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Fast quantum simulation of electronic structure by spectrum amplification

Guang Hao Low, Robbie King, Dominic W. Berry, Qiushi Han, A. Eugene DePrince, Alec White, Ryan Babbush, Rolando D. Somma, Nicholas C. Rubin

arXiv:2502.15882v1quant-phphysics.chem-ph

TL;DR

The paper addresses the high block-encoding cost of fault-tolerant quantum simulation for electronic structure. It combines spectrum amplification with SOS representations and a new DFTHC factorization, achieving lower resource estimates, including a factor-of-4 FeMoco reduction and improvements up to 200 on other systems.

  • Problem

    Ground-state energy estimation remains costly because electronic-structure Hamiltonians require efficient Coulomb-operator compression and low-cost block-encodings.

  • Method

    The paper combines spectrum amplification for sum-of-squares Hamiltonians with classical SOS lower-bound constructions, rectangular square-root block-encodings, and the DFTHC tensor factorization.

  • Results

    A factor of 4 cost reduction is reported for a larger FeMoco active space, with improvements of up to 200 for other benchmark systems.

  • Takeaways & Limitations

    The combined approach brings effective LCU normalization within a factor of two of optimal for all studied systems and shows that the main obstacles to combining spectrum amplification with energy estimation can be overcome.

Abstract

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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 spectrum amplification, which magnifies the spectrum of the low-energy states of Hamiltonians that can be expressed as sums of squares. Spectrum amplification enables estimating ground-state energies with significantly improved cost scaling in the block encoding normalization factor $Λ$ to just $\sqrt{2ΛE_{\text{gap}}}$, where $E_{\text{gap}} \ll Λ$ 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 spectrum 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 CO$_{2}$-fixation catalyst.

I. INTRODUCTION

Electronic-structure simulation is limited by block-encoding costs, motivating spectrum amplification and a new SOS-compatible factorization. The paper combines these tools to reduce ground-state energy-estimation resources while addressing efficient SOS construction and error control.

  • Motivation: Quantum phase estimation offers polynomial-cost electronic-structure simulation, but its cost scales with the Hamiltonian block-encoding normalization set by Coulomb-operator compression.Double factorization and tensor hypercontraction are the two principal compression strategies discussed.
  • Spectrum amplification: Spectrum amplification magnifies low-energy eigenvalues by simulating a square-root Hamiltonian, allowing coarser eigenvalue estimation when the square root is not too costly to simulate.The approach applies to sum-of-squares Hamiltonians with an energy shift making H − E_SOS positive semidefinite.
  • Spectrum amplification: The paper develops rectangular block-encodings and a two-step quantum-walk procedure that retain standard phase-estimation techniques while avoiding the larger factors of earlier square-root methods.Two quantum-walk steps restore the use of standard phase estimation despite the interchange of left and right singular values.
  • SOS representations: Sum-of-squares representations connect ground-state lower bounds to classical optimization relaxations and can produce small gaps for chemistry Hamiltonians.A spin-free level-2 SOS algebra already yields small gaps and improves ground-state energy estimation, while more bespoke algebras trade gap size against block-encoding cost.
  • Factorization and resources: DFTHC combines SOS structure with tensor compression to directly reduce the square-root block-encoding normalization while remaining close to the optimal SOS energy shift.For molecules with up to N = 150 orbitals, reported scalings include block-encoding Toffoli costs of ~N^0.96, λ_sqrt^2 ~N^1.46, and E_gap ~N^0.88.
  • Factorization and resources: Combining DFTHC, BLISS, and spectrum amplification reduces estimated costs by a factor of 4 for a larger FeMoco active space, with improvements of up to 200 for other benchmark systems.The estimates target chemical accuracy, and randomized truncation and optimization conditions provide an alternative treatment of DFTHC approximation errors.

II. QUANTUM CIRCUITS FOR SPECTRUM AMPLIFICATION

Spectrum amplification uses sum-of-squares Hamiltonians and block-encodings of their square roots to amplify sensitivity to low-energy eigenvalues. The construction uses rectangular block-encodings and quantum walks, with lower-energy shifts reducing the effective scaling constant.

