Universal quantum gates, artificial neurons, and pattern recognition simulated by $LC$ resonators

We propose to simulate quantum gates by LC resonators, where the amplitude and the phase of the voltage describe the quantum state. By controlling the capacitance or inductance of resonators, it is possible to control the phase of the voltage arbitrarily. A set of resonators acts as the phase-shift, the Hadamard, and the CNOT gates. They constitute a set of universal quantum gates. We also discuss an application to an artificial neuron. As an example, we study pattern recognition of numbers and alphabets by evaluating the similarity between an input and the reference pattern. We also study colored pattern recognition by using a complex neural network.

Figure 1

(a) Two qubits made of four LC resonators. (b) Phase-shift gate consisting a pair of LC resonators. The capacitance of the state $|1\rangle$ is controlled. (c) Mixing gate. The inductance of the inductor $L_1$ bridging two resonators is controlled. The Hadamard gate is constructed by a combination of the mixing gate and the phase-shift gate. (d) CNOT gate. We bridge the resonators representing $|10\rangle$ and $|11\rangle$ by the inductor $L_1$ (e) Controlled phase-shift gate. We control the capacitance of the resonator representing $|11\rangle$.

Figure 2

Phase-shift gate [Fig. 1] for (a) $\varphi =\pi /4$, (b) $\varphi =\pi /2$ and (c) $\varphi =\pi$. The phase delay is controlled by a time duration of the capacitance perturbation $C_1\left(t\right)$. (1) Time evolution of the perturbed capacitance. (2) Time evolution of the voltage $V_2$. The voltage $V_2$ with (without) the perturbation is represented by a magenta (cyan) curve. (3) Voltage $V_2$ in the final states. (4) Phase delay as a function of time $t$. The horizontal axis is time $t$. Time span is $0<t<50\pi$ for (1), (2), and (4). It is $46\pi <t<50\pi$ for (*3), representing the final state. We set $C_1^0/C_0=0.1$ and $T=10{\omega }_0$ for $0<{\omega }_{0}t<50\pi$. We set ${\omega }_0{t}_1=10\pi$ for all phase-shift gate. We also set ${\omega }_0{t}_2=15\pi$ for the $\pi /4$ phase-shift gate, ${\omega }_0{t}_2=20\pi$ for the $\pi /2$ phase-shift gate and ${\omega }_0{t}_2=30\pi$ for the $\pi$ phase-shift (i.e., Pauli Z) gate,

Figure 3

(a) Mixing gate. (b) NOT gate. [(a1) and (b1)] Time-depending perturbation introduced to the inductance. [(a2) and (b2)] Time evolution of the voltage $V_1$ ($V_2$) at the left (right) LC resonator is represented by a magenta (cyan) curve. (3) Time evolution of $V_1$ and $V_2$ for $0<{\omega }_{0}t<4\pi$, representing the initial state, where $V_2=0$. (4) Time evolution $V_1$ and $V_2$ for $116\pi <t<120\pi$, representing the final state, where the voltage without the perturbation is represented by a black curve. We set $L_1/L_0=0.1$ and $T=10{\omega }_0$ for $0<{\omega }_{0}t<120\pi$. We set ${\omega }_0{t}_1=50\pi$ and ${\omega }_0{t}_2=55\pi$ for the mixing gate and ${\omega }_0{t}_1=55\pi$ and ${\omega }_0{t}_2=65\pi$ for the NOT gate. The horizontal green lines represent the voltage $\pm V^0/\sqrt{2}$.

Figure 4

Equivalent quantum-circuit representation of (a) CZ gate and (b) CCZ gate.

Figure 5

(a) Mean fidelity as a function of randomness in the capacitance $C$, the inductance $L$ and $L_1$. The horizontal axis is the magnitude of randomness $\delta$. (b) Mean fidelity as a function of inaccuracy of $t_1$ and $t_2$. We have taken 1000 times average. See also the caption of Fig. 3.

Figure 6

(a) Schematic for an artificial neuron. It has $m$ real inputs $x_j$ and real internal degree of freedom named weights $w_j$. Output is obtained by calculating the inner product ${\sum }_j{w}_j{x}_j$ and then by applying an activation function $\mathrm{\Phi }\left({\sum }_j{w}_j{x}_j\right)$. (b) Schematic for a complex neuron. It has complex $m$ inputs $x_j$ and complex weights $w_j$ with the inner product ${\sum }_j{w}_j^{*}{x}_j$.

Figure 7

Assignment of binary numbers to (a) $5\times 4$ pixels for the number recognition and (b) $6\times 5$ pixels for the alphabet recognition.

Figure 8

Neural networks for (a) number recognition and (b) alphabet recognition.

Figure 9

(a) Reference patterns and (b) input patterns of numbers. (c) Self similarity $\langle {\psi }_w|{\psi }_w\rangle$ and (d) cross similarity $\langle {\psi }_w|{\psi }_x\rangle$ of the number recognition.

Figure 10

(a) Reference patterns and (b) input patterns of alphabets. (c) Self similarity $\langle {\psi }_w|{\psi }_w\rangle$ and (d) cross similarity $\langle {\psi }_w|{\psi }_x\rangle$ of the alphabet recognition.

Figure 11

Standard quantum circuits for a generation process of the CEW state for (a) three qubits and (b) four qubits. An isolated magenta disk indicates a Z gate. Magenta disks connected by a line indicate a ${\mathrm{C}}^{p-1}{\mathrm{Z}}_{\varphi }$ gate.

Figure 12

(a) Color circle, (b) reference color pattern and (c) input color pattern. The input pattern is made by changing randomly colors of the reference pattern within 20% randomnes.

Figure 13

(a) Standard quantum-circuit model for the calculation of the inner product (82) for an example given by (85) and (86). We use a four-qubit state $\left|n_1{n}_2{n}_3{n}_4\right.〉$ (b) Corresponding electric-circuit simulation. The Hadamard gates are denoted by filled or unfilled black disks connected by a link. The Hadamard bridges with unfilled disks are not necessary since they are irrelevant for the input state $\left|0\right.〉\rangle$ and the output state $\langle \langle 0|$.

Figure 14

Output $y_j$ for each resonator representing $\left|j\right.〉\rangle$. We measure the output $y_0$ for the state $\left|0\right.〉\rangle$, which gives an inner product $3/8=0.375$.

Figure 15

(1) Quantum-circuit representation and (2) graph and hypergraph representation. Z gates correspond to self-loops, which are marked in orange disks in a graph or a hypergraph. CZ gates correspond to edges, which are marked in black lines in a hypergraph. ${\mathrm{C}}^{p-1}\mathrm{Z}$ gates correspond to hyperedges, which are marked in colored ovals in a hypergraph. (a) graph state, (b) for ${\psi }_x$, and (c) for ${\psi }_w$.