Quantum speedup of uncoupled multiqubit open system via dynamical decoupling pulses

Abstract

We present a method to accelerate the dynamical evolution of mutltiqubit open system by employing dynamical decoupling pulses (DDPs) when the qubits are initially in W-type states. Here the qubits are independent and coupled to local Lorentzian reservoirs. It is found that this speedup evolution can be achieved in both the weak-coupling regime and the strong-coupling regime. The essential physical mechanism behind the acceleration evolution is explained as a result of the joint action of the non-Markovianity of reservoirs and the excited-state population of qubits. It is shown that both the non-Markovianity and the excited-state population can be controlled by DDPs to realize the quantum speedup.

Appendices

Appendix 1: Derivation of Eq. (16)

In this appendix, we present a detailed derivation of the quantum state of the N-qubit open system at arbitrary time, i.e., the quantum state given by Eq. (16). Here we assume that the N-qubit system is initially prepared in the W-type states

$$\begin{aligned} \left| \psi _{0}\right\rangle= & {} \alpha _{1}\left| 100\ldots 0\right\rangle +\alpha _{2}\left| 010\ldots 0\right\rangle +\cdots +\alpha _{N}\left| 000\ldots 1\right\rangle \nonumber \\= & {} \sum ^N_{j=1}\alpha _{j}\left| j\right\rangle , \end{aligned}$$

(23)

where the state $\left| j\right\rangle$ means that only the jth qubit is in the excited state $\left| 1\right\rangle$ and the other $N-1$ qubits are in the ground state $\left| 0\right\rangle$. Then the density operator of the initial state can be written as

$$\begin{aligned} \rho _{0}= & {} \left| \psi _{0}\right\rangle \left\langle \psi _{0}\right| \nonumber \\= & {} \sum ^N_{j=1}\left| \alpha _{j}\right| ^{2}\left| j\right\rangle \left\langle j\right| +\sum _{j\ne k}\alpha _{j}\alpha _{k}^{*}\left| j\right\rangle \left\langle k\right| \end{aligned}$$

(24)

In the case of single-qubit system, Eqs. (6) give rise to the following relations,

$$\begin{aligned} \sum _{i=1,2}K_{i}\left| 0\right\rangle \left\langle 0\right| K_{i}^{\dag }= & {} \left| 0\right\rangle \left\langle 0\right| , \end{aligned}$$

(25a)

$$\begin{aligned} \sum _{i=1,2}K_{i}\left| 0\right\rangle \left\langle 1\right| K_{i}^{\dag }= & {} \kappa _{t}\left| 0\right\rangle \left\langle 1\right| ,\end{aligned}$$

(25b)

$$\begin{aligned} \sum _{i=1,2}K_{i}\left| 1\right\rangle \left\langle 0\right| K_{i}^{\dag }= & {} \kappa _{t}\left| 1\right\rangle \left\langle 0\right| , \end{aligned}$$

(25c)

$$\begin{aligned} \sum _{i=1,2}K_{i}\left| 1\right\rangle \left\langle 1\right| K_{i}^{\dag }= & {} \kappa _{t}^{2}\left| 1\right\rangle \left\langle 1\right| +\left( 1-\kappa _{t}^{2}\right) \left| 0\right\rangle \left\langle 0\right| . \end{aligned}$$

(25d)

As for N-qubit system, by the use of Eqs. (25), we can obtain the actions of the Kraus operators on $\left| j\right\rangle \left\langle k\right| (j, k=1,2,3, \cdots , N)$ as follows

$$\begin{aligned} \sum _{\mu _{1},\mu _{2},\ldots \mu _{N}}\left[ \otimes _{i=1}^{N}K_{\mu _{i}}\left( t\right) \right] \left| j\right\rangle \left\langle k\right| \left[ \otimes _{i=1}^{N}K_{\mu _{i}}^{\dag }\left( t\right) \right]= & {} \kappa _{t}^{2}\left| j\right\rangle \left\langle k\right| ,\quad (j \ne k)\end{aligned}$$

(26a)

$$\begin{aligned} \sum _{\mu _{1},\mu _{2},\ldots \mu _{N}}\left[ \otimes _{i=1}^{N}K_{\mu _{i}}\left( t\right) \right] \left| j\right\rangle \left\langle j\right| \left[ \otimes _{i=1}^{N}K_{\mu _{i}}^{\dag }\left( t\right) \right]= & {} \kappa _{t}^{2}\left| j\right\rangle \left\langle j\right| +\left( 1-\kappa _{t}^{2}\right) \nonumber \\&\left| 0\ldots 0\right\rangle \left\langle 0\ldots 0\right| . \end{aligned}$$

(26b)

