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arXiv:2508.02578

Quantum chemistry with provable convergence via randomized sample-based Krylov quantum diagonalization

Samuele Piccinelli, Alberto Baiardi, Barison, Stefano, Max Rossmannek, Almudena Carrera Vazquez, Francesco Tacchino, Stefano Mensa, Edoardo Altamura, Ali Alavi, Mário Motta, Javier Robledo-Moreno, Kirby, William, Kunal Sharma, Antonio Mezzacapo, Ivano Tavernelli

Published Aug 4, 2025• 4 citations
arXivHTMLPDF

Abstract

Quantum algorithms based on classical processing of individual samples have recently emerged as a promising route to perform large-scale quantum-chemical calculations on pre-fault-tolerant and early-fault-tolerant quantum devices. In these algorithms, the quantum computer acts as a sampling engine that generates the subspace in which the Hamiltonian is classically diagonalized. The recently proposed Sample-based Krylov Quantum Diagonalization (SKQD) uses quantum Krylov states as circuits from which samples are collected. Convergence guarantees can be derived for SKQD under assumptions similar to those of quantum phase estimation -- namely, a sufficiently large overlap between the reference and the target ground state -- with the additional requirement that the ground-state wave function be well approximated by a polynomial subset of the full Hilbert space. However, implementations of SKQD for complex many-body Hamiltonians, such as quantum chemistry ones, are limited by the depths of time-evolution circuits needed to generate Krylov vectors. In this work, we introduce a method that combines SKQD with a qDRIFT randomized compilation of the Hamiltonian propagator. The resulting algorithm, termed SqDRIFT, enables executing quantum chemistry experiments on present-day quantum processors while retaining provable convergence guarantees, which we extend to account for the error of the randomized compilation and of finite sampling. We demonstrate its viability by applying SqDRIFT to calculate the electronic ground-state energy of several polycyclic aromatic hydrocarbons, up to system sizes beyond the reach of exact diagonalization.

