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Realization of commonly used quantum gates using perturbed harmonic oscillator

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Abstract

This paper deals with the approximate design of quantum unitary gates using perturbed harmonic oscillator dynamics. The harmonic oscillator dynamics is perturbed by a small time-varying electric field which leads to time-dependent Schrödinger equation. The corresponding unitary evolution after time T is obtained by approximately solving the time-dependent Schrödinger equation. The aim of this work is to minimize the discrepancy between a given unitary gate and the gate obtained by evolving the oscillator in the weak electric field over [0, T]. The proposed algorithm shows that the approximate design is able to realize the Hadamard gate and controlled unitary gate on three-qubit arrays with high accuracy.

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References

  1. Schiff, L.I.: Quantum Mechanics, 3rd edn. McGraw Hill, New York (1955)

    MATH  Google Scholar 

  2. Dirac, P.A.M.: The Principles of Quantum Mechanics, 4th edn, pp. 108–178. Oxford University Press, New York (1958)

    MATH  Google Scholar 

  3. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2001)

    Google Scholar 

  4. Pavicic, M.: Quantum Computation and Quantum Communication: Theory and Experiments. Springer (2006)

  5. Nakahara, M., Ohmi, T.: Quantum Computing: From Linear Algebra to Physical Realizations. CRC Press, Boca Raton (2008)

    Book  Google Scholar 

  6. McMahon, D.: Quantum Computing Explained. Wiley, New York (2007)

    Book  Google Scholar 

  7. Barenco, A., Bennett, C.H., Cleve, R., DiVincenzo, D.P., Margolus, N., Shor, P., Sleator, T., Smolin, J.A., Weinfurter, H.: Elementary gates for quantum computation. Phys. Rev. A 52, 3457–3467 (1995)

    Article  ADS  Google Scholar 

  8. Perelomov, A.M.: Superpositions of the \(SU(1, 1)\). Phys. Lett. A 193, 121–125 (1994)

    Article  MathSciNet  Google Scholar 

  9. Glauber, R.J.: Coherent and incoherent states of radiation field. Phys. Rev. 131, 2766–2788 (1963)

    Article  MathSciNet  ADS  Google Scholar 

  10. Fox, M.: Quantum Optics: An Introduction, vol. 6. Oxford University Press, New York (2005)

    Google Scholar 

  11. Klauder, J.R., Skagerstam, B.: Coherent States. World Scientific, Singapore (1985)

    Book  MATH  Google Scholar 

  12. Perelomov, A.M.: Generalized Coherent States and Their Applications Texts and Monographs in Physics. Springer, New York (1986)

    Book  Google Scholar 

  13. Gazeau, J.P.: Coherent States in Quantum Physics. Wiley-VCH, Berlin (2009)

    Book  Google Scholar 

  14. Zhang, W., Feng, D., Gilmore, R.: Coherent states—theory and some applications. Rev. Mod. Phys. 62, 867–927 (1990)

    Article  MathSciNet  ADS  Google Scholar 

  15. Combescure, M., Robert, D.: Coherent States and Applications, Mathematical Physics. Springer, New York (2012)

    Book  Google Scholar 

  16. Ali, S.T., Antoine, J.P., Bagarello, F., Gazeau, J.P.: Coherent states: a contemporary panorama. J. Phys. A Math. Theor. 45, 240301 (2012)

    Article  MathSciNet  Google Scholar 

  17. Kato, T.: Perturbation Theory for Linear Operators, 2nd edn. Springer, Berlin (1980)

    MATH  Google Scholar 

  18. Kamran, N., Olver, P.J.: Lie algebras of differential operators and Lie-algebraic potentials. J. Math. Anal. Appl. 145, 342–356 (1990)

    Article  MATH  MathSciNet  Google Scholar 

  19. Bach, V.: Schrödinger operators. Encycl. Math. Phys. 1, 487–494 (2006)

  20. Altafini, C.: On the generation of sequential unitary gates from continuous time Schrödinger equations driven by external fields. Quantum Inf. Process. 1, 207–224 (2002)