  • Spectrum amplification: Spectrum amplification applies to sum-of-squares Hamiltonians whose shifted form is positive semidefinite, with the lowest energy Egap determining the amplified low-energy spectrum.The shift ESOS makes H − ESOS positive semidefinite, while Egap = Egs − ESOS measures proximity to frustration freeness.
  • Square-root block-encoding: The square-root Hamiltonian Hsqrt satisfies Hsqrt†Hsqrt = HSA, so singular values of Hsqrt encode square roots of HSA eigenvalues.This construction enables estimating the original energies through singular-value estimation of a square-root operator.
  • Square-root block-encoding: Rectangular block-encodings provide a simpler and cheaper implementation of Hsqrt than approaches based on Hermitian dilations.The paper reports that Hermitian-dilation approaches use more qubits and are at least twice as expensive to block-encode.
  • Quantum walk and phase estimation: The quantum walk converts a block-encoding into eigenphases whose arccosine dependence supplies the phase-estimation advantage of spectrum amplification.The first three quantum-walk operations also provide a block-encoding of HSA/Λ − 1.
  • Quantum walk and phase estimation: A controlled block-encoding combines Chebyshev-polynomial transformations of Oα using preparation and unpreparation of the control state.The control amplitudes provide the appropriate weighting for each T2 term in the linear combination of unitaries.
  • Optimizing the energy shift: Efficient lower-bound methods for the ground-state energy can maximize ESOS, minimize Egap, and thereby reduce the effective spectrum-amplification scaling constant.The approach targets SOS representations near frustration free without requiring the potentially exponential preprocessing needed to obtain an exactly frustration-free representation.

III. SUM-OF-SQUARES REPRESENTATIONS OF THE CHEMISTRY HAMILTONIAN

The section formulates chemistry Hamiltonians as shifted sums of squares and uses semidefinite optimization to obtain lower-energy representations. For electronic structure, a spin-free level-2 SOS algebra offers small gaps while balancing classical and block-encoding costs.

  • SOS formulation: A shifted Hamiltonian can be represented as a sum of squares of Hermitian operator polynomials, with ESOS setting the energy shift.Fixing polynomial degree k defines SOS_k, the subset generated by degree-k polynomials.
  • SOS formulation: The optimal shifted Hamiltonian is determined by a semidefinite program over a chosen SOS generating algebra.The algebra must be rich enough to represent H, while its Gram-matrix factorization yields the SOS terms.
  • Chemistry Hamiltonians: For fermionic Hamiltonians, ladder operators and the identity form a natural generating set whose Gram-matrix constraints encode normal-ordered coefficients.The identity is included because it can arise from fermionic anticommutation relations.
  • Chemistry Hamiltonians: The spin-free level-2 SOS algebra already gives small gaps for relevant spin-free chemistry Hamiltonians and improves ground-state energy estimation.More bespoke algebras can further explore the trade-off between gap minimization and block-encoding cost.
  • Numerical implications: The reported lower bounds are not thermochemical-accuracy estimates, but small SOS ground-state energies—not thermochemical gaps—are the relevant quantity for spectrum amplification.The table compares variational upper energies, SDP lower bounds, their gaps, and a lower bound on effective spectrum amplification.

IV. DOUBLE FACTORIZED-TENSOR-HYPERCONTRACTION SUM-OF-SQUARES: DFTHC-SOS

DFTHC-SOS is a hybrid factorization designed to span the spin-free SOS generators while interpolating between double factorization and tensor hypercontraction. Its circuit combines QROM, rotations, selection, and Majorana operations for block-encoding.

  • DFTHC representation: DFTHC generalizes double factorization and factorized tensor hypercontraction to span the spin-free generating set.The representation introduces freely variable Rank, Bases, and Copies parameters, allowing interpolation between the two factorization regimes.
  • Block-encoding: DFTHC block-encodes SOS generators using LCU constructions after representing fermionic operators with rotated Majorana operators.The construction includes generators associated with double-factorized, tensor-hypercontracted, and spin-free terms.
  • Block-encoding: The circuit uses QROM to load coefficients and rotation data, Rot to prepare basis rotations, and Sel to select SOS generators.The dominant lookup and selection components scale differently with N, R, B, and C.
  • Cost trade-offs: DFTHC balances Rot and Qroam costs through parameter choices and spacetime trade-offs, whereas Rot dominates double factorization and Qroam dominates tensor hypercontraction.The construction computes and uncomputes Rot and Qroam twice and applies Sel twice.
  • Cost trade-offs: For typical DFTHC solutions, Rot, Qroam, and Sel account for about 90% of Toffoli cost, with Sel contributing about 60%.Chemical accuracy is reported for FeMoco when the DFTHC parameters lie roughly between equivalent double-factorization and tensor-hypercontraction parameters.
  • Precision: Chemical precision is attained with bRot between 11 and 18 bits and bcoeff between 6 and 14 bits, depending on the system.These precision ranges are used in the resource estimates.

