acad. Sci 44, 247 (1956).
  • (2) B. Simon, Holonomy, the Quantum Adiabatic Theorem, and Berry’s Phase, Phys. Rev. Lett 51, 2167 (1983).
  • (3) M. V. Berry, Quantal phase factors accompanying adiabatic changes, Proc. R. Soc. London A 392, 45 (1984).
  • (4) P. Zanard and M. Rasetti, Holonomic quantum computation, Phys. Lett. A 264, 94 (1999).
  • (5) G. Falci, R. Fazio, G. M. Palma, J. Siewert, and V. Vedral, Detection of geometric phases in superconducting nanocircuits, Nature (London) 407, 355 (2000).
  • (6) D. M. Tong, E. Sjöqvist, L. C. Kwek, and C. H. Oh, Kinematic Approach to the Mixed State Geometric Phase in Nonunitary Evolution, Phys. Rev. Lett. 93, 080405 (2004).
  • (7) X. X. Yi, L. C. Wang, and T. Y. Zheng, Berry Phase in a Composite System, Phys. Rev. Lett. 92, 150406 (2004).
  • (8) X. X. Yi and W. Wang, Geometric phases induced in auxiliary qubits by many-body systems near their critical points, Phys. Rev. A 75, 032103 (2007).
  • (9) H. Y. Sun, P. L. Shu, C. Li, and X. X. Yi, Feedback control on geometric phase in dissipative two-level systems, Phys. Rev. A 79, 022119 (2009).
  • (10) S. Yin and D. M. Tong, Geometric phase of a quantum dot system in nonunitary evolution, Phys. Rev. A 79, 044303 (2009).
  • (11) H. D. Liu and X. X. Yi, Geometric phases in a scattering process, Phys. Rev. A 84, 022114 (2011).
  • (12) X. K. Song, H. Zhang, Q. Ai, J. Qiu, and F. G. Deng, Shortcuts to adiabatic holonomic quantum computation in decoherence-free subspace with transitionless quantum driving algorithm, New J. Phys. 18, 023001 (2016).
  • (13) J. Pachos and P. Zanardi, Quantum Holonomies for Quantum Computing, Int. J. Mod. Phys. B 15, 1257 (2001).
  • (14) L. M. Duan, J. I. Cirac, and P. Zoller, Geometric manipulation of trapped ions for quantum computation, Science 292, 1695 (2001).
  • (15) L. Faoro, J. Siewert, and R. Fazio, Non-Abelian holonomies, charge pumping and quantum computation with Josephson junctions, Phys. Rev. Lett. 90, 028301 (2003).
  • (16) P. Solinas, P. Zanardi, N. Zangdì, and F. Rossi, Semiconductor-based geometrical quantum gates, Phys. Rev. B 67, 121307(R) (2003).
  • (17) X. B. Wang and K. Matsumoto, Nonadiabatic conditional geometric phase shift with NMR. Phys. Rev. Lett. 87, 097901 (2001).
  • (18) S. L. Zhu and Z. D. Wang, Implementation of universal quantum gates based on nonadiabatic geometric phases. Phys. Rev. Lett. 89, 097902 (2002).
  • (19) E. Sjöqvist, D. M. Tong, L. M. Andersson, B. Hessmo, M. Johansson, and K. Singh, Non-adiabatic holonomic quantum computation, New J. Phys. 14, 103035 (2012).
  • (20) G. F. Xu, J. Zhang, D. M. Tong, E. Sjöqvist, and L. C. Kwek, Nonadiabatic Holonomic Quantum Computation in Decoherence-Free Subspaces, Phys. Rev. Lett. 109, 170501 (2012).
  • (21) G. F. Xu and G. L. Long, Universal Nonadiabatic Geometric Gates in Two-Qubit Decoherence-Free Subspaces, Sci. Rep. 4, 6814 (2014).
  • (22) G. F. Xu and G. L. Long, Protecting geometric gates by dynamical decoupling, Phys. Rev. A 90, 022323 (2014).
  • (23) J. Zhou, W. C. Yu, Y. M. Gao, and Z. Y. Xue, Cavity QED implementation of non-adiabatic holonomies for universal quantum gates in decoherence-free subspaces with nitrogen-vacancy centers, Opt. Express 23, 14027 (2015).
  • (24) Z. Y. Xue, J. Zhou, and Z. D. Wang, Universal holonomic quantum gates in decoherence-free subspace on superconducting circuits, Phys. Rev. A 92, 022320 (2015).
  • (25) G. F. Xu, C. L. Liu, P. Z. Zhao, and D. M. Tong, Nonadiabatic holonomic gates realized by a single-shot implementation, Phys. Rev. A 92, 052302 (2015).
  • (26) P. Z. Zhao, G. F. Xu, and D. M. Tong, Nonadiabatic geometric quantum computation in decoherence-free subspaces based on unconventional geometric phases, Phys. Rev. A 94, 062327 (2016).
  • (27) G. F. Xu, P. Z. Zhao, D. M. Tong, and E. Sjöqvist, Robust paths to realize nonadiabatic holonomic gates, Phys. Rev. A 95, 052349 (2017).
  • (28) Z.-Y. Xue, F.-L. Gu, Z.-P. Hong, Z.-H. Yang, D.-W. Zhang, Y. Hu, and J. Q. You, Nonadiabatic Holonomic Quantum Computation with Dressed-State Qubits, Phys. Rev. Appl. 7, 054022 (2017).
  • (29) A. A. Abdumalikov, J. M. Fink, K. Juliusson, M. Pechal, S. Berger, A. Wallraff, and S. Filipp, Experimental realization of non-abelian geometric gates, Nature (London) 496, 482 (2013).
