Quantum collisional classifier driven by information reservoirs

We investigate the open dynamics of a probe qubit weakly interacting with distinct qubit environments bearing quantum information. We show that the proposed dissipative model yields a binary classification of the reservoir qubits' quantum information in the steady state in the Bloch qubit parameter space, depending on the coupling rates. To describe the dissipation model dynamics, we have adopted the collision model, in which the input information parameters of the reservoir qubits are easily determined. We develop a generalized classification rule based on the results of the micromaserlike master equation where the classification can be described in terms of the Bloch parameters. Moreover, we show that the proposed classification scheme can also be achieved through quantum parameter estimation. Finally, we demonstrate that the proposed dissipative classification scheme is suitable for gradient descent-based supervised learning tasks.

Figure 1

A description of the suggested model. A classical perceptron is seen in (a). Linear classification of a perceptron in the input-output space is shown in (b). The schematic depiction of our proposed quantum classifier is shown in (c).

Figure 2

Temporal and input parameter-dependent steady-state response of the probe qubit by a single information reservoir relative to the number of collisions (nc) based on the $\langle k\rangle$ mean interaction number. Identical reservoir units $|\mathrm{\Psi }(\theta ,\varphi )\rangle$, (a) $\theta =0, \varphi =0$ and (b) $\theta =\pi , \varphi =0$ collided with the probe qubit prepared initially in the $|+\rangle =(|e\rangle +|g\rangle )/\sqrt{2}$ state. For the case of multiple information reservoirs, the binary classification result is obtained according to which hemisphere of the Bloch sphere (c) the steady-state magnetization value of the probe qubit will result. During the evolution, the Bloch vector trajectories with components $\langle {\sigma }_{\nu }\left(n\overline{t}\right)\rangle =\text{Tr}\left[{\varrho }_0\left(n\overline{t}\right){\sigma }_{\nu }\right]$ obtained for $\langle k\rangle =18\times {10}^3$ interactions and depicted as the insets of (a) and (b). In (d), if there is more than one information reservoir, the binary classification result is obtained according to the $\varphi$ amplitude parameter of the Bloch sphere, depending on the $\theta$ amplitude parameter in (c). During evolution, Bloch vector orbitals with components $\langle {\sigma }_{\nu }\left(n\overline{t}\right)\rangle =\text{Tr}\left[{\varrho }_0\left(n\overline{t}\right){\sigma }_{\nu }\right]$ were obtained for $\langle k\rangle =18\times {10}^3$ interactions and are shown as suffixes of (f) and (g). In (e), collision equilibration of noisy probe qubit magnetization by a single information reservoir delineated against the number of collisions (nc). This equilibration is depending on the average number of random interactions in a $T$ time interval. $\theta =0$, is the amplitude parameter of the reservoir units. ${\mathrm{\Gamma }}^{\theta }=2\times {10}^{-5}$, is the decay rate of the probe qubit. (f) The variation of the system qubit's steady-state magnetization. This variation is in the existence of a single information reservoir depending on $\theta$, with fixed $\varphi =0$ interacting with the system qubit. In (g) steady-state magnetization variation of the system qubit interacting with a single information reservoir with different amplitude parameters: blue line: $\theta =\pi /3$ fixed and depending on the $\varphi$; orange line: $\theta =2\pi /3$ fixed and depending on the $\varphi$. $\tau =3$, the probe qubit-reservoir interaction time and $J=0.01$, the coupling strength to the reservoir are dimensionless. They scale with ${\omega }_r$.

Figure 3

The probe qubit steady-state response coupled to the two information reservoirs. Here, the system qubit; $J_1=J/2+\delta J$ and $J_2=J/2-\delta J$ are connected to the reservoirs by their coupling coefficients. $\delta J$ is a fraction of $J$ with $J=0.01$. $\langle k\rangle$, the average number of interactions within the $T=1/{\mathrm{\Gamma }}^{\theta }$ time interval. With identical reservoir units of $|\mathrm{\Psi }({\theta }_i,{\varphi }_i)\rangle , {\theta }_1=0, {\varphi }_1=0$, and ${\theta }_2=\pi , {\varphi }_2=0$, the probe qubit prepared initially in $|+\rangle =(|e\rangle +|g\rangle )/\sqrt{2}$ state collisionally interacted. The longitudinal decay rate of the probe qubit is ${\mathrm{\Gamma }}^{\theta }=2\times {10}^{-5}$. Numerical and analytical verification was obtained for 0.774 (red dashed line) and $-0.492$ (purple dashed line) values, respectively. Other parameters and the initial state of the probe qubit are the same as in the caption of Fig. 2.

