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  • Open Access

Fast Quantum Simulation of Electronic Structure by Spectral Amplification

Guang Hao Low1,*,†, Robbie King1,2, Dominic W. Berry3, Qiushi Han4, A. Eugene DePrince, III1,5, Alec F. White6, Ryan Babbush1, Rolando D. Somma1, and Nicholas C. Rubin1,†,‡

  • *Contact author: guanghaolow@google.com
  • †Contact author: nickrubin@google.com
  • ‡These authors contributed equally to this work.
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Phys. Rev. X 15, 041016 – Published 31 October, 2025

DOI: https://doi.org/10.1103/pb2g-j9cw

Abstract

The most advanced techniques using fault-tolerant quantum computers to estimate the ground-state energy of a chemical Hamiltonian involve compression of the Coulomb operator through tensor factorizations, enabling efficient block encodings of the Hamiltonian. A natural challenge of these methods is the degree to which block-encoding costs can be reduced. We address this challenge through the technique of spectral amplification, which magnifies the spectrum of the low-energy states of Hamiltonians that can be expressed as sums of squares. Spectral amplification enables estimating ground-state energies with significantly improved cost scaling in the block encoding normalization factor Λ to just 2ΛEgap, where Egap≪Λ is the lowest energy of the sum-of-squares Hamiltonian. To achieve this, we show that sum-of-squares representations of the electronic structure Hamiltonian are efficiently computable by a family of classical simulation techniques that approximate the ground-state energy from below. In order to further optimize, we also develop a novel factorization that provides a trade-off between the two leading Coulomb integral factorization schemes—namely, double factorization and tensor hypercontraction—that when combined with spectral amplification yields a factor of 4 to 195 speedup over the state of the art in ground-state energy estimation for models of iron-sulfur complexes and a CO2-fixation catalyst.

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Popular Summary

Simulating molecular systems is widely expected to be one of the first valuable applications of quantum computers, but a primary challenge lies in estimating ground-state energies with low quantum gate costs. Over the past several years, quantum algorithms for this task have improved dramatically, but progress has recently slowed, signaling the need for fresh strategies. In this work, we introduce a new approach that reduces the number of quantum gates needed by combining two ideas: spectral amplification and sum-of-squares representations of the Hamiltonian.

Spectral amplification works by lowering the precision requirements for energy estimation when we are already working in the low-energy regime. However, for this method to be applied, the Hamiltonian—which describes the total energy of a system—must be written in a special mathematical form called a sum of squares. We show how to construct this representation effectively and explain the trade-offs involved. Building on this, we develop a way of factoring Coulomb integrals that fits naturally into the sum-of-squares framework. This not only makes spectral amplification possible but also optimizes other important aspects of the quantum circuits used for estimating ground-state energies.

This is the first demonstration of combining spectral amplification, optimized sum-of-squares representations, and integral compression into a single framework. Looking ahead, we expect these methods to apply broadly, not only in quantum chemistry but also in other quantum and classical simulations where low-energy problems are central.

Article Text

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Outline

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Phys. Rev. X 15, 041016– Published 31 October, 2025
  • Received 6 March 2025
  • Revised 17 July 2025
  • Accepted 9 September 2025
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Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

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