  • (30) G. R. Feng, G. F. Xu, and G. L. Long, Experimental realization of nonadiabatic holonomic quantum computation, Phys. Rev. Lett. 110, 190501 (2013).
  • (31) C. Zu, W. B. Wang, L. He, W. G. Zhang, C. Y. Dai, F. Wang and L. M. Duan, Experimental realization of universal geometric quantum gates with solid-state spins, Nature (London) 514, 72 (2014).
  • (32) S. Arroyo-Camejo, A. Lazariev, S. W. Hell, and G. Balasubramanian, Room temperature high-fidelity holonomic single-qubit gate on a solid-state spin. Nat. Commun. 5, 4870 (2014).
  • (33) C. G. Yale, F. J. Heremans, B. B. Zhou, A. Auer, G. Burkard, and D. D. Awschalom, Optical manipulation of the Berry phase in a solid-state spin qubit, Nat. Photon. 10, 184 (2016).
  • (34) B. B. Zhou, P. C. Jerger, V. O. Shkolnikov, F. J. Heremans, G. Burkard, and D. D. Awschalom, Holonomic Quantum Control by Coherent Optical Excitation in Diamond, Phys. Rev. Lett. 119, 140503 (2017).
  • (35) Y. Sekiguchi, N. Niikura, R. Kuroiwa, H. Kano, and H. Kosaka, Optical holonomic single quantum gates with a geometric spin under a zero field, Nat. Photon. 11, 309 (2017).
  • (36) H. Li, Y. Liu, and G. Long, Experimental realization of single-shot nonadiabatic holonomic gates in nuclear spins, Sci. China Phys. Mech. Astron. 60, 080311 (2017).
  • (37) M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt, Cavity optomechanics, Rev. Mod. Phys. 86, 1391 (2014).
  • (38) M. Ludwig, A. H. Safavi-Naeini, O. Painter, and F. Marquardt, Enhanced Quantum Nonlinearities in a Two-Mode Optomechanical System, Phys. Rev. Lett. 109, 063601 (2012).
  • (39) K. Stannigel, P. Komar, S. J. M. Habraken, S. D. Bennett, M. D. Lukin, P. Zoller, and P. Rabl, Optomechanical Quantum Information Processing with Photons and Phonons, Phys. Rev. Lett. 109, 013603 (2012).
  • (40) A. Schliesser, O. Arcizet, R. Rivi¨¨re, G. Anetsberger, and T. J. Kippenberg, Resolved-sideband cooling and position measurement of a micromechanical oscillator close to the Heisenberg uncertainty limit, Nat. Phys. 5, 509 (2009).
  • (41) Y. C. Liu, Y. F. Xiao, X. Luan, and C. W. Wong, Dynamic Dissipative Cooling of a Mechanical Resonator in Strong Coupling Optomechanics, Phys. Rev. Lett. 110, 153606 (2013).
  • (42) R. W. Peterson, T. P. Purdy, N. S. Kampel, R. W. Andrews, P.-L. Yu, K. W. Lehnert, and C. A. Regal, Laser Cooling of a Micromechanical Membrane to the Quantum Backaction Limit, Phys. Rev. Lett. 116, 063601 (2016).
  • (43) J. D. Thompson, B. M. Zwickl, A. M. Jayich, F. Marquardt, S. M. Girvin, and J. G. E. Harris, Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane. Nature (London) 452, 72 (2008).
  • (44) S. Gröblacher, K. Hammerer, M.R. Vanner, and M. Aspelmeyer, Observation of strong coupling between a micromechanical resonator and an optical cavity field, Nature (London) 460, 724 (2009).
  • (45) S. Weis, R. Riviere, S. Deleglise, E. Gavartin, O. Arcizet, A. Schliesser, and T. J. Kippenberg, Optomechanically induced transparency, Science 330, 1520 (2010).
  • (46) A. H. Safavi-Naeini, T. P. M. Alegre, J. Chan, M. Eichenfield, M. Winger, Q. Lin, J. T. Hill, D. E. Chang, and O. Painter, Electromagnetically induced transparency and slow light with optomechanics, Nature (London) 472, 69 (2011).
  • (47) F. C. Lei, M. Gao, C. Du, Q. L. Jing, and G. L. Long, Three-pathway electromagnetically induced transparency in coupled-cavity optomechanical system, Opt. Express 23, 11508 (2015).
  • (48) V. Fiore, Y. Yang, M. C. Kuzyk, R. Barbour, L. Tian, and H. Wang, Storing optical information as a mechanical excitation in a silica optomechanical resonator, Phys. Rev. Lett. 107, 133601 (2011).
  • (49) Y. D. Wang and A. A. Clerk, Using interference for high fidelity quantum state transfer in optomechanics, Phys. Rev. Lett. 108, 153603 (2012).
  • (50) L. Tian, Adiabatic state conversion and pulse transmission in optomechanical systems, Phys. Rev. Lett. 108, 153604 (2012).
  • (51) T. A. Palomaki, J. W. Harlow, J. D. Teufel, R. W. Simmonds, and K. W. Lehnert, Coherent state transfer between itinerant microwave fields and a mechanical oscillator, Nature (London) 495, 210 (2013).
  • (52) W. Z. Zhang, J. Cheng, and L. Zhou, Quantum control gate in cavity optomechanical system, J. Phys. B 48, 015502 (2015).
  • (53) M. Asjad, P. Tombesi, and D. Vitali, Quantum phase gate for optical qubits with cavity quantum optomechanics, Opt. Express 23, 7786 (2015).
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