Quantum circuit algorithm for topological invariants of second order topological many-body quantum magnets

TL;DR

Problem: calculating many-body topological invariants in interacting quantum magnets is computationally hard. Approach: a quantum-circuit algorithm using adiabatic evolution and Hadamard tests to extract Berry phases along parameter loops for first-order 1D and second-order 2D SPTs. Findings: demonstrated quantized Berry phases of 0 or π corresponding to trivial and HOSPT phases, with path-dependent behavior and degeneracies illustrating gauging subtlety. Limitations: currently enormous circuit depths make implementation on near-term devices impractical, but the framework points toward future fault-tolerant quantum hardware enabling systematic classification of many-body topological matter.

Abstract

Topological quantum matter represents a flexible playground to engineer unconventional excitations. While non-interacting topological single-particle systems have been studied in detail, topology in quantum many-body systems remains an open problem. Specifically, in the quantum many-body limit, one of the challenges lies in the computational complexity of obtaining the many-body ground state and its many-body topological invariant. While algorithms to compute ground states with quantum computers have been heavily investigated, algorithms to compute topological invariants in a quantum computer are still under active development. Here we demonstrate a quantum circuit to compute the many-body topological invariant of a second-order topological quantum magnet encoded in qubits. Our algorithm relies on a quantum circuit adiabatic evolution in transverse paths in parameter space, and we uncover hidden topological invariants depending on the traversed path. Our work puts forward an algorithm to leverage quantum computers to characterize many-body topological quantum matter.

Figures (4)

• Figure 1: (a) An instantaneous eigenstate evolved along a closed parameter path will pick up a Berry phase, which is a measure of how the complex vector changes along the path. This effect is analogous to parallel transport of vectors in curved geometry. (b) Proposed quantum circuit with a Hadamard test on an adiabatic time evolution $U(t_0, t_1)$ which allows to measure $\varphi_B$.
• Figure 2: (a) Schematic of a dimerized Heisenberg model featuring an SPT phase, with alternating bond strengths $J_1$ and $J_2$, and their respective Berry phase for $J_1 > J_2$. (b) We show the Berry phase on the first bond as a function of the interaction strength difference $J_2 - J_1$, with a comparison to the exact solution (in blue). With increasing system size $L$, convergence towards the exact solution improves. A phase transition can be seen where the Berry phase $\varphi_B$ changes from $\pi \rightarrow 0$, corresponding to topologically non-trivial and trivial states.
• Figure 3: Illustration of a two-dimensional tetramerized Heisenberg spin model and gauged twists, on an exemplary $4\times 4$ system. The alternating interaction strengths $J_1$ and $J_2$ are indicated by solid and dashed lines, respectively. This system exhibits a plaquette structure. Plaquettes of type I contain only nearest neighbor interactions of strength $J_1$, plaquettes of type II consist of strengths $J_2$, and plaquettes of type III are mixed. In order to calculate the Berry phase, we gauge a plaquette with four twists $\varphi_j$ on each bond.
• Figure 4: Berry phase calculation on a plaquette as a function of the interaction strength difference $J_2 - J_1$, with a comparison to the exact solution (in blue). With increasing time steps $N$, convergence towards the exact solution improves. A phase transition can be seen where the Berry phase $\varphi_B$ changes from $\pi \rightarrow 0$, corresponding to higher-order topologically non-trivial and trivial states.