Continuous dynamical decoupling and decoherence-free subspaces for qubits with tunable interaction

Abstract

Protecting quantum states from the decohering effects of the environment is of great importance for the development of quantum computation devices and quantum simulators. Here, we introduce a continuous dynamical decoupling protocol that enables us to protect the entangling gate operation between two qubits from the environmental noise. We present a simple model that involves two qubits which interact with each other with a strength that depends on their mutual distance and generates the entanglement among them, as well as in contact with an environment. The nature of the environment, that is, whether it acts as an individual or common bath to the qubits, is also controlled by the effective distance of qubits. Our results indicate that the introduced continuous dynamical decoupling scheme works well in protecting the entangling operation. Furthermore, under certain circumstances, the dynamics of the qubits naturally led them into a decoherence-free subspace which can be used complimentary to the continuous dynamical decoupling.

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Acknowledgements

İ.Y. is supported by the Project RVO 68407700 and RVO 14000 and funding from the project Centre for Advanced Applied Sciences, Registry No. CZ.02.1.01/0.0/0.0/16 019/0000778, supported by the Operational Programme Research, Development and Education, co-financed by the European Structural and Investment Funds and the state budget of the Czech Republic. F.F.F. has been supported by the Brazilian agencies FAPESP under Grant No. 2017/07787-7, by CNPq under Grant No. 302280/2017-0, and by INCT-IQ. G.K. is supported by the BAGEP Award of the Science Academy and the GEBIP program of the Turkish Academy of Sciences (TUBA). G.K. is also supported by the Scientific and Technological Research Council of Turkey (TUBITAK) under Grant No. 117F317.

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A: Derivation of the master equation

A: Derivation of the master equation

We introduce the details of the derivation of the master equation that governs the time evolution of our two-qubit system. First of all, we assume that each source of error, induced by the environment, is present separately. In other words, both independent environments for each of the qubits and the common environment, which the qubits together couple when they are close, are assumed to be present during the whole evolution. In this case, the total Hamiltonian is given by Eq. (1), with \(H_{\mathrm{int}}\) defined by Eq. (6) and \(H_{\mathrm{env}}\) defined by:

$$\begin{aligned} H_{\mathrm{env}}=\sum _{k} \omega _{k} {a_{k}}^{\dagger }a_{k}+\sum _{s=1}^{2}\sum _{k} \omega _{k}^{(s)}{a_{k}^{(s)}}^{\dagger }a_{k}^{(s)}, \end{aligned}$$
(13)

where \(\omega _{k}\) is the frequency of the kth normal mode of the environment that introduces common errors, with \(a_{k}\) and \({a_{k}}^\dagger \) being the annihilation and creation operators, respectively. In the second term, \(\omega _{k}^{(s)}\) is the frequency of the kth normal mode of the sth independent environment with respective annihilation (\(a_{k}^{(s)}\)) and creation (\({a_{k}^{(s)}}^{\dagger }\)) operators. It is important to emphasize that each error source is present independently of others since, in such a case, the total environment Hamiltonian could be written as a linear combination of the Hamiltonians of three distinct environments.

Having defined the total Hamiltonian, we utilize the Redfield master equation approach to calculate time evolution of the reduced two-qubit system:

$$\begin{aligned} \frac{{\hbox {d}}\tilde{\rho }_{S}\left( t\right) }{{\hbox {d}}t} = -\int _{0}^{t}Tr_{\mathrm{env}}\left\{ \left[ \tilde{H}_{\mathrm{int}}\left( t\right) ,\left[ \tilde{H}_{\mathrm{int}}\left( t^{\prime }\right) ,\rho _{R}\tilde{\rho }_{S}\left( t\right) \right] \right] \right\} {\hbox {d}}t^{\prime }. \end{aligned}$$
(14)

Here,

$$\begin{aligned} \tilde{H}_{\mathrm{int}}(t)=U_{\mathrm{env}}^\dagger (t) U_{\mathrm{gate}}^\dagger (t) U_{c}^\dagger (t) {H}_{\mathrm{int}}U_{c}(t) U_{\mathrm{gate}}(t) U_{\mathrm{env}}(t), \end{aligned}$$
(15)

where \(U_{\mathrm{env}}(t) = \exp (-iH_{\mathrm{env}}t)\), \(U_{\mathrm{gate}}(t)\) is the unitary evolution operator related to the time-dependent Hamiltonian \(H_{\mathrm{gate}}(t)\), given by Eq. (2), and \(U_c(t)\) is given by Eq. (9). In addition, \(\rho _E=\frac{1}{Z}\exp (-\beta H_{\mathrm{env}})\), where Z is the partition function \(Z = \mathrm{Tr_{env} }\exp (-\beta H_{\mathrm{env}})\) and \(\beta =1/k_BT\) with \(k_B\) being the Boltzmann constant, and T is the absolute temperature of the environment. Finally, \(\tilde{\rho }_{S}\left( t\right) = U_{\mathrm{gate}}^\dagger \left( t\right) U_{c}^\dagger \left( t\right) {\rho }_{S}\left( t\right) U_{c} \left( t\right) U_{\mathrm{gate}} \left( t\right) \) where \({\rho }_{S}(t)\) is the density operator in the Schrodinger picture.