Figure 4

The system's steady-state response as a function of the linear change of the input parameters. A change in the steady-state magnetization of a system qubit connected to various surroundings that convey information and are parametrized by $\theta$ for (a) and (b). When the $\theta =0^{\circ },{10}^{\circ }...{180}^{\circ }$ azimuthal angles are present, $19\times 19=361$ dots are presented, each representing steady-state magnetization in the first and second environments. System to reservoirs coupling are fixed and equal to $J_1=J_2=0.01$. In both (a) and (b), the magnetization is plotted versus $\mathrm{\Theta }=\pi -({\theta }_1+{\theta }_2)$ for practical scaling as described in the text. (a) The change in the steady-state magnetization of the system qubit as a function of the average number of interactions $\langle k\rangle$ (16000) throughout a time interval $T=1/{\mathrm{\Gamma }}^{\varphi }$. (b) How the steady-state magnetization of the system qubit varies based on three distinct average interaction values of $\langle k\rangle$. (c) Linearly separated binary classification pattern of the system qubit in the steady state coupled to two information reservoirs with equal strengths. The pattern is obtained against the variation of $\theta$ parameters in range $0<{\theta }_{1,2}<\pi$, with the randomly chosen 32 amplitude parameter pairs. For (a)–(c) the remaining parameters and the initial state of the probe qubit are the same as in the caption of Fig. 2. For (d) and (e), the system qubit's steady-state magnetization varies according to the coupling coefficients $J_1=J/2+\delta J$ and $J_2=J/2-\delta J$, where $\delta J$ is a portion of $J$ with $J=0.01$. The average number of interactions $\langle k\rangle$ therein the times $T_1=1/{\mathrm{\Gamma }}^{\theta }$ and $T_2=1/{\mathrm{\Gamma }}^{\varphi }$. With identical reservoir units of $|\mathrm{\Psi }({\theta }_i,{\varphi }_i)\rangle$ with for (d) ${\theta }_1={\theta }_2=2\pi /3, {\varphi }_1=\pi /2, {\varphi }_2=3\pi /2$ and for (e) ${\theta }_1={\theta }_2=\pi /3, {\varphi }_1=\pi /2$, the probe qubit prepared initially in $|\mathrm{\Psi }(\theta ,\varphi )\rangle$ with $\theta =0, \varphi =0$ state collisionally interacted. The energy dissipation rate of the probe qubit is ${\mathrm{\Gamma }}^{\theta }=9.09\times {10}^{-6}$ and the dephasing time is ${\mathrm{\Gamma }}^{\varphi }=7.41\times {10}^{-6}$. (f) The system qubit's linearly separable classification pattern in steady state. The couplings between the information reservoirs and the system qubit were assumed to be equivalent. 32 amplitude parameter pairs were classified randomly in the ranges of $0<{\varphi }_1<\pi , \pi <{\varphi }_2<2\pi$, and ${\theta }_1={\theta }_2=\pi /3$. The interaction rate $r$ and the interaction duration $\tau$ are assumed as $r=0.16, \tau =3.00$, respectively, for (d)–(f).

Figure 5

The steady-state probe qubit's QFI depends on three distinct amplitude parameter pairs-${\theta }_1,{\theta }_2$-in the presence of two information reservoirs. QFI calculated against $0\le \delta \theta <\pi$. ${\theta }_2=11\pi /12$ for the amplitude parameter of the second reservoir and ${\theta }_1=\pi /12$ for the amplitude parameter of the first reservoir are fixed values that correspond to the second and third situations, respectively. Based on $\delta \theta$, as shown in the legend, the other parameters are determined. The probe qubit's coupling to both reservoirs is configured to be equal and $J1=J2=0.01$. The interaction rate is assumed to be $r=0.20$ and the interaction duration is assumed to be $\tau =3.00$.

Figure 6

The steady-state probe qubit's QFI depends on three distinct amplitude parameter pairs-${\varphi }_1,{\varphi }_2$-in the presence of two information reservoirs. Here, ${\theta }_1={\theta }_2=\pi /6$ amplitude parameter pairs are set to match to the other situations. QFI calculated against $0\le \delta \varphi <2\pi$. ${\varphi }_2=0.00$ for the amplitude parameter of the second reservoir and ${\varphi }_1=\pi$ for the amplitude parameter of the first reservoir are fixed values that correspond to the second and third situations, respectively. Based on $\delta \varphi$, as shown in the legend, the other parameters are determined. The probe qubit's coupling to both reservoirs is configured to be equal and $J1=J2=0.01$. The interaction rate is assumed to be $r=0.20$ and the interaction duration is assumed to be $\tau =3.00$.

Figure 7

GD outcomes based on three distinct learning rates' coupling coefficients. Here, the initial values $\langle {\sigma }_z^1\rangle =0.94, \langle {\sigma }_z^2\rangle =-0.10, g_1=0.002, g_2=0.05$, and ${\langle {\sigma }_z^0\rangle }_{des}^{ss}=0.42$, respectively.

Figure 8

The cost function's 3D surface plot according to $J_1$ and $J_2$. Here, the initial values for the dashed line are the learning rate $\eta =2.6\times {10}^{-5}, \langle {\sigma }_z^1\rangle =0.94, \langle {\sigma }_z^2\rangle =-0.10, J_1=0.002, J_2=0.05$, and ${\langle {\sigma }_z^0\rangle }_{des}^{ss}=0.42$, respectively.