Quantum complexity of symmetry-protected topological phases of matter

Abstract

Classical computers have been instrumental to our understanding of quantum phases of matter. However, their ability to simulate quantum many-body systems is fundamentally limited – there are quantum systems that are inherently more challenging to simulate. We study the limitations of classical simulations in the context of symmetry-protected topological (SPT) phases of matter. We identify obstructions to efficiently simulating SPT phases and develop our understanding of their intrinsic quantum information-theoretic structures. More specifically, we define the concepts of a symmetry-protected sign problem and symmetry-protected magic to study the quantum complexity of bosonic SPT phases. We demonstrate that certain SPT phases possess these properties as a consequence of their long-range correlations and anomalous symmetry action at a boundary. We also consider the quantum complexity of fermionic SPT phases by employing bosonization dualities, which map fermionic SPT phases to more familiar bosonic SPT phases. In the process, we develop a bosonization duality within the framework of tensor networks and provide an algorithm for bosonizing a fermionic tensor network state using its local tensors. We then focus on a class of interacting fermionic SPT phases classified by group supercohomology. We construct an exactly solvable Hamiltonian for each two- and three-dimensional supercohomology SPT phase. We also identify explicit finite-depth quantum circuits capable of disentangling the ground statesof our models. We illustrate how the structures of the circuit can give rise to anomalous topological order on the boundary of the three-dimensional models.