Simulation of memristive synapses and neuromorphic computing on a quantum computer

One of the major approaches to spike-based neuromorphic computing is using memristors as analog synapses. We propose unitary quantum gates that exhibit memristive behaviors, including Ohm's law, pinched hysteresis loop and synaptic plasticity. Hysteresis depending on the quantum phase and long-term plasticity that encodes the quantum state are observed. We also propose a three-layer neural network with the capability of universal quantum computing. Quantum state classification on the memristive neural network is demonstrated. These results pave the way towards quantum spiking neural network built on unitary processes. We obtain these results in numerical simulations and experiments on the superconducting quantum computer ibmq_vigo.

Figure 1

(a) Presynaptic and postsynaptic biological neurons. (b) A memristor. In the quantum regime, we use qubits to represent the input/output current and resistance of the memristor. (c) Memristive gate $M_{\theta }$ decomposed into elementary quantum gates, where $R_{\mathrm{z}}=e^{-i\frac{\theta }{2}Z}$ and $R_{\mathrm{x}}=e^{-i(\frac{\pi }{2}-\theta )X}$. (d) Memristive quantum neural network.

Figure 2

Hysteresis loops of memristive gates. Here ${\langle Z_{\mathrm{C}}\rangle }_{\mathrm{in}}$ and ${\langle Z_{\mathrm{C}}\rangle }_{\mathrm{out}}$ represent the voltage and current, respectively. The phase $\eta =1$ in (a) and $\eta =i$ in (b).

Figure 3

Hysteresis loops of $({\langle Z_{\mathrm{C}}\rangle }_{\mathrm{in}}\left(t\right),{\langle Z_{\mathrm{R}}\rangle }_{\mathrm{out}}\left(t\right))$. Here ${\langle Z_{\mathrm{C}}\rangle }_{\mathrm{in}}$ and ${\langle Z_{\mathrm{R}}\rangle }_{\mathrm{out}}$ represent the voltage and conductance, respectively. The phase $\eta =1$ in (a) and $\eta =i$ in (b).

Figure 4

(a) The circuit in Fig. 1 displayed using rotation gates with angles, $R_x\left(\varphi \right)=e^{-i\frac{\varphi }{2}X}$, $R_y\left(\varphi \right)=e^{-i\frac{\varphi }{2}Y}$, and $R_z\left(\varphi \right)=e^{-i\frac{\varphi }{2}Z}$. (b) Circuit of the memristive gate $M_0$. (c) Circuit of the memristive gate $M_{\theta }$ optimized for the implementation on ibmq_vigo. (d) Circuit of the modified memristive gate with a measurement on the current qubit and a feedback gate on the resistance qubit. (e) Circuit of the modified memristive gate with an additional controlled-NOT gate, which is used in the experiment on ibmq_vigo. (f) Circuit of the quantum state classification experiment on ibmq_vigo.

Figure 5

(a) Classical and (b),(c) quantum long-term plasticity based on memristive gates. Thin solid curves represent numerical results, and filled circles represent experimental results. Empty circles denote initial values in the experiments. In (b) and (c), we take the same values of the parameter $\theta$. Dashed horizontal lines denote values in the input state of current qubits, i.e., the steady state. $P=X,Y,Z$ are Pauli operators. The quantum state is successfully encoded when three ${\langle P_{\mathrm{R}}\rangle }_{\mathrm{out}}$ converge to dashed lines.

Figure 6

Universal quantum computing operations on the neural network, including (a) write operation, (b) read operation, (c) single-qubit gate, and (d) two-qubit gate. Red arrows denote the time sequence.

Figure 7

Single-qubit gate. Red numbers denote the time sequence.

Figure 8

(a) Neural network for the classification of two-qubit states. (b) Probability of the output bit 0 given optimal parameters. Dashed and solid boxes represent the theoretical and experimental results, respectively.

Figure 9

Values of the distance in the optimisation computing for the classification of (a) Bell states, (b) ten-qubit Ising-model states with all parameters, (c) ten-qubit Ising-model states with reduced parameters, and (d) three-qubit Ising-model states. Blue dots denote the distance returned in each step. Dashed lines denote the quantum upper bound of the distance. Solid lines denote classical values, i.e., $D(q(•|{\mathrm{\Phi }}_{\mathrm{ghz}}),q(•|{\mathrm{\Phi }}_{+}))$.