    Article  MathSciNet  Google Scholar 

  21. Zhang, Y., Kauffman, L.H., Ge, M., Baxterizations, Y.: Universal quantum gates and Hamiltonians. Quant. Inf. Process. 4, 159–197 (2005)

  22. Altafini, C.: Parameter differentiation and quantum state decomposition for time varying Schrödinger equations. Rep. Math. Phys. 52(3), 381–400 (2003)

  23. Hirota, O.: Some remarks on a conditional unitary operator. Phys. Lett. A 155(6–7), 343–347 (1991)

    Article  MathSciNet  ADS  Google Scholar 

  24. Parthasarathy, H.: Mathematical Ideas for Signal Processing Application. I.K. International Publishing House, New Delhi (2013)

  25. Shepherd, D.J.: On the role of Hadamard gates in quantum circuits. Quantum Inf. Process. 5, 161–177 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  26. Mohammad, N.S., Mehmandoost, M.: Realization of quantum Hadamard gate by applying optimal control fields to a spin qubit. In: 2nd International Conference on Mechanical and Electronics Engineering, vol. 2, pp. 292–296 (2010)

  27. Kumar, P.: Direct implementation of an N-qubit controlled-unitary gate in a single step. Quantum Inf. Process. 12, 1201–1223 (2013)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  28. Altintas, A.A., Ozaydin, F., Yesilyurt, C., Bugu, S., Arik, M.: Constructing quantum logic gates using q-deformed harmonic oscillator algebras. Quantum Inf. Process. 12, 1035–1044 (2014)

    Article  MathSciNet  ADS  Google Scholar 

Download references

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Correspondence to Kumar Gautam.

Appendices

Appendix 1

Referring to Eq. (6)

$$\begin{aligned} W(t) = I-i\epsilon \int _{0<t_1<T}\widetilde{V}(t_1)\hbox {d}t_1 -\epsilon ^2\int _{0<t_2<t_1<T}\widetilde{V}(t_1)\widetilde{V}(t_2)\hbox {d}t_2\hbox {d}t_1 +O\left( \epsilon ^3\right) \end{aligned}$$
(33)

The gate error energy is given by (refer “Appendix 1”)

$$\begin{aligned} {\mathbb {E}}&= \left\| U_\mathrm{d}-U(T)\right\| ^2 = \left\| U_\mathrm{d}-\hbox {e}^{-iH_0 T}W(T)\right\| ^2= \left\| {\widetilde{{U}_\mathrm{d}}}-W(T)\right\| ^2\\ {\mathbb {E}}&= \left\| {\widetilde{{U}_\mathrm{d}}}+ i\epsilon \int _{0<t_1<T}\widetilde{V}(t_1)\hbox {d}t_1 +\epsilon ^2\int _{0<t_2<t_1<T}\widetilde{V}(t_1)\widetilde{V}(t_2)\hbox {d}t_2\hbox {d}t_1 +O\left( \epsilon ^3\right) \right\| ^2 \end{aligned}$$

where

$$\begin{aligned} {\widetilde{{U}_\mathrm{d}}} = \hbox {e}^{iH_0t} U_d-I \end{aligned}$$

Expanding this Frobenius norm, we get

$$\begin{aligned} \Vert U_d-U(T)\Vert ^2&= \left\| {\widetilde{U_\mathrm{d}}}\right\| ^2 \nonumber \\&\quad +\, \epsilon ^2 \int _{{0<t_1},{t_2<T}}{\mathrm {Tr}}\left[ \widetilde{V}(t_1)\widetilde{V}(t_2)\right] \hbox {d}t_1\hbox {d}t_2\nonumber \\&\quad +\,i\epsilon {\mathrm {Tr}}\left[ {\widetilde{U_\mathrm{d}^*}}\int _{0<t_1<T}\widetilde{V}(t_1)\hbox {d}t_1\right] \nonumber \\&\quad -\, i\epsilon {\mathrm {Tr}}\left[ {\widetilde{U_\mathrm{d}}}\int _{0<t_1<T}\widetilde{V}(t_1)\hbox {d}t_1\right] \nonumber \\&\quad +\,\nonumber \epsilon ^2{\mathrm {Tr}}\left[ {\widetilde{U_\mathrm{d}^*}} \int _{0<t_2<t_1<T}\widetilde{V}(t_1)\widetilde{V}(t_2)\hbox {d}t_2\hbox {d}t_1\right] \nonumber \\&\quad +\,\epsilon ^2{\mathrm {Tr}}\left[ {\widetilde{U_\mathrm{d}}} \int _{0<t_2<t_1<T}\widetilde{V}(t_2)\widetilde{V}(t_1)\hbox {d}t_2\hbox {d}t_1\right] \nonumber \\&\quad +\,O\left( \epsilon ^3\right) \end{aligned}$$
(34)