A. Numerical optimization

The numerical optimization fits DFTHC approximations while regularizing block-encoding normalization and SOS gaps. Gradient-based GPU optimization is used to search over factorization parameters and related symmetry shifts.

  • Optimization objective: DFTHC simultaneously targets electronic-structure SOS representations with favorable gaps, block-encoding costs, and normalization factors.The approach finds approximate two-body tensors without directly solving the SOS semidefinite program.
  • Resource accounting: For FeMoco54, the cost-breakdown instance uses N = 54, (R, B, C) = (10, 27, 27), and a total qubit cost of 1131.The qubit total exploits reuse between temporary inner-state-preparation ancillas and persistent RPREP qubits.
  • Optimization objective: The optimization minimizes two-body tensor error while regularizing the normalization factor Λ and the SOS gap Egap.The one-body terms can be chosen freely so that the exact one-body Hamiltonian contribution is retained despite two-body approximation error.
  • Symmetry shifting: The objective also incorporates symmetry shifting through BLISS, which preserves spectra in a fixed particle-number sector.The work considers first- and second-order annihilators constructed from particle-number conservation.
  • Resource accounting: A single Adam optimization step is benchmarked on one Nvidia L4 GPU with a learning rate decreasing geometrically from 10^-1 to typically 10^-4, 10^-5, or 10^-6.The runtime table reports these measurements for typical examples and hyperparameters.
  • Optimization procedure: The nonlinear program uses Adam gradient descent with automatic differentiation and GPU acceleration; each step’s dominant tensor contraction costs O(RCN^4) multiplications.The initialization uses simple random parameter values rather than chemically motivated starting conditions.

B. Selection of good solutions

The study selects DFTHC solutions by scanning factorization parameters and correlation-energy errors, while recognizing that nonlinear optimization makes performance fluctuate across ranks and initial conditions.

  • B. Selection of good solutions: Good solutions are validated using the CCSD(T) correlation-energy difference ϵcorr relative to the original Hamiltonian as a proxy for exact correlation-energy error.Approximate solutions are sufficient when they remain chemically accurate.
  • B. Selection of good solutions: Candidate DFTHC solutions are generated by scanning rank and factorization parameters, including DF-like and THC-like limits, then retaining points below the correlation-energy threshold.For FeMoCo-54, the selection threshold is ϵth ≤0.3mHa.
  • B. Selection of good solutions: CCSD(T) energies are used to recompute Egap, although CCSD(T) is not an upper bound on the estimated ground-state energy and the correction is expected to be small relative to Egap.Spin sectors are selected per system to obtain converged calculations.
  • B. Selection of good solutions: Correlation-energy error does not improve monotonically with rank because the nonlinear DFTHC optimization is sensitive to initial conditions.Randomized restarts provide statistics for more robust conclusions.

V. RESOURCE ESTIMATES

Resource estimates evaluate DFTHC+BLISS+SA across chemically diverse systems by balancing factorization accuracy, spectrum-amplification gaps, and quantum implementation costs. The protocol is reported as substantially cheaper than prior approaches, while its accuracy depends on preserving a faithful Hamiltonian representation.

