## Quantum Neural Networks

Optical parametric oscillators (OPO) with degenerate signal and idler frequencies, ωs=ωi=ωp/2, generate squeezed vacuum states at pump rates below the oscillation threshold. Squeezed vacuum states are linear superposition states of canonical coordinate eigenstates |X＞, so that they realize a quantum parallel search for different lX> eigenstates. At above the threshold, the degenerate OPOs go through spontaneous symmetry breaking and randomly pick up either 0-phase or π-phase coherent fields. We interpret such bi-phase oscillation at above the threshold as classical Ising spin-up and -down states, so that such spontaneous symmetry breaking is considered as decision making process of the Ising spin. It is important to note that the degenerate OPOs at below the threshold occupy simultaneously spin-up and spin-down states as superposition. This means the degenerate OPOs behave as analog quantum devices at below the threshold and as digital classical devices at above the threshold, as shown in Fig.1.

If N degenerate OPOs are mutually coupled via optical delay line circuits or measurement feedback circuits, the phase decison process of each OPO is no more governed by spontaneous symmetry breaking, but instead N OPOs collectively pick up a unique spin configuration due to the quantum correlation among N OPOs as shown in Fig.2. We can determine the amplitudes and phases of mutual coupling constants among N OPOs in order to make the selected spin configuration be a ground state of a given Ising problem. Before the final decision is made at the threshold, N OPOs perform a quantum parallel search with their squeezed vacuum states and establish a seed for the final decision making through formation of quantum entanglement or discord between N OPOs. This is a principle of degenerate OPO based coherent Ising machines.

In general, quantum computational resources, such as superposition and quantum correlation for parallel search and quantum interference for suppressing classical chaos, lose their abilities as a dissipative coupling to external reservoirs increases, as shown in Fig.3. On the other hand, classical computational resources, such as spontaneous/collective symmetry breaking for decision making and exponential amplitude amplification of a solution state, lose their capabilities as a dissipative coupling to external reservoirs decreases as shown in Fig.3. We need the quantum and classical computational resources simultaneously in order to realize an efficient computing device. A degenerate OPO is a unique device which posesses such quantum and classical behaviors simultaneously at room temperatures.

A few topics of our current efforts include:

- Entanglement and quantum discord in coherent Ising machines and XY machines
- Optical all-to-all coupling in coherent Ising machines and XY machines using Floquet engineering
- Quantum suppression of classical chaos in coherent SAT solvers
- Fig 1 Vacuum state squeezing and spontaneous symmetry breaking in degenerate OPO

Fig 1: Vacuum state squeezing and spontaneous symmetry breaking in degenerate OPO