To write an effective master equation in order to be solved numerically, we rewrite the interaction Hamiltonian as

$$\begin{aligned} H_{\mathrm{int}}=\sum _{n=1}^3 \varvec{\lambda }\cdot {\varvec{S}}_n(t)\otimes B_n+\varvec{\lambda ^*} \cdot {\varvec{S}}_n^{\dagger }(t) \otimes B_n^{\dagger }, \end{aligned}$$
(16)

where \(n=\{1,2,3\}\) represents each one of the three independent baths (one individual for each qubit plus a collective one for both qubits) and

$$\begin{aligned} {\varvec{S}}_1(t)= & {} \xi (t)\left[ {\varvec{\sigma }}^{(1)} + {\varvec{\sigma }}^{(2)}\right] ,\nonumber \\ {\varvec{S}}_2(t)= & {} \left[ 1-\xi (t)\right] {\varvec{\sigma }}^{(1)},\nonumber \\ {\varvec{S}}_3(t)= & {} \left[ 1-\xi (t)\right] {\varvec{\sigma }}^{(2)}, \end{aligned}$$
(17)

with \(B_n = \sum _k g_k^{n}a_k^{n}\), and \({\varvec{\lambda }} = \hat{z}\) for a dephasing environment and \({\varvec{\lambda }}=(\hat{x} + i \hat{y})\) for the amplitude damping. In the interaction picture, we can finally write the interaction Hamiltonian as:

$$\begin{aligned} \tilde{H}_{\mathrm{int}}\left( t\right) = \sum _{n=1}^3 \Lambda _n\left( t\right) \tilde{B}_n(t)+\Lambda ^{*}_n\left( t\right) \tilde{B}_n^{\dagger }\left( t\right) , \end{aligned}$$
(18)

where

$$\begin{aligned} \Lambda _n\left( t\right)= & {} U_{\mathrm{gate}}^\dagger \left( t\right) U_{c}^\dagger \left( t\right) \left[ {\varvec{\lambda } \cdot {\varvec{S}} }_n(t)\right] U_{c}\left( t\right) U_{\mathrm{gate}}\left( t\right) , \end{aligned}$$
(19)
$$\begin{aligned} \tilde{B}_n(t)= & {} U_{\mathrm{env}}^{\dagger }\left( t\right) (B_n) U_{\mathrm{env}}\left( t\right) . \end{aligned}$$
(20)

Replacing \(\tilde{H}_{\mathrm{int}}\left( t\right) \) in the master equation, we can write it in a more clear way:

$$\begin{aligned} \frac{\hbox {d}\tilde{\rho }_{S}\left( t\right) }{\hbox {d}t}&=\sum _{n=1}^3 \left[ \tilde{\rho }_{S}\left( t\right) \Lambda _n^{*}\left( t^{\prime }\right) ,\Lambda _n\left( t\right) \right] \mathcal{G}_{1}\left( t-t^{\prime }\right) \nonumber \\&\quad +\left[ \tilde{\rho }_{S}\left( t\right) \Lambda _n\left( t^{\prime }\right) ,\Lambda _n^{*}\left( t\right) \right] \mathcal{G}_{2}\left( t-t^{\prime }\right) +\left[ \Lambda _n^{*}\left( t\right) ,\Lambda _n\left( t^{\prime }\right) \tilde{\rho }_{S}\left( t\right) \right] \mathcal{G}_{1}^{*}\left( t-t^{\prime }\right) \!\!\nonumber \\&\quad +\left[ \Lambda _n\left( t\right) ,\Lambda _n^{*}\left( t^{\prime }\right) \tilde{\rho }_{S}\left( t\right) \right] \mathcal{G}_{2}^{*}\left( t-t^{\prime }\right) , \end{aligned}$$
(21)

where

$$\begin{aligned} {\mathcal G}_1(t)= & {} \int _0^\infty \hbox {d}\omega J(\omega )h(\omega )\exp (-i\omega t)\nonumber \\ {\mathcal G}_2(t)= & {} \int _0^\infty \hbox {d}\omega J(\omega )\exp (i\omega t)[h(\omega )+1], \end{aligned}$$
(22)

with \(h(\omega ) = \frac{1}{\exp (\beta \omega ) -1}\), and \(J(\omega ) = \omega \exp (-\omega /\omega _c)\) where \(\omega _c\) is the cutoff frequency. Note that to describe the environment spectrum, as usual, we assume that the number of environmental normal modes per unit frequency becomes very large. We also assume that all environments are identical since \({\mathcal G}_1(t)\) and \({\mathcal G}_2(t)\) are same for all baths. Further details can be found in Ref. [31], where the calculation has been developed for the case of time-independent interaction Hamiltonians.

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Yalçınkaya, İ., Çakmak, B., Karpat, G. et al. Continuous dynamical decoupling and decoherence-free subspaces for qubits with tunable interaction. Quantum Inf Process 18, 156 (2019). https://doi.org/10.1007/s11128-019-2271-0

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Keywords

  • Open quantum systems
  • Dynamical decoupling
  • Quantum control
  • Quantum gates