  • V. RESOURCE ESTIMATES: DFTHC+BLISS+SA resource estimates cover Iron-Sulfur complexes, two FeMoco active spaces, CPD1-P450, and a CO2-fixation Ruthenium catalyst.The systems span an order of magnitude in orbital and electron counts.
  • V. RESOURCE ESTIMATES: The resource search balances gap, rotation, coefficient-oracle, correlation-energy, block-encoding, and time costs across DFTHC parameters.Reported quantities include Λ, λeff, space and time resources, and improvement factors.
  • V. RESOURCE ESTIMATES: A smaller Λ combined with a small Egap indicates that spin-free DFTHC representations can support spectrum-amplifiable Hamiltonians.The result identifies compressed representations where spectrum amplification can reduce phase-estimation costs.
  • V. RESOURCE ESTIMATES: At least 3× lower costs are obtained overall, including a 4× improvement for FeMoco-76 and 10× to 195× improvements for smaller Iron-Sulfur and CO2 systems.FeMoco-76 is estimated at 9.99×10^8 Toffoli gates.
  • V. RESOURCE ESTIMATES: FeMoco-76 is estimated to require 4.5 million physical qubits for 8.6 hours under the stated quantum error-correction model.The estimate uses four CCZ factories, physical error rate 0.001, 1-microsecond cycles, and 10-microsecond reaction time.
  • V. RESOURCE ESTIMATES: The DFTHC protocol must choose parameters large enough to preserve a faithful original-Hamiltonian representation while optimizing phase-estimation cost.High-spin correlation-energy metrics can differ substantially from those for low-spin, potentially multireference targets.

VI. DISCUSSION

The paper combines spectrum amplification with DFTHC to reduce ground-state energy estimation costs while balancing SOS gaps against block-encoding complexity. It also identifies remaining optimization and scalability boundaries for DFTHC and SA.

  • Spectrum amplification with DFTHC reduces ground-state energy estimation costs for second-quantized electronic structure Hamiltonians.
  • The new integral compression scheme keeps effective LCU normalization within a factor of two of the optimal value across all studied systems.
  • A factor of 4 reduction is achieved for the larger FeMoco active space compared with prior THC plus block-invariant symmetry shifting.
  • Further refinements include space-time-aware parameter optimization, semidefinite-program preprocessing, and smaller SOS expansions.
  • Applying SA beyond chemistry requires balancing growing SOS complexity against the increased block-encoding cost.

2. Quantum walk on rectangular matrices

The rectangular-matrix quantum-walk construction provides a simpler block-encoding route for the non-Hermitian square root and preserves phase-estimation advantages through alternating walks. This reduces query requirements relative to the Hermitian construction.

  • The singular values of the rectangular square root match those of the spectrum-amplifiable Hamiltonian.
  • Rectangular block-encodings provide a more straightforward representation of the non-Hermitian Hamiltonian square root.
  • Alternating W1 and W2 quantum walks allows standard phase estimation to estimate energies.
  • The alternating construction uses half as many SELECT and PREPARE queries as the Hermitian square-root approach.
  • The product W2W1 can also be used for phase estimation through the Chebyshev-polynomial relation.

Appendix B: DFTHC block-encoding cost

The DFTHC block-encoding combines tensor-factorized SOS generators with indexed state preparation, alias sampling, qubitization, and spin-controlled operations. Its implementation balances multiple lookup and selection costs while keeping the construction efficient.

  • DFTHC uses SOS generators from spin-free, double-factorization, and tensor-hypercontraction components.
  • The construction contains N + RC(B + 1) coefficients and N + RB real unit vectors across its generators.
  • An indexed register maps composite variables G, r, and c to coefficients and operator choices for controlled block-encoding.
  • Coherent alias sampling prepares the weighting state used to index controlled operator block-encodings.
  • Preparation failures increase the normalization factor by less than about 0.1%.
  • The spin-control construction adds 4 Toffolis to each operator block-encoding and 4N Toffolis for four controlled swaps.

Appendix C: Robust selection of DFTHC solutions

DFTHC solution selection uses statistical correlation-energy behavior rather than assuming monotonic improvement with rank. Threshold-based selection improves robustness, though it may require larger ranks and remains tied to chosen error models.

  • The optimal DFTHC rank is difficult to identify because correlation-energy error fluctuates and does not improve monotonically with rank.
  • Selection uses σcorr ≤ 0.7mHa and a mean correlation-energy error much smaller than the threshold.
  • The robust threshold criterion can require a significantly larger rank than the lowest-cost acceptable solution.
  • The mean correlation-energy error empirically concentrates near zero up to statistical accuracy, permitting σPEA = 1.4mHa.

Appendix D: Truncation to finite bits-of-precision

The appendix analyzes randomized finite-precision representations for DFTHC block-encodings, showing that truncation errors decrease rapidly with precision while adding little normalization cost.