Note that

$$\begin{aligned} {\widetilde{V}}^*(t)&={\widetilde{V}}(t) \end{aligned}$$
(35)
$$\begin{aligned} \left( {\widetilde{V}}(t_1){\widetilde{V}}(t_2)\right) ^{*}&=\widetilde{V}(t_2)\widetilde{V}(t_1) \end{aligned}$$
(36)

We calculate the variational derivative with respect to \(\widetilde{V}(t)\) of the last function taking into account energy constraint using Lagrange’s multiplier. The energy constraint

$$\begin{aligned} E = \int _0^T {\mathrm {Tr}}\left[ \textit{AV}^2(t)\right] \hbox {d}t \end{aligned}$$

must be expressed in terms of \(\widetilde{V}(t)\). Using

$$\begin{aligned} \widetilde{V}(t) = \hbox {e}^{iH_0t} V(t)\hbox {e}^{-iH_0t}, \end{aligned}$$

this constraint becomes

$$\begin{aligned} E = \int _0^T {\mathrm {Tr}}\left[ A(t)\widetilde{V}^2(t)\right] \hbox {d}t \end{aligned}$$

where \(A(t)= \hbox {e}^{iH_0t} A \hbox {e}^{-iH_0t}\). The quantity to be minimized is

$$\begin{aligned} \Vert U_\mathrm{d}-U(T)\Vert ^2-\lambda \int _0^T {\mathrm {Tr}}\left[ A(t)\widetilde{V}^2(t)\right] \hbox {d}t \end{aligned}$$
(37)

where \(\lambda \) is the Lagrange multiplier. We set the variational derivative of the above equation with respect to \(\widetilde{V}(t)\) to zero. The coefficient of \(\delta \widetilde{V}(t_2)\) is

$$\begin{aligned}&2\epsilon ^2\int _0^T {\widetilde{V}(t_1)}\hbox {d}t_1+i\epsilon U_\mathrm{d}^{*}-i\epsilon U_\mathrm{d} +\epsilon ^2 {\widetilde{{U}_\mathrm{d}^{*}}}\int _{t_2}^T {\widetilde{V}(t_1)}\hbox {d}t_1\\&\quad \qquad +\,\epsilon ^2\left( \int _0^{t_2} {\widetilde{V}(t_1)}\hbox {d}t_1\right) {\widetilde{{U}_\mathrm{d}^{*}}} +\epsilon ^2\left( \int _{t_2}^T {\widetilde{V}(t_1)}\hbox {d}t_1\right) U_\mathrm{d}\\&\quad \qquad +\,\epsilon ^2\left( \int _0^{t_2} {\widetilde{V}(t_1)}\hbox {d}t_1\right) U_\mathrm{d} -\lambda \left( A(t_2)\widetilde{V}(t_2) +\widetilde{V}(t_2)A(t_2)\right) = 0 \end{aligned}$$

Differentiate with respect to \(t_2\) and replace it by t,

$$\begin{aligned} -\epsilon ^2 {\widetilde{{U}_\mathrm{d}^*}} \widetilde{V}(t)+\epsilon ^2\widetilde{V}(t)U_\mathrm{d}^{*} -\epsilon ^2 \widetilde{V}(t)U_\mathrm{d}+\epsilon ^2 U_\mathrm{d}\widetilde{V}(t)- \lambda \left( A\widetilde{V}^{\prime }(t)+\widetilde{V}^{\prime }(t)A\right) =0 \end{aligned}$$

We have assumed that A(t) is constant operator in order to simplify the calculations and replacing \(\epsilon \) by 1, \(\widetilde{V}\) by V and \({\widetilde{{U}_\mathrm{d}}}\) by \(U_\mathrm{d}\).

