Enhancing Scalability of Quantum Eigenvalue Transformation of Unitary Matrices for Ground State Preparation through Adaptive Finer Filtering

TL;DR

This work tackles the bottleneck of ground-state preparation for large Hamiltonians on quantum devices by introducing Adaptive Finer Filtering (AFF), an approach that combines an adaptive sequence of QETU-based eigenspace filters with spectrum profiling to overcome small spectral gaps and state degeneracy. AFF preserves a fixed polynomial degree while progressively stretching the spectrum across multiple filtering stages, achieving a circuit-depth scaling of γ_AFF = O(Δ^{−1}) and reducing reliance on precise cut-off values μ. When paired with phase-estimation methods such as Robust Phase Estimation (RPE) or Quantum Complex Exponential Least Squares (QCELS), AFF yields substantial improvements in ground-state overlap and energy accuracy in TFIM simulations, albeit with increased total simulation time due to spectrum profiling steps. Overall, the approach offers a scalable path to ground-state preparation on quantum hardware, with clear opportunities for integration with pre-processing and classical state-preparation techniques to further enhance practicality and resilience to noise.

Abstract

Hamiltonian simulation is a domain where quantum computers have the potential to outperform their classical counterparts. One of the main challenges of such quantum algorithms is increasing the system size, which is necessary to achieve meaningful quantum advantage. In this work, we present an approach to improve the scalability of eigenspace filtering for the ground state preparation of a given Hamiltonian. Our method aims to tackle limitations introduced by a small spectral gap and high degeneracy of low energy states. It is based on an adaptive sequence of eigenspace filtering through Quantum Eigenvalue Transformation of Unitary Matrices (QETU) combined with spectrum profiling. By combining our proposed algorithm with state-of-the-art phase estimation methods, we achieved good approximations for the ground state energy with local, two-qubit gate depolarizing probability up to $10^{-4}$. To demonstrate the key results in this work, we ran simulations with the transverse-field Ising Model on classical computers using Qiskit. We compare the performance of our approach with the static implementation of QETU and show that we can consistently achieve three to four orders of magnitude improvement in the absolute error rate.

Figures (7)

• Figure 1: Quantum Eigenvalue Transformation of Unitary Matrices (QETU) Circuit in compact notation where $U$ is the multi-qubit gate applying the time evolution operator $U=e^{-iH}$, acting on all the system qubits and the X-Rotation gates are applied to the ancilla qubit. Symmetric phases $(\phi_0, \phi_1, \dots \phi_1, \phi_0) \in \mathds{R}^{\eta+1}$ are optimized for a given target polynomial $F(a)$.
• Figure 2: Example polynomial $F(a)$ of degree 30, as an approximation of the even step function with cut-off value $\mu = 0.75$. We used Chebyshev polynomials of the second kind as basis and employed the convex optimization library "cvxpy" to optimize coefficients of the Chebyshev polynomials.
• Figure 3: Hadamard test circuit used to compute the Fourier moments in Eq. \ref{['eq:fourier_moments']}. The identity $I$ is inserted to compute $\text{Re}\braket{\psi|\mathop{\mathrm{e}}\nolimits^{-i k H}|\psi}$ and the conjugated phase gate $S^{\dag}$ is used to compute $\text{Im}\braket{\psi|\mathop{\mathrm{e}}\nolimits^{-i k H}|\psi}$.
• Figure 4: Example plots of cumulative distribution functions (CDF) from the first (top) and second (bottom) filtering stages, plotted next to its first and second derivatives. Fourier moments were acquired through Hadamard sampling (Fig. \ref{['fig:hadamard_circuit']}), using $10^3$ shots and simulated with a local depolarizing probability of $10^{-3}$ for the two-qubit gates. Red horizontal lines represent precision bounds for the spectrum profiling $\xi_1, \xi_2$, and green vertical lines represent new upper/lower spectrum bounds resulting from the spectrum profiling. The Hamiltonian is the TFIM with system size $L=6$ and parameters $J=1$, $g=1$. For the top graph: $(D, \beta, \lambda_{\text{LB}}, \lambda_{\text{UB}}) = (7, 5, -10, 10)$ and the input state $\ket{\psi}$ was taken as the result of the initial filtering by QETU with $\eta = 14$, $\mu = 0.95$, applied to a randomized state $\ket{\psi_{\text{init}}}$. For the bottom graph: $(D, \beta, \lambda_{\text{LB}}, \lambda_{\text{UB}}) = (7, 5, -8.78, -1.43)$ and the input state $\ket{\psi}$ was taken as the result of the secondary filtering by QETU with $\eta = 14$, $\mu = 0.92$, applied to the result of the initial filtering.
• Figure 5: Ground state energy estimation using Adaptive Finer Filtering (AFF), combined with Robust Phase Estimation rpe (left) and QCELS qcels (right). For comparison, the performance of statically applying QETU with the same number of repetitions $M=3$ and combining it with RPE / QCELS is plotted alongside in each graph. Simulations were conducted with $10^4$ (of which around $1\%$ corresponds to successful amplification) shots on the ancilla qubit per real and imaginary part of the targeted phase. The TFIM model with system size $L=6$ and system parameters $J=1, g=1$ (Spectral gap $\Delta \approx 0.263$) was used. Here, $p_{\text{Depolar}}$ represents the local depolarizing probability of two-qubit gates. The depolarizing probability of single-qubit gates is taken as $p_{\text{Depolar}}/10$.

Theorems & Definitions (6)

• Definition 1: Relative Amplification
• Definition 2: Simulation Time
• Definition 3: Maximal Simulation Time
• Definition 4: Total Simulation Time
• Definition 5: Circuit Depth
• Proposition