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Quantum computing enhanced computational catalysis

Vera von Burg, Guang Hao Low, Thomas Häner, Damian S. Steiger, Markus Reiher, Martin Roetteler, Matthias Troyer

arXiv:2007.14460v2quant-phcs.ETphysics.chem-ph

TL;DR

Accurate electronic energies are difficult to obtain for chemically relevant many-electron systems because classical methods face exponential scaling and dynamical-correlation challenges. The paper estimates quantum resources for catalytic intermediates and transition states, introducing double-factorized qubitization for a ruthenium-catalyzed CO2-to-methanol reaction. The approach reaches approximately 10^10-Toffoli-gate costs for 52–65 orbitals, while larger active spaces and realistic hardware remain important challenges.

  • Problem

    Classical electronic-structure calculations face exponential many-electron scaling, while small active spaces neglect dynamic correlation and can compromise chemically relevant accuracy.

  • Method

    The paper combines accurate quantum-energy error bounds, double-factorized qubitization, active-space analysis, and hardware-aware resource estimates for a ruthenium CO2-fixation catalytic cycle.

  • Results

    52–65 orbitals require on the order of 10^10 Toffoli gates using about 4000 qubits, while chemical-accuracy estimates scale as approximately N^3.25 at fixed atom count.

  • Takeaways & Limitations

    Double-factorized qubitization substantially reduces molecular-energy simulation costs, but reliable computational catalysis requires quantum hardware able to represent larger active spaces and dynamical correlations.

  • Takeaways & Limitations

    Small active spaces can omit dynamic electron correlation and compromise chemical accuracy, while larger active-space selection becomes ambiguous.

Abstract

from arXiv · show

The quantum computation of electronic energies can break the curse of dimensionality that plagues many-particle quantum mechanics. It is for this reason that a universal quantum computer has the potential to fundamentally change computational chemistry and materials science, areas in which strong electron correlations present severe hurdles for traditional electronic structure methods. Here, we present a state-of-the-art analysis of accurate energy measurements on a quantum computer for computational catalysis, using improved quantum algorithms with more than an order of magnitude improvement over the best previous algorithms. As a prototypical example of local catalytic chemical reactivity we consider the case of a ruthenium catalyst that can bind, activate, and transform carbon dioxide to the high-value chemical methanol. We aim at accurate resource estimates for the quantum computing steps required for assessing the electronic energy of key intermediates and transition states of its catalytic cycle. In particular, we present new quantum algorithms for double-factorized representations of the four-index integrals that can significantly reduce the computational cost over previous algorithms, and we discuss the challenges of increasing active space sizes to accurately deal with dynamical correlations. We address the requirements for future quantum hardware in order to make a universal quantum computer a successful and reliable tool for quantum computing enhanced computational materials science and chemistry, and identify open questions for further research.

I. INTRODUCTION

The paper frames accurate molecular electronic energies as a major quantum-computing opportunity for chemistry, where classical methods face exponential scaling and correlation challenges. It studies quantum-resource requirements for catalytic reaction analysis using improved algorithms and more realistic hardware assumptions.

  • Motivation: Exact electronic-structure calculations face exponential growth with molecular-orbital count, restricting full configuration interaction to comparatively small molecules.This is the curse of dimensionality that motivates quantum approaches.
  • Motivation: Quantum computing encodes molecular states using qubits that scale linearly with orbital count, potentially enabling full-configuration-interaction solutions for larger orbital spaces.
  • Scope: The work revisits quantum-computing-assisted reaction-mechanism elucidation for carbon-dioxide conversion to methanol, a prototypical catalytic reaction.
  • Algorithmic advances: The study applies refined qubitization to molecular systems using double-factorized two-electron integrals, reducing estimated Toffoli costs by more than an order of magnitude.For a 1 mHartree nitrogen-fixation energy estimate, the cost is reduced to 1.2 × 10^10 Toffoli gates from 2.3 × 10^11 using single factorization.
  • Hardware assumptions: The resource analysis adopts 10 µs fault-tolerant gate times and nearest-neighbor connectivity to represent realistic mid-term hardware assumptions.These assumptions replace the earlier study’s 100 ns gates and all-to-all connectivity.

II. A HOMOGENEOUS CARBON DIOXIDE FIXATION CATALYST

The study examines a ruthenium catalyst that captures and transforms carbon dioxide to methanol, using previously characterized structures while assessing electronic-structure reliability and correlation. Functional choice can substantially alter predicted relative energies and even compound ordering.