  • Randomized rounding: Randomized rounding independently explores rotation and coefficient precision without scanning all hyperparameter combinations for every sample.The strategy uses unbiased rounding for rotation angles and coefficients, with 64 samples used to estimate mean changes and standard deviations.
  • Error behavior: O(2^-brot,-bcoeff) scaling is observed for σtrunc as rotation and coefficient precision increase.The mean truncation error concentrates rapidly toward zero under the randomized rounding procedure.
  • Error budget: 0.830 mHa is the truncation-error budget imposed by the linear error bound for the considered resource estimates.The FeMoCo-54 solutions are treated as typical for selecting the required precision.
  • Rotation precision: (N − 1)bbits is the Toffoli cost for discretizing each sequence of N − 1 Majorana-rotation angles.Each angle is rounded using brot bits, with the final bit sampled from a Bernoulli distribution.
  • Normalization impact: The residual correction from finite rotation precision is exponentially small, and the resulting change in block-encoding normalization is also small.The higher-accuracy one-body block-encoding process can be nested if the desired accuracy is not yet achieved.

Appendix E: Estimating E∗ gap with DFTHC

This appendix estimates the SOS gap for large molecular systems using DFTHC solutions and scans the regularization parameter when the target gap is unavailable beforehand.

  • Gap estimation: E*gap is estimated for XVIII-100 and XVIII-150 by scanning Ereg instead of setting it to an unavailable target value.The scan uses the range of Ereg reported in Table IV and evaluates resulting DFTHC solutions with CCSD(T) energies.
  • Selection rule: The smallest observed Egap is selected as the estimate of E*gap because the gap remains robust under DFTHC optimization.The authors report that the optimized gaps do not attain unphysical values.
  • Scan data: Table XII scans Ereg and R, retaining solutions with |ϵcorr| ≤ 0.3 mHa and plotting their correlation-energy errors and Egap values.The dashed line marks chemical accuracy at 1.6 mHa, while E*gap comes from the solution with the smallest Egap.

Appendix F: SOS Representation and moment method conversion

The appendix converts a semidefinite-program solution into a sum-of-squares Hamiltonian and obtains Hsqrt through a Cholesky decomposition of the resulting positive semidefinite operator.

  • Semidefinite program: The SOS construction begins from a standard-form semidefinite program whose primal and dual variables encode operator constraints.The primal variable lies in a cone, while the dual formulation uses the corresponding dual cone.
  • Dual bound: The dual SDP provides a lower bound on the primal objective for any primal-feasible matrix.This lower-bound property supplies the scalar used to form the positive semidefinite operator in the SOS construction.
  • SOS conversion: The operator ˆ˜B is formed after mapping the relevant linear constraints into operators, and its positive semidefiniteness defines the dual SOS Hamiltonian.Constraints involving the constant primal variable and contractions require special treatment under the operator map.
  • Square-root construction: Hsqrt is obtained by Cholesky decomposition of the positive semidefinite operator ˆ˜B.The construction requires care because a generic operator mapping need not preserve the SOS form.
  • Spin-free chemistry representation: Spin-free rank-2 particle-hole and hole-particle generators are supplemented with rank-1 generators because the initial SOS is not flexible enough.The resulting SOS coefficients are matched to the one- and two-electron integrals of the spin-free chemistry Hamiltonian.

Appendix H: Correction for divergent derivative

The appendix corrects uncertainty propagation when inverse functions have divergent derivatives near zero, then shows that the resulting correction is negligible for the paper’s typical parameters.

  • Motivation: The usual linear error-propagation formula can predict an unreasonably large improvement near a divergent derivative such as √x at small x.The appendix derives corrections using a quadratic expansion of the inverse function.
  • Square-root example: For y = √x, squaring an estimate of y adds a variance-dependent correction to the estimated mean of x.The corrected treatment avoids an uncertainty that fails to approach zero as the true value becomes small.
  • Distributional correction: Excess kurtosis increases the correction through the fourth standardized moment κ of the error distribution.The general inverse-function expansion incorporates distributional kurtosis into the propagated uncertainty.
  • Regime of validity: Λ ≫ Emin makes the correction negligible under the parameter regime considered for spectrum amplification.The appendix solves for the relevant estimation uncertainty and expands for small Hamiltonian-estimation error.
  • Numerical estimate: 2.5 × 10^-7 is the reported fractional correction for Λ ∼ 100 Ha, λeff ∼ 10 Ha, and σ′Ek ∼ 1 mHa.This estimate supports neglecting the correction for the typical parameters examined.
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