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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Appendices
Appendix 1
Referring to Eq. (6)
The gate error energy is given by (refer “Appendix 1”)
where
Expanding this Frobenius norm, we get
Note that
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
must be expressed in terms of \(\widetilde{V}(t)\). Using
this constraint becomes
where \(A(t)= \hbox {e}^{iH_0t} A \hbox {e}^{-iH_0t}\). The quantity to be minimized is
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
Differentiate with respect to \(t_2\) and replace it by t,
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 \(U_\mathrm{d}\).
Appendix 2
Referring to Eq. (9)
The time-dependent Schrödinger equation in Eq. (39) leads to
where \(\langle m |x U(t)| n\rangle = \sum _{r=0}^{N-1} x_{mr} U_{rn} (t)\)
where
Let \(U_{mn} (t) = \hbox {e}^{-iE_mt}W_{mn}(t).\) We get from the above,
So
Iterating Eq. (40) twice, we obtain
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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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DOI: https://doi.org/10.1007/s11128-015-1059-0