Explicit demonstration of initial state construction in artificial neural networks using NetKet and IBM Q experience platform

Abstract

Quantum neural networks have gained significant interest in recent times for the representation of many-body states and classical simulation of quantum computation. Here, we discuss the methods to generate specific initial states in the Restricted Boltzmann Machines analogous to those used as the initial states while performing quantum computation in quantum computers such as IBM Q. We validate our approach by applying the Pauli X gate to the single-qubit and two-qubit initial states using NetKet and compare the results to that obtained by performing the same task in the IBM quantum computer. We find that using this approach, the RBM neural networks can represent desired quantum states with high accuracy. Thus, this method is promising to mimic quantum computation classically in neural networks by using specific initial states.

Supplementary material

1.1 Derivation of the changes in weight parameters to implement the Pauli X gate

Following the notation used in [4], we find that the $X_l$ gate should flip the spin of the $l_{th}$ qubit and should not affect the other qubits. Thus, the RBM amplitudes should satisfy

$$\begin{aligned} \langle \mathscr {B}|X_l|\Psi _{W}\rangle = \langle B_1 \cdots \bar{B}_{l} \cdots B_N|\Psi _W\rangle \end{aligned}$$

(4)

Since $\bar{B}_l$= $-B$ as a $+1$ spin should be flipped to $-1$ spin and vice versa, the equations that should be satisfied are

$$\begin{aligned} -B_l W_{lk} + b_k= & {} B_{l}W'_{lk}+ b'_k \end{aligned}$$

(5a)

$$\begin{aligned} -B_la_l= & {} B_la'_l +C \end{aligned}$$

(5b)

whose solution is

• 1. $W'_{lk}=-W_{lk}$

• 2. $b'_{k}=b_{k}$

• 3. $a'_{l}=-a_{l}$

• 4. $C=0$

• 5. All other weight parameters are left unchanged

1.2 Preparation on states in IBM Q experience platform

The methods to generate desired initial states (on which the Pauli X gates are applied in tables VII and VIII of the paper) in the IBM quantum computer are represented pictorially in the following figures. The results are obtained by running the experiment using the ibmqx2 backend with 8192 number of slots (Fig. 6).

Fig. 6

Preparation of states in the IBM network: Each qubit starts in the $|0\rangle$ state. The desired states are prepared by applying various gates to each of the qubits. Here, we have performed a spin z measurement of each of the qubits at the end of the circuit to verify that the qubits are in the desired state