Quantum-machine-learning channel discrimination

In the problem of quantum channel discrimination, one distinguishes between a given number of quantum channels, which is done by sending an input state through a channel and measuring the output state. This work studies applications of variational quantum circuits and machine-learning techniques for discriminating such channels. In particular, we explore (i) the practical implementation of embedding this task into the framework of variational quantum computing, (ii) training a quantum classifier based on variational quantum circuits, and (iii) applying the quantum kernel estimation technique. For testing these three channel discrimination approaches, we considered a pair of entanglement-breaking channels and the depolarizing channel with two different depolarization factors. For the approach (i), we address solving the quantum channel discrimination problem using widely discussed parallel and sequential strategies. We show the advantage of the latter in terms of better convergence with less quantum resources. Quantum channel discrimination with a variational quantum classifier (ii) allows one to operate even with random and mixed input states and simple variational circuits. The kernel-based classification approach (iii) is also found effective as it allows one to discriminate depolarizing channels associated not with just fixed values of the depolarization factor, but with ranges of it. Additionally, we discovered that a simple modification of one of the commonly used kernels significantly increases the efficiency of this approach. Finally, our numerical findings reveal that the performance of variational methods of channel discrimination depends on the trace of the product of the output states. These findings demonstrate that quantum machine learning can be used to discriminate channels, such as those representing physical noise processes.

Figure 1

A schematic of the parallel strategy. Provided with the opportunity of $p$ applications of Bob's channel ${\mathrm{\Phi }}_y$, Alice prepares a composite state ${\rho }_{PR}^{\mathrm{in}}$ of $p$ qubits of the register $P$ and $r$ qubits of the register $R$. The qubits of $P$ are sent through the channels ${\mathrm{\Phi }}_y$, while the qubits of $R$ remain untouched. In the scheme, the lines coming from ${\rho }_{PR}^{\mathrm{in}}$ indicate the subsystems of the corresponding registers. At the end, all $p+r$ qubits are measured with the POVM $\mathrm{\Pi }$.

Figure 2

A schematic of the sequential strategy. Provided with the opportunity of $p$ applications of Bob's channel ${\mathrm{\Phi }}_y$, Alice prepares the composite state ${\rho }_{PR}^{\mathrm{in}}$ of one qubit of the register $P$ and $r$ qubits of the register $R$. The qubit of $P$ is sent through the channels ${\mathrm{\Phi }}_y$, while the qubits of $R$ remain untouched. After the $j\mathrm{th}$ application of the channel ${\mathrm{\Phi }}_y$, Alice modifies the whole state by applying the channel ${\mathcal{E}}_j$. In the scheme, the lines coming from ${\rho }_{PR}^{\mathrm{in}}$ indicate the subsystems of the corresponding registers. At the end, all $1+r$ qubits are measured with the POVM $\mathrm{\Pi }$.

Figure 3

A variational quantum circuit implementing the parallel channel discrimination strategy from Fig. 1. The initial state ${\rho }_{PR}^{\mathrm{in}}$ is prepared from ${|0\rangle }_P^{\otimes p}\otimes {|0\rangle }_R^{\otimes r}$ via the unitary transformation $\mathcal{U}\left({\mathbf{\theta }}_{\mathit{0}}\right)$, while the unitary $\mathcal{U}\left({\mathbf{\theta }}_{\mathit{1}}\right)$ is used to rotate the measurement basis. If the number of allowed applications of the channel ${\mathrm{\Phi }}_y$ is $p$ and the number of ancillary qubits is $r$, the technique requires $(p+r)$ qubits. Note that $r$ could be set to zero.

Figure 4

A variational quantum circuit implementing the sequential channel discrimination strategy from Fig. 2. Alice's channels $\mathcal{E}={\left\{{\varepsilon }_j\right\}}_{j=1}^{p-1}$ are replaced by the parametrized unitaries ${\left\{U\left({\mathit{\theta }}_j\right)\right\}}_{j=1}^{p-1}$, where $p$ is the number of allowed applications of the channel ${\mathrm{\Phi }}_y$. The input state ${\rho }_{PR}^{\mathrm{in}}$ is prepared from ${|0\rangle }_P\otimes {|0\rangle }_R^{\otimes r}$ via the unitary transformation $U\left({\mathit{\theta }}_0\right)$, while $U\left({\mathit{\theta }}_p\right)$ is used to rotate the measurement basis. This method necessitates $(1+r)$ qubits, with one qubit in the register $P$ and $r$ qubits in the register $R$. In analogy with the parallel strategy, $r$ might be set to zero.

Figure 5

A layer of the hardware-efficient Ansatz with 4 input qubits and 12 variational parameters. Here, $R_{\sigma }\left(\theta \right)=e^{-ı\theta \sigma }$ with $\sigma \in \{X,Y,Z\}$ specifying the Pauli operator and $\theta \in [0,2\pi )$ being the optimization parameters.

