Computational complexity of Berry phase estimation in topological phases of matter

TL;DR

This work investigates the quantum computational complexity of Berry phase estimation (BPE) in topological phases, revealing a sharp separation between regimes with and without a guiding state. It introduces a new quantum algorithm that computes the Berry phase by combining two adiabatic evolutions with different runtimes, enabling estimation of $\theta_B$ modulo $2\pi$ with inverse-polynomial precision in polynomial time when a guiding state is available (placing BPE with guiding state in $\mathsf{BQP}$). It then defines a novel complexity class, $\mathsf{dUQMA}$, to capture BPE without a guiding state and proves $\mathsf{dUQMA}$-completeness for BPE with an energy threshold, as well as $\mathsf{P}^{\mathsf{dUQMA[log]}}$-hardness and containment in $\mathsf{P}^{\mathsf{PGQMA[log]}}$ for the general case. The results establish a quantum advantage for topological invariant estimation and connect Berry-phase computation to natural quantum analogue complexity classes, offering a roadmap for further complexity-theoretic analysis of topological phenomena. The work also discusses extensions to other invariants, potential classical algorithms, and reductions to physically motivated Hamiltonians. Overall, the paper provides foundational links between topological matter and quantum computational complexity, with broad implications for quantum algorithms and complexity theory.

Abstract

The Berry phase is a fundamental quantity in the classification of topological phases of matter. In this paper, we present a new quantum algorithm and several complexity-theoretical results for the Berry phase estimation (BPE) problems. Our new quantum algorithm achieves BPE in a more general setting than previously known quantum algorithms, with a theoretical guarantee. For the complexity-theoretic results, we consider three cases. First, we prove $\mathsf{BQP}$-completeness when we are given a guiding state that has a large overlap with the ground state. This result establishes an exponential quantum speedup for estimating the Berry phase. Second, we prove $\mathsf{dUQMA}$-completeness when we have $\textit{a priori}$ bound for ground state energy. Here, $\mathsf{dUQMA}$ is a variant of the unique witness version of $\mathsf{QMA}$ (i.e., $\mathsf{UQMA}$), which we introduce in this paper, and this class precisely captures the complexity of BPE without the known guiding state. Remarkably, this problem turned out to be the first natural problem contained in both $\mathsf{UQMA}$ and $\mathsf{co}$-$\mathsf{UQMA}$. Third, we show $\mathsf{P}^{\mathsf{dUQMA[log]}}$-hardness and containment in $\mathsf{P}^{\mathsf{PGQMA[log]}}$ when we have no additional assumption. These results advance the role of quantum computing in the study of topological phases of matter and provide a pathway for clarifying the connection between topological phases of matter and computational complexity.

Figures (3)

• Figure 1: The problem that we study in this paper: what is the (quantum) computational complexity of estimating the Berry phase? How can we estimate the Berry phase with quantum computers?
• Figure 2: The Berry phase estimation problem is to decide whether $\theta_B$ is in the red region or the blue region in $[0,2\pi)$ for the given $a,b$.
• Figure 3: Schematic of the Berry phase estimation algorithm. Two independent quantum phase estimation (QPE) procedures are run on the adiabatic propagators $U(T)$ and $U(\alpha T)$, each with input $\ket{\psi(0)}$ prepared from the guiding state $\ket{c}$. The outcomes $m_T$ and $m_{\alpha T}$ encode the phases $\varphi_1 \equiv \theta_B + \theta_D$ and $\varphi_\alpha \equiv \theta_B + \alpha\theta_D \pmod{2\pi}$, respectively. These classical outputs are then combined in post-processing to cancel the dynamical phase and reconstruct the Berry phase $\theta_B$.

Theorems & Definitions (26)

• Definition 1: BQP
• Definition 2: QMA
• Definition 3: UQMA aharonov2022pursuit
• Definition 4: PGQMA aharonov2022pursuit
• Definition 5: Berry phase estimation with a guiding state
• Theorem 1
• Theorem 2: Quantum algorithm for Berry phase estimation
• Definition 6: Berry phase estimation with energy threshold
• Definition 7: dUQMA
• Theorem 3