High-Fidelity Scalable Quantum State Preparation via the Fusion Method

TL;DR

This work tackles scalable preparation of specific eigenstates in quantum many-body systems by extending the Rodeo Algorithm with a fusion method that preconditions inputs via adiabatic ramps and builds the full system from smaller blocks. The fusion approach, combining adiabatic preconditioning with coherent RA refinements in a binary fusion scheme, delivers robust exponential convergence across system sizes for the XX spin-1/2 model, outperforming both pure adiabatic and unmodified RA at infidelities below about $10^{-3}$. Tensor-network simulations validate the approach and show practical cost savings, highlighting the method's suitability for 1D and quasi-1D geometries on NISQ and post-NISQ devices. Overall, the paper presents a general hybrid paradigm—preconditioning approximate solutions followed by fast purification—that enables scalable quantum state preparation, with potential extensions to higher dimensions and alternative lattice constructions.

Abstract

Robust and efficient eigenstate preparation is a central challenge in quantum simulation. The Rodeo Algorithm (RA) offers exponential convergence to a target eigenstate but suffers from poor performance when the initial state has low overlap with the desired eigenstate, hindering the applicability of the original algorithm to larger systems. In this work, we introduce a fusion method that preconditions the RA state by an adiabatic ramp to overcome this limitation. By incrementally building up large systems from exactly solvable subsystems and using adiabatic preconditioning to enhance intermediate state overlaps, we ensure that the RA retains its exponential convergence even in large-scale systems. We demonstrate this hybrid approach using numerical simulations of the spin- 1/2 XX model and find that the Rodeo Algorithm exhibits robust exponential convergence across system sizes. We benchmark against using only an adiabatic ramp as well as using the unmodified RA, finding that for state preparation precision at the level of $10^{-3}$ infidelity or better there a decisive computational cost advantage to the fusion method. These results together demonstrate the scalability and effectiveness of the fusion method for practical quantum simulations.

Figures (4)

• Figure 1: (a) Rodeo Algorithm circuit. Ancilla qubits are initialized in $|1\rangle$ and are used as control qubits for controlled time evolution. Each ancilla corresponds to a time sample $t_j$. A phase kickback from the controlled unitaries is propagated through the ancillas and a successful Rodeo Algorithm application is categorized by measurement of all $|1\rangle$ states on the ancillas after the algorithm choi2021rodeo. Specifically, if $|\psi _i\rangle$ is an eigenstate of $\hat{H}$ with energy $E_t,$ the phase kickback from the unitary will be exactly canceled by the phase gate, such that the ancilla always measures one. Contrastingly, when $|\psi _i\rangle$ is not the exact targeted eigenstate, the orthogonal components of the input state will cause phase kickbacks that do not get canceled by the phase gate, and lead to a decrease in the success probability. (b) Illustration of the fusion approach. Instead of directly preparing a many-body quantum state, we subdivide the system, in this case, in a binary fashion. Each "fusion step" consists of an (i) adiabatic preconditioning and the (ii) RA purification. (iii) Fusion steps are repeated until the full system is reconstructed.
• Figure 2: Fusion method results for the XX chain for finding the ground state of a chain of length $L$ by fusing two chains of length $L/2$. The expected unitless cost for each method, $J\kappa$, is plotted for various infidelities. Yellow lines with circle markers are for pure adiabatic evolution (method 1), red lines with square markers are for the unmodified rodeo algorithm (method 2), and blue lines with triangle markers are for the hybrid method (method 3). Color saturation represents the merged system size $L \in \{4,8,16,32,64,128,256\}$, with lightest saturation for $L=4$ and darkest saturation for $L=256$. Results for small infidelities are extrapolated for the largest system sizes $L=128,256$ from the first two datapoints. Results for the unmodified RA are an upper bound on its cost.
• Figure 3: General scheme of the performance of hybrid methods.
• Figure 4: Ground state preparation in the quarter-filled sector of the XX model. Yellow lines with circle markers are for pure adiabatic evolution (method 1), red lines with square markers are for the unmodified rodeo algorithm (method 2), and blue lines with triangle markers are for the hybrid method (method 3). Color saturation represents the merged system size $L \in \{8,16,32,64,128\}$.