Figure 6

The explicit variational quantum circuits trained for discriminating the entanglement-breaking channels (15) with $p=2$ channel applications. On the left shown is the circuit for the parallel strategy and on the right is for the sequential strategy. Note that the channels to be discriminated map two-qubit states into one-qubit states. Therefore, for the sequential strategy, after the first application of the channel ${\mathrm{\Phi }}_y$, one has to add an extra qubit in some state (in our case, $|0\rangle$).

Figure 7

Probability of successful discrimination ${\mathrm{p}}_{\mathrm{s}}$ between a pair of depolarizing channels with ${\alpha }_0$ and ${\alpha }_1$ achieved with the parallel (left panel) and sequential (right panel) strategies. With different colors shown are the results for specific number of Ansatz layers $l$. Marks connected by solid lines stand for the average probabilities obtained after 10 independent runs, and shaded areas are to show the standard deviation. The black solid line indicates the maximum achievable probability for the parallel strategy ${\mathrm{p}}_{\diamond }^{\mathrm{par}}$ for $p=2$ channel evaluations; for the sequential strategy, this line overlaps with the purple curve corresponding to $l=14$.

Figure 8

A schematic of the variational classifier of quantum channels. Bob prepares the states ${\mathrm{\Phi }}_y\left[\rho \right]\otimes \rho$ for Alice who then applies the unitary $U\left(\mathit{\theta }\right)$ and measures the resultant state in the computational basis. The measurement results are further used to compute the prediction value $p$ as yielded by Eq. (20). To train the circuit, Alice minimizes the square distances (21) between the predictions $p$ and the true labels $y$.

Figure 9

The accuracy of quantum channel discrimination as obtained on the test set for depolarizing channels with ${\alpha }_0$ and ${\alpha }_1$. The left, center, and right panels show the accuracy of $U_1$-, $U_2$-, and $U_3$-based classifiers defined in (22, 23, 24), respectively. The training and test sizes are $N_{\text{train}}=N_{\text{test}}=1000$.

Figure 10

The accuracy of quantum classifiers trained to discriminate depolarizing channels corresponding to the intervals of the depolarization factor $I_1, I_2, I_3$, and $I_4$ and defined by Eqs. (28, 29, 30, 31), respectively. The corresponding classification accuracies are $A_1=0.999, A_2=1.0, A_3=0.67$, and $A_4=0.519$. The input state is ${\rho }^{\mathrm{in}}=|+\rangle \langle +|$, and the sizes of the training and test sets are $N_{\text{train}}=100$ and $N_{\text{test}}=1000$. The vertical axis shows the normalized prediction value determined by (27). The color intensity features the density of data points.

Figure 11

Trace of the product (left) and the diamond distance (right) for the parallel discrimination strategy versus depolarization factors $({\alpha }_0,{\alpha }_1)$ for $p=2$ channel applications. Herein, ${\mathrm{\Phi }}_y\equiv \mathrm{\Phi }\left({\alpha }_y\right)$ is the depolarizing channel (18), and ${\rho }_y=\mathrm{\Phi }\left({\alpha }_y\right)\left[\rho \right]$.

Figure 12

The Pearson correlation coefficients between the trace of the product, diamond distance, and the average successful discrimination probabilities. On the left are the results for the parallel strategy, and on the right are for the sequential strategy. Each data point is obtained by averaging out five independent runs. By solid lines, data points are fitted by the functions of $l$, the number of layers of the hardware-efficient Ansatz. We used $f\left(l\right)=l^{-1/a}$ for fitting the trace of the product, and $g\left(l\right)=1-e^{-bl}$ for the diamond distance.

Figure 13

The accuracy of the kernel-based classifier obtained on the test set for the intervals $I_4$ defined in (31). From left to right given are the accuracy for the classifiers based on the kernel (32) for different $n\in \{1,2,3,4\}$. The corresponding classification accuracies are $A_1=0.522, A_2=0.708, A_3=0.941$, and $A_4=0.981$. The input state is ${\rho }^{\mathrm{in}}=|+\rangle \langle +|$, and the sizes of the training and test sets are $N_{\text{train}}=100$ and $N_{\text{test}}=1000$. The vertical axis shows the normalized prediction value defined in (27). The color intensity features the density of data points.

Figure 14

The accuracy of the kernel-based classifier obtained on the test set for the intervals $I_1$ defined in (28). From left to right given are the accuracy for the classifiers based on the kernel (32) for different $n\in \{1,2,3,4\}$. The corresponding classification accuracies are $A_1=0.595, A_2=0.847, A_3=0.869$, and $A_4=0.858$. The input states ${\rho }^{\mathrm{in}}$ are random and mixed, and the sizes of the training and test sets are $N_{\text{train}}=100$ and $N_{\text{test}}=1000$. The vertical axis shows the normalized prediction value defined in (27). The color intensity features the density of data points.