Tensor networks for studies of topological phase transitions

Quantum mechanics is one of the fastest developing areas of modern physics, the progress of which is being made both in theoretical understanding and description of phenomena, but also in their experimental realization. Although its foundations date back to the beginning of the 20th century, the technological quantum revolution did not take place until the 21st century and the last years especially, culminating in the awarding of the Nobel Prize in Physics in 2022 `for experiments with entangled photons, establishing the violation of Bell inequalities and pioneering quantum information science` to, among others, the Austrian physicist, prof. Anton Zeilinger. Currently, one of the greatest and most interesting endeavours of quantum physics is to create a quantum computer that can handle problems that cannot be solved on the largest classical supercomputers. One of the paradigms of quantum computation is the topological quantum computation, where quantum logic gates would be performed by braiding exotic particles called anyons. Unlike many known elementary particles responsible for the building and interaction of matter: bosons (like the commonly known photon) and fermions (like electrons)- anyons are the collective emerging effect of strongly correlated quantum systems, called topologically ordered systems. Remarkably, in 2021 it was possible to experimentally realize topologically ordered systems in quantum simulators. To do this, atoms are caught individually in traps and their interactions are precisely adjusted to simulate the desired physics. One of these experiments makes use of so-called Rydberg atoms, whose properties make them an ideal platform for quantum simulators. With this fellowship, my goal is to make an important step towards simulating these exotic phases of matter and quantum systems. To this end, we will develop a set of methods that enable the theoretical investigation of the dynamical preparation of topologically ordered systems in realistic experiments. The dynamical preparation of a quantum state means that an experiment starts with an easy-to-control initial state with no (or very little) quantum entanglement. Then the interactions between the atoms are slowly switched on, enabling the state to adjust to the changes of the varying parameters. If the preparation takes place too quickly, defects will arise in the system, which will negatively affect the quality of the final state. A major contribution of the project will be to develop numerical methods that allow to optimize the preparation time and to study the topological properties of the final state. The results of the project will thus allow us to find the optimal balance between a short preparation time and a small number of defects, and thus realize the best possible topological states in experiments.

The project delivered on its main objective: developing new computational methods to study and optimize topologically ordered quantum systems, and applying them to models directly relevant to quantum computing and experiment. A central result was the improvement of optimization techniques for two-dimensional tensor networks (PEPS). These are mathematical structures that efficiently represent quantum states with their entanglement pattern. By adopting automatic differentiation (AD), a technique familiar in modern AI and machine learning, we overcame key limitations that previously made these techniques difficult to use. Specifically we derived corrected formulas for matrix operations, which resulted in more reliable calculations. We also introduced a smarter way to structure the underlying code using implicit differentiation techniques. Rather than laboriously ensuring every computational step is differentiable, we separate the optimization itself from a compact set of equations that capture its outcome, and differentiate only those. The result is code that is simpler to write, faster and more stable, and produces significantly better approximations of quantum ground states. These advances have implications not only for ground state optimization, but also for time evolution, excitations, and broader machine learning applications. We applied these methods to two physically relevant models. The first is the Toric Code (a paradigmatic model of topological quantum computation) subject to two parallel magnetic fields. We studied how the system transitions out of its topological phase as exotic particles, its elementary excitations called anyons, condense into the ground state, and found how the mathematical representation of the state (PEPS representation) naturally adapts depending on which type of anyon condensation drives the transition. These findings are being prepared for publication. Second model describes Rydberg atoms on the ruby lattice. It is directly relevant to experiments, already performed in Harvard in 2021, and it possesses the same topological order as the toric code model. We studied the phase diagram of this system, identifying the parameter values at which the system undergoes a phase transition from a trivial phase to a topologically ordered phase and further to another ordered phase. Analysis of this second transition is ongoing and collection of results have to be finalized before they can be published. Together, these results represent a significant step toward understanding and optimizing topological states in realistic quantum simulators, as set out in the original proposal.