Efficient Berry Phase Calculation via Adaptive Variational Quantum Computing Approach

TL;DR

This work addresses computing geometric Berry phases for strongly correlated topological systems on near-term quantum devices. It introduces an adaptive variational quantum dynamics framework (AVQDS) combined with adaptive variational ground-state preparation (AVQITE) to perform cyclic adiabatic evolution with compact circuits. The method accurately reproduces Berry phases for a four-site SSHH model in both noninteracting and interacting regimes, including topological phase transitions, while dramatically reducing circuit depth and gate counts compared to fixed Trotter evolution. The results highlight the potential of variational quantum algorithms to efficiently simulate geometric properties and topological invariants in complex quantum many-body systems, with clear pathways for hardware deployment and future extensions.

Abstract

We present an adaptive variational quantum algorithm to estimate the Berry phase accumulated by a nondegenerate ground state under cyclic, adiabatic evolution of a time-dependent Hamiltonian. Our method leverages cyclic adiabatic evolution of the Hamiltonian and employs adaptive variational quantum algorithms for state preparation and evolution, optimizing circuit efficiency while maintaining high accuracy. We benchmark our approach on dimerized Fermi-Hubbard chains with four sites, demonstrating precise Berry phase simulations in both noninteracting and interacting regimes. Our results show that circuit depths reach up to 106 layers for noninteracting systems and increase to 279 layers for interacting systems due to added complexity. Additionally, we demonstrate the robustness of our scheme across a wide range of parameters governing adiabatic evolution and variational algorithm. These findings highlight the potential of adaptive variational quantum algorithms for advancing quantum simulations of topological materials and computing geometric phases in strongly correlated systems.

Figures (7)

