A unified framework for describing open quantum systems interconnected by bosonic quantum fields (such as optical or microwave fields) has recently been developed. This is enabling the development of engineering strategies to design physical systems for the purpose of quantum enhanced information processing applications. Our group is working on tools for working with quantum feedback networks, model reduction methods for dealing with the inherent complexity of such systems as well as concrete applications that demonstrate the power of our approach.

## Modeling of Quantum Networks

In the quest for creating “quantum enhanced” systems for information processing many currently pursued design strategies face difficulties in scaling significantly beyond a few dozen qubits. The dominant design paradigm relies on near perfect quantum components and a vast overhead of classical external control. We are working on a more integrated framework which treats quantum and hybrid quantum-classical systems on equal footing.

### Quantum Hardware Description Language

We have recently defined a Quantum Hardware Description Language (QHDL) capable of describing networks of interconnected open quantum systems. QHDL is compiled to symbolic and numerical system models by a custom software tool suite named QNET. This allows us to rapidly iterate over quantum network designs and derive the associated equations of motion.

### Automated Quantization of Optical Networks

Networks of open quantum systems with feedback have become an active area of research for applications such as quantum control, quantum communication and coherent information processing. A canonical formalism for the interconnection of open quantum systems using quantum stochastic differential equations (QSDEs) has been developed by Gough, James and co-workers and has been used to develop practical modeling approaches for complex quantum optical, microwave and optomechanical circuits/networks.

Our work fills a significant gap in existing methodology by showing how trapped modes resulting from feedback via coupled channels with finite propagation delays can be identified systematically in a given network. Our method is based on the Blaschke-Potapov multiplicative factorization theorem for inner matrix-valued functions, which has been applied in the past to analog electronic networks (Tabak et al. in EPJ Quantum Technology 3:3, 2016). Our results provide a basis for extending the Quantum Hardware Description Language (QHDL) framework for automated quantum network model construction (Tezak et al. in Philos. Trans. R. Soc. A, Math. Phys. Eng. Sci. 370(1979):5270-5290, 2012) to efficiently treat scenarios in which each interconnection of components has an associated signal propagation time delay.

In the figure below, we show the color wheel graph of the frequency response of a sample passive network. The roots of the function portrayed correspond to modes of the network.

### Distributed Quantum Systems

We are at the cusp of a technological revolution based on the development of devices that actively operate at the quantum level. These include quantum information processing devices and quantum sensors. To achieve their full potential, these devices are linked in networks, where they interface also with classical or semi-classical components. We aim to implement a complete software framework for the modeling, simulation, and optimization of quantum networks. We have developed the QNET package to analytically describe quantum networks, based on the mathematical SLH framework due to (Gough and James (2009). This includes a computer algebra system for quantum mechanics, and automated model reduction. Interfacing with efficient simulation packages from quantum trajectories such as QSD and QDYN, allows us to accurately simulate the dynamics even of large scale networks. We are combining gradient-based optimal control and quantum trajectory simulations to determine how to operate quantum networks with the highest possible fidelity under realistic noise conditions. Applications include the distribution of entanglement in a quantum network, and the preparation of non-classical states for quantum-enhanced sensors. This work is in collaboration with Rad Balu and Kurt Jacobs at the Army Research Lab.

## Reduced Models for Optical Networks

### Semiclassical Nonlinear Networks

In collaboration with researchers at Hewlett Packard Labs we have developed a model reduction technique for describing networks of nonlinear oscillators in the semi-classical regime. This allows us to design ultra-low power photonic systems capable of classical computation and investigate the effect of quantum noise on their error rates.

Building on this method, we have proposed all-optical circuits for digital and analog information processing tasks. As an example of an analog application we have proposed neuromorphic circuits capable of supervised learning.

The system is hierarchically composed of tunable linear amplifiers, analog phase memories and thresholding non-linear circuits which can be used to construct more general quantum feedback networks for nonlinear information processing.

One of the goals of photonics is to provide a low-power alternative to electronic computing. In the low-power limit, however, the effects of photon “shot noise” become important and degrade the performance of a device. To model this, in principle, we need to perform full quantum simulations, which are impractical for the large circuits we need to do anything useful.

In a collaboration with Charles Santori at HP Labs, we showed that classical photonic networks, based on Kerr resonators with photons per cavity, could be accurately simulated using the truncated Wigner method, a semiclassical approximation whose computation time scales linearly with circuit size. We wrote code applying the Wigner method to arbitrary quantum networks, and simulated optical latches, flip-flops and digital counters (Ref. [3]) \cite{Santori2014}. With the Wigner method, one cannot model fully quantum behavior, but we could make strong statements about quantum limits to low-power classical photonic computing.

### Material-based optical devices: Free-carrier oscillations

The next step was to extend the Wigner method to optical cavities with free-carrier nonlinearities, since in most materials, free-carrier dispersion is orders of magnitude stronger than the Kerr effect. This was a challenge because the carriers are defined by fermionic operators, but the right bosonization does the trick. In the end, one derives a set of stochastic differential equations that resembled the Kerr equations from the HP paper, but had additional noise terms due to free-carrier excitation and decay, which are incoherent processes. These equations are useful when studying phase-sensitive amplifiers and latches and limit-cycle behavior associated with the free-carrier Hopf bifurcation (Refs. [4, 5])

Present and future work involves extending these methods to novel materials and interactions with much stronger optical nonlinearities, e.g. broadband plasmonic nonlinearities with thin-film materials or excitons in 2D materials and quantum wells.

### Reduced Models for Dissipative Quantum Systems

We are also working on novel simulation methods that exploit the semi-classical localization of dissipative open quantum systems to reduce the computational complexity.

This is achieved by means of an adaptive model transformation capable of dividing the description of quantum states into a low-dimensional quasi-classical part coupled to a lower complexity quantum state. This approach is exact and naturally tailored to simulating coupled quantum systems with varying degrees of dissipation.

### Dimensionality Reduction by Machine Learning

Simulating a network of quantum devices requires exponential resources in the system size. However, for many applications the devices are operated in regimes where the dynamics are effectively lower-dimensional. We attempt to find the hidden lower-dimensional structure and generate a reduced model that is easier to simulate using machine learning techniques.

One strategy is to construct a locally invertible map between the full space and the low dimensional space on patches (i.e. charts) of simulated data. In the figure, we illustrate how local tangent space alignment (LTSA) can be used to map a high-dimensional system, in this case a cavity with a strong Kerr nonlinearity, to a low dimensional space where the trajectory points can be readily visualized.