Solving quantum master equations with deep quantum neural networks

Deep quantum neural networks may provide a promising way to achieve a quantum learning advantage with noisy intermediate-scale quantum devices. Here, we use deep quantum feed-forward neural networks capable of universal quantum computation to represent the mixed states for open quantum many-body systems and introduce a variational method with quantum derivatives to solve the master equation for dynamics and stationary states. Owning to the special structure of the quantum networks, this approach enjoys a number of notable features, including an efficient quantum analog of the back-propagation algorithm, resource-saving reuse of hidden qubits, general applicability independent of dimensionality and entanglement properties, as well as the convenient implementation of symmetries. As proof-of-principle demonstrations, we apply this approach to both one-dimensional transverse field Ising and two-dimensional $J_1\text{−}{J}_2$ models with dissipation, and show that it can efficiently capture their dynamics and stationary states with a desired accuracy.

Figure 1

Schematic illustration of the DQFNN (deep quantum feed-forward neural network) method in solving quantum master equations. This DQFNN has $d$ layers in total, with an output (bottom) layer that indicates the target quantum system, and $d-1$ hidden layers that simulate the environment and conduct interaction in the target quantum system. A quantum perceptron is defined as an arbitrary unitary operator acting on qubits from neighboring layers [22], and the DQFNN is parametrized by the product of these unitary operators. We use the output state as the variational ansatz state of a quantum open system. The qubits in the output layer are measured and the measurement results are used by a quantum optimization algorithm to update the network parameters.

Figure 2

(a) A DQFNN with layer structure (2,2,3,3,5) is used to solve the dissipative transverse field Ising model [see Eq. (10)]. Here, each quantum perceptron consists of a unitary acting on two-input and one-output qubits and this unitary is parametrized by a simple quantum circuit (black-dotted box), which contains twelve Euler rotations and six controlled-not gates. (b) Numerical results for the steady state of the dissipative Ising model with five spins. The expectation values of three different observables $\overline{{\sigma }^x}=\frac{1}{5}\left({\sum }_k{\sigma }_k^x\right)$, $\overline{{\sigma }^z}=\frac{1}{5}\left({\sum }_k{\sigma }_k^z\right)$, and ${\sigma }_1^x{\sigma }_2^x$ are calculated by both the DQFNN and exact diagonalization (ED) methods. Here, we set $J=1$ as the energy unit and the dissipation ratio is $\gamma =1$. For the DQFNN approach, the learning rate $\lambda$ is chosen to be $\lambda =0.01\times 0.{999}^{N_s}$, where $N_s$ is the number of steps.

Figure 3

Numerical results for the time dynamics of the dissipative Ising model with five spins under the periodic boundary condition. The time step is chosen to be $5\times {10}^{-3}$ and the sampling size for every step is $5\times {10}^4$. Here we plot each point for every 50 steps. (a), (b), and (c) plot the time dependences of $\overline{\langle {\sigma }^z\rangle }$, $\overline{\langle {\sigma }^x\rangle }$, and $\langle {\sigma }_1^x{\sigma }_2^x\rangle$, respectively. (d) plots the real and imaginary parts of $\delta \mathcal{L}$, which measures the convergence to the steady state.

Figure 4

Numerical results for the dissipative 2D $J_1\text{−}{J}_2$ model. Here, the simplest two-by-two square lattice is considered and we use a DQFNN with layer structure (4,4,4) to obtain the results. The parametrization of the DQFNN is similar to the case for the 1D dissipative Ising model. (a), (b), and (c) show respectively $\overline{\langle {\sigma }^z\rangle }$, $\overline{\langle {\sigma }^x\rangle }$, and $\langle {\sigma }_0^x{\sigma }_1^x\rangle$ as a function of iteration steps with the dissipation rate chosen to be $\gamma =1.0$. The red-dotted lines represent their corresponding values for the steady state (SS) obtained from exact diagonalization. (d) The real part of $\delta \mathcal{L}$, which converges to zero after around 1000 iteration steps [$\text{Re}\left(\delta \mathcal{L}\right)<{10}^{-2}$], indicating that the output state of the DQFNN indeed converges to the steady state.

Figure 5

Illustration that the DQFNN can represent any single-qubit gate and the nearest two-qubit operation. (a) shows that any single-qubit gate can be generated. The blue and the black circuits represent the two target qubits and the blue boxes $U_1\left(\theta \right)$ represent the arbitrary single-qubit gate, which can be applied respectively in their perceptrons. The swap gate is applied in their perceptron with the white circuit qubit in the state $\frac{1}{\sqrt{2}}|0\rangle +|1\rangle$. After tracing out the previous layer, we obtain the output density matrix with a single-qubit gate applied. (b) shows that any nearest two-qubit gate can be achieved in our DQFNN: in the first perceptron swap the state of the blue qubit to the next layer and then apply the two-qubit gate $U_2\left(\theta \right)$, and in the second perceptron swap the black qubit to the next layer. We can obtain the output density matrix with the two-qubit gate applied.