Based on Eqs. (24), (26) and (12), the reduced density matrix of the N-qubit system at the time t can be expressed as

$$\begin{aligned} \rho _{t}= & {} \sum _{\mu _{1},\mu _{2},\ldots \mu _{N}}\left[ \otimes _{i=1}^{N}K_{\mu _{i}}\left( t\right) \right] \left[ \sum _{j}\left| \alpha _{j}\right| ^{2}\left| j\right\rangle \left\langle j\right| +\sum _{j\ne k}\alpha _{j}\alpha _{k}^{*}\left| j\right\rangle \left\langle k\right| \right] \left[ \otimes _{i=1}^{N}K_{\mu _{i}}^{\dag }\left( t\right) \right] \nonumber \\= & {} \sum _{j}\left| \alpha _{j}\right| ^{2}\bigg [ \kappa _{t}^{2}\left| j\right\rangle \left\langle j\right| +\left( 1-\kappa _{t}^{2}\right) \left| 0\ldots 0\right\rangle \left\langle 0\ldots 0\right| \bigg ]+\kappa _{t}^{2}\sum _{j\ne k}\alpha _{j}\alpha _{k}^{*}\left| j\right\rangle \left\langle k\right| \nonumber \\= & {} \kappa _{t}^{2}\rho _{0}+\left( 1-\kappa _{t}^{2}\right) \left| 0\ldots 0\right\rangle \left\langle 0\ldots 0\right| \nonumber \\= & {} P_{t}\left| \psi _{0}\right\rangle \left\langle \psi _{0}\right| +\left( 1-P_{t}\right) \left| 0\ldots 0\right\rangle \left\langle 0\ldots 0\right| , \end{aligned}$$

(27)

which exactly is the expression given by Eq. (16).

Appendix 2: The non-Markovianity measure

The non-Markovianity measure we employ here is based on the amount of information exchanged between the open system and its reservoir, which is defined as [73, 74]

$$\begin{aligned} \Gamma= & {} \max _{\rho _{1,2}(0)}\int _{\sigma (t)>0}\hbox {d}t\sigma (t), \end{aligned}$$

(28)

where $\sigma (t)$ denotes the rate of change of the trace distance

$$\begin{aligned} \sigma (t)= & {} \frac{\hbox {d}}{\hbox {d}t}\mathcal {D}\left[ \rho _{1}(t),\rho _{2}(t)\right] , \end{aligned}$$

(29)

and the trace distance is defined as

$$\begin{aligned} \mathcal {D}\left[ \rho _{1}(t),\rho _{2}(t)\right]= & {} \frac{1}{2}\text {Tr}\left| \rho _{1}(t)-\rho _{2}(t)\right| . \end{aligned}$$

(30)

Physically, $\sigma (t)<0 (\Gamma =0)$ holds for all dynamical semigroups and all time-dependent Markovian processes, while $\sigma (t)> 0 (\Gamma >0)$ holds for non-Markovian dynamics. It is should be noted that the maximum in Eq. (28) is taken over all initial-state pairs of the system. Here we only consider the non-Markovianity in a pair of qubit–reservoir. It has been proved that, in the case of a qubit, the optimal initial-state pairs are always antipodal points on the Bloch sphere [75]. So we can assume $\rho _{1,2}(0) =\left| \psi _{1,2}\left( 0\right) \right\rangle \left\langle \psi _{1,2}\left( 0\right) \right| , \left| \psi _{1}\left( 0\right) \right\rangle =\cos \theta \left| 1\right\rangle +\sin \theta \exp \left( i\varphi \right) \left| 0\right\rangle$ and $\left| \psi _{2}\left( 0\right) \right\rangle = \sin \theta \left| 1\right\rangle -\cos \theta \exp \left( i\varphi \right) \left| 0\right\rangle$ with $\theta \in \left[ 0,\pi /2\right] ,\varphi \in \left[ 0,2\pi \right]$. During the evolution time t, the optimal initial-state pair $\rho _{1}(0)$ and $\rho _{2}(0)$ evolve to the final-state pair $\rho _{1}(t)$ and $\rho _{2}(t)$. According to Eq. (4), we can obtain the simple expression of the trace distance between two final states at the time t

$$\begin{aligned} \mathcal {D}\left( \rho _{1}(t),\rho _{2}(t)\right)= & {} \sqrt{\cos ^{2}(2\theta )P_{t}^{2}+\sin ^{2}(2\theta )P_{t}}. \end{aligned}$$

(31)

What is more, as the Bloch sphere governed by Eq. (4) is rotation symmetrical with respect to the z axis, the most strongest oscillating direction is probably either the pole direction or any direction in the equatorial plane. In other words, the optimal initial-state pair should be either $\left\{ \left| 0\right\rangle ,\left| 1\right\rangle \right\} (\theta =0)$ or $\left\{ \left| +\right\rangle ,\left| -\right\rangle \right\} (\theta =\pi /4)$ with $\left| \pm \right\rangle =\left( \left| 0\right\rangle \pm e^{i\varphi }\left| 1\right\rangle \right) /\sqrt{2}$ [76–80].

Substituting Eq. (31) and Eq. (29) into Eq. (28), we can obtain the simple expression of the measure of non-Markovianity within the driving time $\tau$

$$\begin{aligned} \Gamma= & {} \text {max}\{\Gamma _{\theta =0},\Gamma _{\theta =\pi /4}\}, \end{aligned}$$

(32)

where the two non-Markovianity parameters $\Gamma _{\theta =0}$ and $\Gamma _{\theta =\pi /4}$ are defined by

$$\begin{aligned} \Gamma _{\theta =0}= & {} \int _{0,\dot{P}_{t}>0}^{\tau }\dot{P}_{t}\hbox {d}t,\end{aligned}$$

(33a)

$$\begin{aligned} \Gamma _{\theta =\pi /4}= & {} \int _{0,\dot{P}_{t}>0}^{\tau }\frac{\dot{P}_{t}}{2\sqrt{p_{t}}}\hbox {d}t. \end{aligned}$$

(33b)