• Figure 1: Hadamard test circuits for measuring the Berry phase. (a) Circuit for Berry phase measurement using the cyclic adiabatic operator $U_\text{loop}$. The ancillary qubit (upper horizontal line) is initialized in $\vert0\rangle$, with Hadamard gates $H$ applied to it. The register for physical system qubits (lower horizontal line) is initially in the reference product state $\vert\Psi_0\rangle$. The unitary operator $U_\mathrm{G}$ prepares the ground state, $\vert G(0)\rangle=U_\mathrm{G}\vert\Psi_0\rangle$, while the ancilla-controlled application of $U_\text{loop}$ enables cyclic adiabatic evolution. For the cyclic adiabatic evolution operator $U_\text{loop}$ defined in Eq. \ref{['eq:Uloop']}, measuring the ancilla qubit in the computational basis gives the probability $P_{\vert0\rangle}=\frac{1}{2}(1+\cos\varphi_\text{B})$ of observing the ancilla in state $\vert0\rangle$. (b) Berry phase measurement using a variational quantum circuit. The circuit employs parametrized unitaries $U_\mathrm{G}[\boldsymbol{\theta}^1]$ and $U_\mathrm{loop}[\boldsymbol{\theta}^2]$, generated via AVQITE and AVQDS, respectively. The angles of both parameterized unitaries evolve during the cyclic adiabatic evolution. After the adiabatic evolution, the circuit measures the overlap between the states $U_\mathrm{G}[\boldsymbol{\theta}^1(0)]\vert\Psi_0\rangle$ and $U_\text{loop}[\boldsymbol{\theta}^2(2\pi)]U_\mathrm{G}[\boldsymbol{\theta}^1(2\pi)]\vert\Psi_0\rangle$. For sufficient accurate adiabatic evolution, these two states differ only by a phase $\varphi_\mathrm{qc}$, leading to the measurement probability $p_{\vert0\rangle} = \frac{1}{2} \left(1 + \cos \varphi_\mathrm{qc} \right)$. Note that the practical implementation of the controlled-parameterized unitaries associated with the ancilla in (b) introduce only as many controlled single-qubit rotation gates as the number of variational parameters in the unitaries Libbi2022.
• Figure 2: Parameter-loop evolution illustrating dynamical-phase cancellation and Berry-phase addition. Forward half (blue arrow): $U_{\circlearrowleft}(0, T/2)$ with $t:0\to T/2$ and $\lambda:0\to\pi$. Return half (red arrow): $U_{\circlearrowleft}(T/2, 0)=U_{\circlearrowleft}(0,T/2)^\dagger$ executed with time-reversed propagation, $t:T/2\to 0$, while $\lambda:\pi\to 2\pi$ in the same loop orientation. Annotations indicate $-\varphi_{\mathrm D}/2 + \varphi_{\mathrm D}/2 = 0$ (dynamical-phase cancellation) and $\varphi_{\mathrm B}/2 + \varphi_{\mathrm B}/2 = \varphi_{\mathrm B}$ (Berry-phase addition).
• Figure 3: Benchmarking of the AVQDS approach for calculating the Berry phase of the noninteracting SSH model. (a) The energy $E(\lambda)$ of the quantum state evolved by AVQDS for $T = 20$ (black line), $T = 100$ (orange line), and $T = 200$ (blue line), compared to exact diagonalization (ED) results (red line), for a dimerization parameter of $\delta=-0.3$. For comparison, the first three ED excited energies $E_{1,2,3}(\lambda)$ are plotted in red with distinct dashed line styles. (b) Corresponding infidelity $1 - f$, showing improved accuracy with increasing $T$. The maximum infidelity across all $\lambda$ is $5.0\times 10^{-2}$ for $T=20$, $2.8\times 10^{-3}$ for $T=100$, and $4.2\times 10^{-4}$ for $T=200$. (c) Contributions to the global phase $\varphi_\mathrm{G}$: dynamical phase $\varphi_{\mathrm{G},1}(\lambda)$ (solid lines) and geometric phase contribution $\varphi_{\mathrm{G},2}(\lambda)$ (dashed lines), calculated by solving Eq. \ref{['eq:phiG']} using a fourth-order Runge-Kutta method. The red dashed line marks $\varphi_{\mathrm{G},i} = 0$. (d) Comparison of the Berry phase computed via Eq. \ref{['eq:phiB2']} with $\varphi_\mathrm{B}$ obtained from ED (red dashed line), demonstrating the robustness of the AVQDS approach. The Berry phase should be read at the end of the loop, $\lambda=2\pi$, where the curves for different $T$ converge, rather than across the entire $\lambda$ range. (e) Number of CNOT gates as a function of $\lambda$ with maximum counts of 496, 884, and 668 for $T = 20$, $T=100$, and $T = 200$, respectively. (f) Circuit depth (number of layers of unitaries) during cyclic state evolution, reaching 47 layers for $T = 20$, 92 layers for $T = 100$, and 68 layers for $T = 200$.
• Figure 4: Robustness of the AVQDS algorithm in Berry phase calculations. (a) Computed Berry phase $\varphi_\mathrm{B}$ as a function of the cyclic state evolution time $T$ for $\delta = -0.9$ (black circles) and $\delta = -0.5$ (orange diamonds). The ED result for $T=200$ is shown as a red dashed line for reference. Accurate results can already be obtained for small $T = 1.6$ ($T = 2.0$) for $\delta=-0.9$ ($\delta=-0.5$), despite strong nonadiabaticity in this regime. (b) Computed Berry phase as a function of time step size $\delta t$ using a fixed $T = 20$ for $\delta = -0.9$ (black circles) and $\delta = -0.5$ (orange diamonds), compared with the ED result (red dashed line). The Berry phase is accurately obtained for $\delta t \leq 0.5$, but significant deviations appear beyond this threshold. (c) Computed Berry phase as a function of McLachlan distance cutoff $L^2_\text{cut}$ using a fixed $T = 20$ for $\delta = -0.9$ (black circles) and $\delta = -0.5$ (orange diamonds), alongside the ED result (red dashed line). Accurate results are maintained for $L^2_\text{cut} \leq 10^{-1}$ ($10^{-2}$) for $\delta=-0.9$ ($\delta=-0.5$) before strong deviations emerge. (d-f) Corresponding maximum infidelities during the cyclic state evolutions, $\max_t[1 - f(t)]$, for the data in panels (a-c). Apart from the $T$-dependence, the infidelity remains low below the critical parameter values but increases significantly above them. (g-i) Corresponding CNOT gate counts for the data in panels (a-c). The CNOT count remains stable with increasing $T$ (g), decreases with increasing $\delta t$ for $\delta t=-0.9$ but remains stable for $\delta t=-0.5$ until the accuracy threshold (h), and decreases by a factor of 4 (3) for $\delta=-0.9$ ($\delta=-0.5$) as $L^2_\text{cut}$ increases to its critical value (i).
• Figure 5: Quantum circuit complexity and fidelity for Berry phase calculations across a topological phase transition in the noninteracting SSH model. (a) Computed Berry phase $\varphi_\mathrm{B}$ as a function of the dimerization parameter $\delta$ for evolution times $T = 20$ (black circles), $T = 100$ (orange diamonds), and $T = 200$ (blue triangles), compared to exact diagonalization (ED) results (gray squares). All AVQDS calculations accurately reproduce the ED results and resolve the phase transition at $\delta = 0$. (b) Maximum infidelity during the cyclic adiabatic evolution as a function of $\delta$ for the three values of $T$. The infidelity peaks near the phase transition at $\delta = 0$ from both sides and decreases with increasing $T$, staying below $3.6\times 10^{-2}$ for $T = 200$. Inset: Lowest five many-body energies $E_{0\ldots4}$ vs. dimerization $\delta$ under a twisted boundary with $\lambda=\pi$. The gap $\Delta=E_1-E_0$ narrows as $\delta\!\to\!0$, explaining the rise of nonadiabatic effects and the need for longer evolution times near the transition at $\delta=0$. (c) Number of CNOT gates as a function of $\delta$ for the three different $T$. The highest CNOT count occurs for $T = 100$ in the nontrivial topological region ($\varphi_\mathrm{B} = \pi$), while all three cases require comparable CNOT numbers in the trivial phase region. The nontrivial phase requires up to four times more CNOT gates than the trivial phase region. (d) Circuit depth, measured in number of layers, as a function of $\delta$ for the three different $T$. The maximum depths are 51 layers for $T = 20$, 106 layers for $T = 100$, and 89 layers for $T = 200$. Note that for $\delta=\pm 1$, the model reduces to isolated dimers and dangling edges (for $\delta=-1$), leading to much simpler circuits.