DOI
10.48550/arxiv.2508.02578

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Quantum algorithms based on classical processing of individual samples have recently emerged as a promising route to perform large-scale quantum-chemical calculations on pre-fault-tolerant and early-fault-tolerant quantum devices. In these algorithms, the quantum computer acts as a sampling engine that generates the subspace in which the Hamiltonian is classically diagonalized. The recently proposed Sample-based Krylov Quantum Diagonalization (SKQD) uses quantum Krylov states as circuits from which samples are collected. Convergence guarantees can be derived for SKQD under assumptions similar to those of quantum phase estimation -- namely, a sufficiently large overlap between the reference and the target ground state -- with the additional requirement that the ground-state wave function be well approximated by a polynomial subset of the full Hilbert space. However, implementations of SKQD for complex many-body Hamiltonians, such as quantum chemistry ones, are limited by the depths of time-evolution circuits needed to generate Krylov vectors. In this work, we introduce a method that combines SKQD with a qDRIFT randomized compilation of the Hamiltonian propagator. The resulting algorithm, termed SqDRIFT, enables executing quantum chemistry experiments on present-day quantum processors while retaining provable convergence guarantees, which we extend to account for the error of the randomized compilation and of finite sampling. We demonstrate its viability by applying SqDRIFT to calculate the electronic ground-state energy of several polycyclic aromatic hydrocarbons, up to system sizes beyond the reach of exact diagonalization.
Approximating the ground state of many-body systems is a key computational bottleneck underlying important applications in physics and chemistry. The most widely known quantum algorithm for ground state approximation, quantum phase estimation, is out of reach of current quantum processors due to its high circuit-depths. Subspace-based quantum diagonalization methods offer a viable alternative for pre- and early-fault-tolerant quantum computers. Here, we introduce a quantum diagonalization algorithm which combines two key ideas on quantum subspaces: a classical diagonalization based on quantum samples, and subspaces constructed with quantum Krylov states. We prove that our algorithm converges in polynomial time under the working assumptions of Krylov quantum diagonalization and sparseness of the ground state. We then demonstrate the scalability of our approach by performing the largest ground-state quantum simulation of impurity models using a Heron quantum processors and the Frontier supercomputer. We consider both the single-impurity Anderson model with 41 bath sites, and a system with 4 impurities and 7 bath sites per impurity. Our results are in excellent agreement with Density Matrix Renormalization Group calculations.
Sample-based Krylov quantum diagonalization (SKQD) is a paradigmatic example of a quantum-centric supercomputing workflow that combines the convergence structure of Krylov quantum diagonalization with classical sampling-based post-processing. It provides convergence guarantees assuming that the important computational-basis configurations can be sampled from a set of Krylov states with sufficient probability. We analyze this assumption under depolarizing noise, deriving shot-count resource estimates that expose an exponential depth penalty, and perform experiments on current noisy hardware with the one-dimensional single-impurity Anderson model on a 20-site (40-qubit) instance. Device noise breaks the practical convergence predicted by the noiseless SKQD analysis, but approximate compilation techniques can compress the Krylov time-evolution circuits before execution. The compressed circuits accumulate less hardware noise, recover the expected energy convergence, and reduce both the quantum shot budget and the classical subspace dimension, remaining beneficial even when classical configuration recovery is applied.
The estimation of low energies of many-body systems is a cornerstone of the computational quantum sciences. Variational quantum algorithms can be used to prepare ground states on pre-fault-tolerant quantum processors, but their lack of convergence guarantees and impractical number of cost function estimations prevent systematic scaling of experiments to large systems. Alternatives to variational approaches are needed for large-scale experiments on pre-fault-tolerant devices. Here, we use a superconducting quantum processor to compute eigenenergies of quantum many-body systems on two-dimensional lattices of up to 56 sites, using the Krylov quantum diagonalization algorithm, an analog of the well-known classical diagonalization technique. We construct subspaces of the many-body Hilbert space using Trotterized unitary evolutions executed on the quantum processor, and classically diagonalize many-body interacting Hamiltonians within those subspaces. These experiments demonstrate exponential convergence towards an estimate of the ground state energy, and show that quantum diagonalization algorithms are poised to complement their classical counterparts at the foundation of computational methods for quantum systems. The estimation of low energies of many-body systems is a cornerstone of the computational quantum sciences. This paper demonstrates on a superconducting quantum processor that the Krylov quantum diagonalization algorithm is poised to complement its classical counterparts at the foundation of computational methods for quantum systems.
Recently, sample-based quantum diagonalization (SQD) has emerged as a promising approach to compute ground and excited states of problem Hamiltonians.This method classically diagonalizes a Hamiltonian in a subspace that is spanned by samples obtained from a quantum computer. However, by its nature, SQD suffers from a fundamental sampling problem, as some basis states that are required for a targeted accuracy may only be sampled extremely rarely. To alleviate this limitation, we introduce the SQD-AA algorithm that combines SQD with amplitude amplification (AA). SQD-AA uses AA to sequentially reduce probabilities of already measured bitstrings, thus making the observation of new ones more likely. We observe a reduction in the total query complexity of more than a factor 100 for algebraically and exponentially decaying model distributions, and analytically show a quadratic advantage for the latter. Moreover, we evaluate real molecules in an early fault-tolerant scenario and compare SQD-AA to SQD and iterative quantum phase estimation (iQPE). For all considered examples, we observe the lowest total number of T-gates for SQD-AA while only requiring circuits that are 3-4 orders of magnitude shallower than those needed for iQPE. Given this substantial reduction in circuit depth compared to iQPE while saving 2 orders of magnitude in total runtime compared to SQD, we expect a significant regime in early fault-tolerance where SQD-AA runs feasibly, but iQPE circuits are too deep to execute confidently.
We propose a class of randomized quantum Krylov diagonalization (rQKD) algorithms capable of solving the eigenstate estimation problem with modest quantum resource requirements. Compared to previous real-time evolution quantum Krylov subspace methods, our approach expresses the time evolution operator ${e}^{\ensuremath{-}i\stackrel{\ifmmode \hat{}\else \^{}\fi{}}{H}\ensuremath{\tau}}$ as a linear combination of unitaries and subsequently uses a stochastic sampling procedure to reduce circuit depth requirements. While our methodology applies to any Hamiltonian with fast-forwardable subcomponents, we focus on its application to the explicitly double-factorized electronic-structure Hamiltonian. To demonstrate the potential of the proposed rQKD algorithm on near-term quantum devices, we provide numerical benchmarks for a variety of molecular systems with circuit-based state-vector simulators including the effects of sampling noise, achieving ground-state energy errors of less than $1 \mathrm{kcal} {\mathrm{mol}}^{\ensuremath{-}1}$ with circuit depths orders of magnitude shallower than those required for low-rank deterministic Trotter-Suzuki decompositions.
Quantum subspace diagonalization and quantum Krylov algorithms offer a feasible, pre- or early-fault tolerant alternative to quantum phase estimation for using quantum computers to estimate the low-lying spectra of quantum systems. However, despite promising proof-of-principle results, such methods suffer from high sensitivity to noise (including intrinsic sources such as sampling noise), making their utility for realistic industry-relevant problems an open question. To improve the potential applicability of such methods, we introduce a new variant of thresholding for noisy generalized eigenvalue problems that arise in quantum subspace diagonalization that has the potential to better control sensitivity to noise. Our approach leverages eigenvector-preserving transformations (rotations) of the generalized eigenvalue problem prior to thresholding. We study this effect in practical settings by applying this rotation thresholding scheme to an iterative quantum Krylov algorithm for several chemical systems, including the industry-relevant Fe(III)-NTA chelate complex. We develop a particular heuristic to select the rotation angle from noisy data and find for certain systems and noise regimes that the samples required to reach a target error for ground state estimation can be reduced by a factor of up to 100. Furthermore, with oracle access to the optimal transformation, more dramatic improvements are possible and we observe reductions in sample requirements by up to 10410^4104, motivating the continued development of methods that can realize these improvements in practice. While we develop our approach in the context of quantum subspace diagonalization, the improved thresholding scheme we develop could be advantageous in any context where one must solve noisy, ill-conditioned generalized eigenvalue problems.