Quantum approximate optimization algorithm for Bayesian network structure learning

Abstract

Bayesian network structure learning is an NP-hard problem that has been faced by a number of traditional approaches in recent decades. Currently, quantum technologies offer a wide range of advantages that can be exploited to solve optimization tasks that cannot be addressed in an efficient way when utilizing classic computing approaches. In this work, a specific type of variational quantum algorithm, the quantum approximate optimization algorithm, was used to solve the Bayesian network structure learning problem, by employing $3n(n-1)/2$ qubits, where n is the number of nodes in the Bayesian network to be learned. Our results showed that the quantum approximate optimization algorithm approach offers competitive results with state-of-the-art methods and quantitative resilience to quantum noise. The approach was applied to a cancer benchmark problem, and the results justified the use of variational quantum algorithms for solving the Bayesian network structure learning problem.

Appendices

Appendix A: Kraus operators for noise simulation

Following the formalism of Kraus operators [14], each quantum channel $\varepsilon$ is defined given a set of matrices $E_i$, which are applied to a quantum state, where i runs through all the operators considered for a given channel and k is the number f operators that define the noise channel. Thus, given a state $\psi$, the resulting state after applying a quantum channel is

$$\begin{aligned} \varepsilon (\psi ) = \sum _k E_i \psi E_i^\dagger , \end{aligned}$$

(14)

where the Kraus operators must meet the restriction $\sum _k E_i E_i^\dagger = 1$. Note that all the operators defined in this section are one-qubit operations ($E_i^{(1)}$), and that all the channels are parameterized by $\omega \in [0, 1]$, which regulates the probability of occurrence of the respective noise.

1.1 A.1 Amplitude dam** channel

This quantum operation describes the energy dissipation of the quantum states. The operation over a one-qubit system is defined as

$$\begin{aligned} \varepsilon _{AD}(\psi ) = E_1 \psi E_1^\dagger + E_2 \psi E_2^\dagger , \end{aligned}$$

where $E_1$ and $E_2$ are defined as

$$\begin{aligned} E_1^{(1)} = \begin{pmatrix} 1 &{} 0\\ 0 &{} \sqrt{1-\omega } \end{pmatrix},\quad E_2^{(1)} = \begin{pmatrix} 0 &{} \sqrt{\omega }\\ 0 &{} 0 \end{pmatrix}. \end{aligned}$$

The $E_2$ operation converts a $|{1}\rangle$ to a $|{0}\rangle$, representing the physical process of the environment energy lost. $E_1$ leaves $|{0}\rangle$ unaltered but decreases the amplitude of a $|{1}\rangle$ state. Physically, this occurs because some energy was not lost to the environment, and thus the quantum state is more likely to be in the $|{0}\rangle$ state, rather than in the $|{1}\rangle$ state.

1.2 A.2 Phase dam** channel

This quantum operation describes the loss of quantum information without loss of energy. The operation over a one-qubit system is defined as

$$\begin{aligned} \varepsilon _{PD}(\psi ) = E_1 \psi E_1^\dagger + E_2 \psi E_2^\dagger , \end{aligned}$$

where $E_1$ and $E_2$ are defined as

$$\begin{aligned} E_1^{(1)} = \begin{pmatrix} 1 &{} 0\\ 0 &{} \sqrt{1-\omega } \end{pmatrix},\quad E_2^{(1)} = \begin{pmatrix} 0 &{} 0\\ 0 &{} \sqrt{\omega } \end{pmatrix}. \end{aligned}$$

In this case, the $E_1$ operator acts in the same way as in the case of the amplitude dam** noise channel, leaving $|{0}\rangle$ unchanged, but reducing the amplitude of $|{1}\rangle$. However, in this case, $E_2$ also reduces the amplitude of the $|{1}\rangle$ state, but does not change it to $|{0}\rangle$.

1.3 A.3 Depolarizing channel

This quantum operation describes the depolarization of a qubit. That is, with certain probability $\omega$ a quantum state is replaced by the mixed state I/2. The operation over a one-qubit system is defined as

$$\begin{aligned} \varepsilon _D(\psi ) = \omega \frac{I}{2} + (1-\omega )\psi . \end{aligned}$$

(15)

Despite the fact that Eq. (15) does not involve any Kraus operators, it is possible to define the depolarizing channel with the following Kraus operators: [14],

$$\begin{aligned} \begin{aligned} E_1^{(1)}&= \sqrt{1-\frac{3\omega }{4}} I_2,\quad E_2^{(1)} = \sqrt{\frac{\omega }{4}} \sigma ^{x}, \\ E_3^{(1)}&= \sqrt{\frac{\omega }{4}} \sigma ^{y},\quad \quad \quad E_4^{(1)} = \sqrt{\frac{\omega }{4}} \sigma ^{z}, \end{aligned} \end{aligned}$$

(16)

where $\sigma ^{x}$, $\sigma ^{y}$ and $\sigma ^{z}$ are the Pauli operators, and $\omega = 1$ implies the output state $\varepsilon _D(\psi )$ to be the mixed state I/2.

The application of the Kraus operators defined in Eq. (16) for two-qubits quantum system is defined as:

Appendix B: Complementary results for noise simulation

Table 2 shows the detailed mean and standard deviation of the experimental results represented in Fig. 8.

Table 2 Mean best costs ($\mu$) and standard deviations ($\sigma$) found for different values of the $\omega$ parameter over 50 executions of the QAOA approach for the BNSL problem, and the mean number of iterations ($\mu$) and standard deviations ($\sigma$) until convergence for 50 executions. AD, PD and DE represent amplitude dam**, phase dam**, and depolarizing simulated errors, respectively