$$\begin{aligned} \lambda \textit{AV}^{\prime }+\lambda V^{\prime } A = \left( U_\mathrm{d}-U_\mathrm{d}^{*}\right) V+V\left( U_\mathrm{d}^{*}-U_\mathrm{d}\right) \end{aligned}$$
(38)

Appendix 2

Referring to Eq. (9)

$$\begin{aligned} i\frac{\hbox {d}}{\hbox {d}t}\left\langle m |U(t)| n\right\rangle = \left\langle m | H_0 U(t)| n\right\rangle + \epsilon \left\langle m |V(t)U(t)| n\right\rangle \end{aligned}$$
(39)

The time-dependent Schrödinger equation in Eq. (39) leads to

$$\begin{aligned} i\frac{\hbox {d}U_{mn} (t)}{\hbox {d}t} = E_m U_{mn} (t) - \epsilon q E(t)\left\langle m |xU(t)| n\right\rangle \end{aligned}$$

where \(\langle m |x U(t)| n\rangle = \sum _{r=0}^{N-1} x_{mr} U_{rn} (t)\)

$$\begin{aligned} i\frac{\hbox {d}U_{mn} (t)}{\hbox {d}t}&= E_m U_{mn} (t) - \epsilon q E(t)\sum _{r=0}^{N-1} x_{mr} U_{r n} (t)\\ x_{mn}&= \langle m |x| n\rangle = \int _{-\infty }^\infty {xu_m(x)u_n(x)\hbox {d}x} \end{aligned}$$

where

$$\begin{aligned} x&= \frac{a+a^\dag }{\sqrt{2}}\\ x_{mn}&=\frac{1}{\sqrt{2}}\langle m |(a+a^\dag )|n\rangle \\ x_{mn}&= \frac{1}{\sqrt{2}}\left( {\sqrt{n} \delta _{m,n-1}+ \sqrt{m} \delta {n,m-1}}\right) \end{aligned}$$

Let \(U_{mn} (t) = \hbox {e}^{-iE_mt}W_{mn}(t).\) We get from the above,

$$\begin{aligned} W_{mn}^{\prime }(t) = i\epsilon q E(t)\sum _{r=0}^{N-1} x_{mn}\hbox {e}^{-iE_rt}W_{rn}(t) \end{aligned}$$

So

$$\begin{aligned} W_{mn}(T) = \delta _{mn}+i\epsilon q \sum _{r=0}^{N-1} \int _0^T E(t_1)x_{mr} \hbox {e}^{-iE_rt}W_{rn}(t_1)\hbox {d}t_1 \end{aligned}$$
(40)

Iterating Eq. (40) twice, we obtain

$$\begin{aligned} W_{mn}(T)&= \delta _{mn}+ i\epsilon q \int _0^T x_{mn} E(t_1) \hbox {e}^{-iE_nt_1}\hbox {d}t_1 \\&\quad -\,\epsilon ^2 q^2 \int _{0<t_2<t_1<T} E(t_1)x_{mr} \hbox {e}^{-iE_rt_1} E(t_2)x_{rn} \hbox {e}^{-iE_nt_2}\hbox {d}t_2\hbox {d}t_1 +O\left( \epsilon ^3\right) \\ W_{mn}(T)&= \delta _{mn}+ i\epsilon q x_{mn}\int _0^T E(t_1) \hbox {e}^{-iE_nt_1}\hbox {d}t_1\\&\quad -\,\epsilon ^2 q^2\sum _{r=0}^{N-1} x_{mr}x_{rn}\int _{0<t_2<t_1<T} E(t_1)E(t_2) \hbox {e}^{-i\left( E_rt_1 + E_nt_2\right) }\hbox {d}t_2\hbox {d}t_1 +O\left( \epsilon ^3\right) \end{aligned}$$

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Gautam, K., Rawat, T.K., Parthasarathy, H. et al. Realization of commonly used quantum gates using perturbed harmonic oscillator. Quantum Inf Process 14, 3257–3277 (2015). https://doi.org/10.1007/s11128-015-1059-0

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