Improved Rodeo Algorithm Performance for Spectral Functions and State Preparation

TL;DR

The paper tackles the sensitivity of the Rodeo Algorithm (RA) to time-sampling in quantum state purification and spectrum estimation. It introduces generalized superiterations—geometric time sequences governed by a single parameter $\alpha$—and shows that adaptive optimization of $\alpha$ yields near-optimal exponential suppression of undesired spectral components across gapped Hamiltonians, supported by analytic insights linked to Bernoulli convolutions and Pisot-number theory. Numerical experiments on XX and TFIM models illustrate robust performance improvements, including near-critical regimes and poor initial overlaps, outperforming Gaussian-random sampling in shot-by-shot reliability. The work provides a practical, model-agnostic protocol and public optimization tools that can significantly reduce runtime and hardware demands for quantum simulations.

Abstract

The Rodeo Algorithm is a quantum computing method for computing the energy spectrum of a Hamiltonian and preparing its energy eigenstates. We discuss how to improve the performance of the rodeo algorithm for each of these two applications. In particular, we demonstrate that using a geometric series of time samples offers a near-optimal optimization space for a given total runtime by studying the Rodeo Algorithm performance on a model Hamiltonian representative of gapped many-body quantum systems. Analytics explain the performance of this time sampling and the conditions for it to maintain the established exponential performance of the Rodeo Algorithm. We finally demonstrate this sampling protocol on various physical Hamiltonians, showing its practical applicability. Our results suggest that geometric series of times provide a practical, near-optimal, and robust time-sampling strategy for quantum state preparation with the Rodeo Algorithm across varied Hamiltonians without requiring model-specific fine-tuning.

Figures (6)

• Figure 1: Examples results shown for $N=10$, $\Delta _{\text{min}} = 0.1$, $\Delta _{\text{max}} = 1$. Solid lines correspond to time samples in the generalized superiteration subspace, while dashed lines correspond to the global optimum across all time samples with $N\leq 10$ and total time less than or equal to the indicated constraint.
• Figure 2: XX chain results for $\ket{\psi} = \ket{e_1}$ with $N=100$ time samples and a system size of $L=10$. We define the fidelity $F = |\braket{\psi '|\psi _0}|^2$ where $\ket {\psi _0}$ is the true zero-magnetization sector ground state. Solid colored lines represent fixed-$\alpha$ generalized superiterations that are fit to have total time $T$. The solid black line (Adaptive $\alpha$) corresponds to optimizing over $\alpha$ at each value of $T$. The red line with discrete markers corresponds to the averaged performance of the Gaussian random RA, with the distribution's variance optimized for each point.
• Figure 3: XX chain with $N=100$ time samples and a system size of $L=10$. We begin with a fusion ansatz. Minimization over $\alpha$ is done assuming it is monotonically decreasing.
• Figure 4: Optimal $\alpha$ across $T$ for the XX chain for the RA ansatz $\ket {e_1}$ and the fusion ansatz.
• Figure 5: TFIM results. (a) Fidelity versus total time for various values of $h/J$. All solid lines indicate $h/J=1$, dotted lines correspond to $h/J= 0.5$, and dashed lines correspond to $h/J = 3$, as indicated by the legend of (b). We plot the superiterations with $\alpha = 2$ for reference in the golden lines. The red datapoints indicate the RRA performance while the black lines correspond to the optimal generalized superiteration performance. (b) Value of the optimal $\alpha$ as a function of total time. $\alpha$ is capped at $2$, hence the plateau at small $T$ where the true optimum is greater than 2.