## Final Report Summary - SQSNP (Simulating quantum systems numerically and physically)

The fundamental goal of this project, Simulating Quantum Systems Numerically and Physically (SQSNP), was to improve our ability to predict a priori the emergent behaviour of large, many-body systems. This project aimed to take a two-pronged approach to expanding the realm of solvability, using both advanced numerical techniques as well as studying physical implementations recently made possible by experimental advances, and the aims of the project are split into these two directions.

The majority of the work performed throughout the project was directed at this first aim, to produce and perfect new numerical tools for understanding many-body quantum systems. The project was successful in developing novel simulation techniques based on tensor networks. Tensor network techniques have emerged over the past decade or two as successful variational and direct methods for simulating strongly-correlated systems with little approximation – however intense research has been ongoing to improve the efficiency of the methods in two- and higher-dimensional quantum systems.

The major breakthrough of this project has been the successful development of new interpretations and applications of the tensor network formulism to both the Monte Carlo and series expansion approaches. In the traditional Monte Carlo approach, possible configurations of a system are randomly sampled individually through an algorithm such as the famous Metropolis algorithm. For simple systems, just a small number of samples (or individual configurations) may broadly represent the physics typical to the whole system, and in many cases Monte Carlo has been extremely successful. For strongly-correlated systems, where for instance the order of the system may be much more subtle, many more configurations or samples may be necessary to produce an adequate signal-to-noise ratio to measure useful physics, and in some cases such as where the “sign problem” rules, the difficulty grows exponentially in the system size.

In SQSNP, a brand-new Monte Carlo approach has been developed where each “sample” of the algorithm corresponds to a tensor network representing a very large number of individual configurations, thus overcoming many of the shortfalls of the traditional Monte Carlo approach. For instance, the accuracy scaling with computation cost can now scale exponentially better to that in-line with longstanding tensor network techniques (such as the density matrix renormalization group, DMRG, approach, which is vastly more precise than traditional Monte Carlo in 1D quantum systems). This new approach also may be useful for representing systems that normally suffer from the sign problem, since relatively simple tensor networks can represent configurations much more applicable to the physics of those systems than the standard product basis employed in Monte Carlo. In the future, the approach can be combined with newer enhancements such as continuous-time Monte Carlo and so forth, and the approach has much potential.

The formalism developed throughout this period also lead to insights in how to apply tensor networks in other contexts. For instance, it is natural to express an interacting system in a series expansion and we are, for the first time, learning how to fully utilize and exploit this idea in the tensor network formulism. Early in the project, ideas were successfully developed about using tensor network solvers to facilitate traditional series expansion (notably, the numerical linked-cluster expansion). The more “modern” approach, developed by Dr. Ferris and local ICFO collaborators post-doc Dr. Shi-Ju Ran and PhD student Emanuele Tiirrito, is to work the series expansion directly into the tensor network by representing the exact state as an easy-to-write yet difficult-to-compute tensor network, and starting with current numerical approximations as the “first order” of the expansion technique.

The second major aim of this project was devoted towards physical simulation of quantum systems via analogue quantum simulators. This workload was planned towards the end of the two-year project period (to use the newly developed numerical tools), and as the project was cut short by six-months, a smaller fraction of this work was completed. None-the-less, longstanding and ongoing collaborations within ICFO and abroad means the project has had a positive, if smaller, impact in this field. This, together with the continuing work of Dr. Ran and Mr. Tirrito, means the fruit of this research project will continue to be borne by ICFO in the future.

The majority of the work performed throughout the project was directed at this first aim, to produce and perfect new numerical tools for understanding many-body quantum systems. The project was successful in developing novel simulation techniques based on tensor networks. Tensor network techniques have emerged over the past decade or two as successful variational and direct methods for simulating strongly-correlated systems with little approximation – however intense research has been ongoing to improve the efficiency of the methods in two- and higher-dimensional quantum systems.

The major breakthrough of this project has been the successful development of new interpretations and applications of the tensor network formulism to both the Monte Carlo and series expansion approaches. In the traditional Monte Carlo approach, possible configurations of a system are randomly sampled individually through an algorithm such as the famous Metropolis algorithm. For simple systems, just a small number of samples (or individual configurations) may broadly represent the physics typical to the whole system, and in many cases Monte Carlo has been extremely successful. For strongly-correlated systems, where for instance the order of the system may be much more subtle, many more configurations or samples may be necessary to produce an adequate signal-to-noise ratio to measure useful physics, and in some cases such as where the “sign problem” rules, the difficulty grows exponentially in the system size.

In SQSNP, a brand-new Monte Carlo approach has been developed where each “sample” of the algorithm corresponds to a tensor network representing a very large number of individual configurations, thus overcoming many of the shortfalls of the traditional Monte Carlo approach. For instance, the accuracy scaling with computation cost can now scale exponentially better to that in-line with longstanding tensor network techniques (such as the density matrix renormalization group, DMRG, approach, which is vastly more precise than traditional Monte Carlo in 1D quantum systems). This new approach also may be useful for representing systems that normally suffer from the sign problem, since relatively simple tensor networks can represent configurations much more applicable to the physics of those systems than the standard product basis employed in Monte Carlo. In the future, the approach can be combined with newer enhancements such as continuous-time Monte Carlo and so forth, and the approach has much potential.

The formalism developed throughout this period also lead to insights in how to apply tensor networks in other contexts. For instance, it is natural to express an interacting system in a series expansion and we are, for the first time, learning how to fully utilize and exploit this idea in the tensor network formulism. Early in the project, ideas were successfully developed about using tensor network solvers to facilitate traditional series expansion (notably, the numerical linked-cluster expansion). The more “modern” approach, developed by Dr. Ferris and local ICFO collaborators post-doc Dr. Shi-Ju Ran and PhD student Emanuele Tiirrito, is to work the series expansion directly into the tensor network by representing the exact state as an easy-to-write yet difficult-to-compute tensor network, and starting with current numerical approximations as the “first order” of the expansion technique.

The second major aim of this project was devoted towards physical simulation of quantum systems via analogue quantum simulators. This workload was planned towards the end of the two-year project period (to use the newly developed numerical tools), and as the project was cut short by six-months, a smaller fraction of this work was completed. None-the-less, longstanding and ongoing collaborations within ICFO and abroad means the project has had a positive, if smaller, impact in this field. This, together with the continuing work of Dr. Ran and Mr. Tirrito, means the fruit of this research project will continue to be borne by ICFO in the future.