Complexity of estimating Berry phase in 2-local quantum systems revealed
On the Computational Complexity of Guided Berry Phase Estimation
Computational Complexity
Summary
This work studies how hard it is to figure out a special quantum property called the Berry phase for systems made of interacting quantum bits (qubits). It shows that when you have a guiding state close to the system's lowest energy state, deciding this phase is just as tough as the hardest problems a quantum computer can solve. The authors prove this for different types of qubit interactions and lattice layouts, and also show that for simpler systems with only one qubit interacting at a time, the Berry phase can be calculated efficiently. This marks a clear difference in difficulty depending on how qubits interact.
What this means in practice
- •For quantum hardware designers: Understanding the computational limits for estimating Berry phases guides the design of quantum devices relying on controlled qubit interactions in 2D layouts.
- •For quantum algorithm developers: This result clarifies when computational resources are sufficient to simulate geometric quantum effects, informing algorithm development for error mitigation and state characterization.
A theory result. No direct application yet.
Authors
Gabriel Waite
Abstract
We prove that deciding the Berry phase for parameterised 2-local qubit Hamiltonians is BQP-complete when presented with a classical description of a guiding state, promised to overlap with the ground state of the system. Our results extend to systems with weighted Heisenberg interactions and when restricted to a 2D square or triangular lattice geometry. The techniques we develop leverage the Schrieffer--Wolff transformation, typically used in the construction of perturbative gadget reductions for local Hamiltonian problems, extending it to parameterised families of Hamiltonians. We demonstrate that there exists a choice of parameterised simulator Hamiltonians whose Berry phase well-approximates that of a parameterised target family. Using the perturbative gadget reduction framework of Oliveira and Terhal and of Schuch and Verstraete, we adapt the arguments to parameterised interactions and demonstrate the error bounds in the resulting simulation can be controlled. Additionally, we provide an explicit proof that families of 1-local Hamiltonians have a Berry phase that can be efficiently computed to inverse-polynomial precision. This establishes a complexity transition between 1-local and 2-local Hamiltonian families.