Figure 6

Demonstration of implementing an arbitrary unitary operation between two arbitrary qubits in the same layer by applying a series of the unitary gate from (a) to (e). Considering an input density matrix ${\rho }^{L-1}$ in the $(L-1)\mathrm{th}$ layer, we denote the two target qubits by black and blue discs, respectively. Without loss of the generality, we take the first and the $k\mathrm{th}$ qubit as the target qubits. The rest of the qubits in the $(L-1)\mathrm{th}$ layer are drawn in grey, and the qubits in the $L\mathrm{th}$ layer are drawn in white and initialized to $\frac{1}{\sqrt{2}}\left(\right|0\rangle +|1\rangle )$. (a) Using two swap gates to transfer the black qubit to the last qubit in the $(L-1)\mathrm{th}$ layer. Consequently, the last qubit in the $(L-1)\mathrm{th}$ layer is swapped to the corresponding position in $L\mathrm{th}$ layer as shown in (b). Repeat this operation until the black qubit and the blue qubit are in the same perceptron, as shown in (c), and we apply the two-qubit unitary gate. After that, in (d), we apply a swap gate between the states of the black qubit and the blue qubit. Then we repeat the swap operations similarly as in (b) until we transfer the black qubit to the $L\mathrm{th}$ layer at its corresponding position. It is easy to find that, after these operations, the rest of the qubits in the $(L-1)\mathrm{th}$ layer have been swapped to the corresponding position in the $L\mathrm{th}$ layer, and the states of qubits in the $(L-1)\mathrm{th}$ layer are all taken as $\frac{1}{\sqrt{2}}\left(\right|0\rangle +|1\rangle )$ in (e). After tracing out the $(L-1)\mathrm{th}$ layer, we obtain the output density matrix in the $L\mathrm{th}$ layer with an arbitrary two-qubit unitary gate acting on the two target qubits (f).

Figure 7

The quantum circuits for obtaining $S$ and $\vec{f}$ used in the DQFNN approach. Here, H denotes the Hadamard gate and C-SWAP represents the controlled-swap gate defined in Eq. (C1). Each circuit is composed of two DQFNNs to generate $\rho \left(\mathrm{\Theta }\right)$ and $\rho \left({\mathrm{\Theta }}^{\prime }\right)$ and one ancillary qubit for measuring. (a) The circuit for measuring $\text{Re}\left\{\text{Tr}\left[\rho \left(\mathrm{\Theta }\right)\rho \left({\mathrm{\Theta }}^{\prime }\right)\right]\right\}$, where $|\psi \rangle =\left(\right|0\rangle +|1\rangle )/\sqrt{2}$. (b) The circuit for measuring $\text{Tr}\left[\rho \left(\mathrm{\Theta }\right){\widehat{O}}_1\rho \left({\mathrm{\Theta }}^{\prime }\right)\right]$. C-${\widehat{O}}_1$ is the controlled-${\widehat{O}}_1$ gate defined in Eq. (C9), where $|\psi \rangle =\left(\right|0\rangle +|1\rangle )/\sqrt{2}$ for measuring the real part of the quantity and $|\psi \rangle =\left(\right|0\rangle +i|1\rangle )/\sqrt{2}$ for measuring the imaginary part of the quantity. (c) The circuit for measuring $\text{Tr}\left[\rho \left(\mathrm{\Theta }\right){\widehat{O}}_1\rho \left({\mathrm{\Theta }}^{\prime }\right){\widehat{O}}_2\right]$, where $|\psi \rangle$ is initialized by the same strategy as in (b).

Figure 8

Numerical demonstration of the robustness of the DQFNN against the measurement noise on solving the steady state of the transverse Ising model with transverse field $h=1.1$ and the dissipation rate $\gamma =1.0$. (a), (b), and (c) show respectively $\overline{\langle {\sigma }^z\rangle }$, $\overline{\langle {\sigma }^x\rangle }$, and $\langle {\sigma }_1^x{\sigma }_2^x\rangle$ as a function of evolution time. The DQFNN has the structure (2,2,3,3,5) with different measurement noise levels $E_r=0.01,0.05,0.1,0.2,0.4,0.8$. (d) plots the real parts of $\delta \mathcal{L}$, which converge to ${10}^{-2}$ as the amplitudes of noise are $E_r=0.01$ and $E_r=0.1$, while they get lost for $E_r=0.4$ and $E_r=0.8$.

Figure 9

Numerical demonstration of the robustness of the DQFNN against the measurement noise and decoherence noise on finding the steady state of the transverse Ising model with transverse field $h=1.1$ and the dissipation rate $\gamma =1.0$. (a) We take the measurement noise as an example. Here, we simulate three different structures of the DQFNN, namely, (2,2,2), (2,2,2,2,4), and (2,2,3,3,5). The noise amplitude ranges from 0.01 to 10.24. We calculate the fidelity between the DQFNN result as converged and the physical steady state solved by ED. As it is shown in the figure, when the noise amplitude $E_r$ is larger than 0.2 the training process is lost and cannot find the steady-state accuracy. (b) We take the decoherence noise as an example. Here we simulate the (2,2,2), (3,3,3), and (2,2,2,2,4) structures of the DQFNN. The noise amplitude $\epsilon$ is ranging from 0.001 to 0.5. Like the behavior in (a), as $\epsilon >0.04$, the fidelity decays dramatically.