  • Catalytic system: The selected ruthenium catalyst converts carbon dioxide to methanol, providing a prototypical homogeneous-catalysis case with relevance to heterogeneous catalysis.
  • Structures analyzed: Eight intermediates and transition states were selected from an existing DFT-based mechanistic analysis of the Leitner-group catalyst.
  • Scope and assumptions: The isolated-complex analysis neglects solvent effects and nuclear-dynamics contributions needed for free-energy calculations.
  • Electronic structure: The ruthenium complexes show mostly dynamic electron correlation rather than pronounced multiconfigurational character, motivating larger quantum computers for complete state representation.The authors identify a need for a few thousand logical qubits to capture relevant dynamical correlations accurately.
  • DFT uncertainty: More than 50 kJ/mol separates relative energies obtained with different density-functional approximations, and the functional choice can reverse the ordering of structures V and VIII.Structure optimization has a comparatively small effect on relative electronic energies.

III. QUANTUM COMPUTING ENHANCED COMPUTATIONAL CATALYSIS

Computational catalysis requires highly accurate electronic energy differences, yet classical approximations and active-space methods face substantial accuracy and scaling limits. Quantum computing is positioned as an embedded step for evaluating electronic structures in catalytic mechanisms.

  • Motivation: Reaction rates depend exponentially on transition-state energy differences, making accurate electronic energies especially important for bond-breaking and bond-making processes.Electronic energy differences are identified as the most crucial contribution among the free-energy terms relevant to reaction rates.
  • Challenges: Approximate electronic-structure methods lack comparable accuracy and feasibility for general electronically complex structures.Well-established efficient classical methods mainly apply to closed-shell single-determinant structures, not general strong-correlation problems.
  • Quantum-computing role: The catalytic workflow embeds quantum computing among structure exploration, electronic-structure evaluation, and reaction-mechanism analysis.The protocol presents quantum computing as a key step that can replace or augment traditional methods such as CASSCF, DMRG, or FCIQMC.
  • Challenges: Active-space methods can leave residual energy uncertainty because they neglect dynamic correlation from most virtual orbitals.Convergence may also be difficult to establish for DMRG and FCIQMC, while rigorous useful error estimates are generally hard to obtain.
  • Quantum-computing role: Quantum computing can encode a quantum state with qubit requirements scaling linearly in the number of molecular orbitals, potentially enabling full-CI solutions in larger orbital spaces.The stated advantage concerns orbital spaces inaccessible to traditional computation because of exponential state-representation scaling.

IV. QUANTUM ALGORITHMS FOR CHEMISTRY

The section develops bounded-error quantum phase estimation for electronic energies, emphasizing controlled errors and reduced non-Clifford gate costs. Its central algorithmic strategy is qubitization using structured molecular Hamiltonian representations.

  • Accuracy and cost requirements: Reliable quantum algorithms must provide controlled electronic-energy errors, because uncontrolled approximations would undermine the effort required for error-corrected quantum hardware.The section contrasts this requirement with residual or difficult-to-control uncertainties in VQE, DMRG, and FCIQMC.
  • Quantum phase estimation: Quantum phase estimation estimates eigenenergies by repeatedly applying a Hamiltonian-dependent unitary, with standard deviation ΔE = O(α/n).An arbitrary trial state collapses to eigenstate k with probability p_k = |⟨ψ_k|ψ_trial⟩|^2.
  • Accuracy and cost requirements: The algorithm budgets phase-estimation error as 0.9ΔE and walk-operator compilation error as ΔW ≤ 0.1ΔE/α.The latter contributes a systematic energy bias of αΔW.
  • Quantum phase estimation: Qubitization implements a unitary walk operator instead of directly simulating time evolution, using a normalizing constant α ≥ ||H||.The walk operator is W = e^{i sin^-1(H/α)} in the qubitization formulation described here.
  • Accuracy and cost requirements: Non-Clifford gates receive particular optimization attention because error-corrected T gates can cost 100 to 10000 times more than two-qubit Clifford gates.A Toffoli gate may be implemented using four T gates.

V. EFFICIENT ENCODING OF DOUBLE-FACTORIZED ELECTRONIC STRUCTURE

Double factorization makes electronic Hamiltonians sparser and lowers qubitization costs by exploiting low-rank structure and smaller matrix norms. Truncation schemes further reduce representation size, but their error control is not uniformly rigorous.

  • Double-factorized representation: Double factorization reduces qubitization cost by combining a sparse Hamiltonian representation with partial diagonalization and a smaller normalizing constant.The representation targets both fewer Hamiltonian terms and reduced simulation normalization.
  • Low-rank structure: Rank factorization reduces the two-electron Hamiltonian representation from O(N^4) terms to O(RN^2), with typical molecular rank R ∼ N log N.The factorization follows from symmetry-constrained decomposition of the two-electron tensor into symmetric matrices.
  • Low-rank structure: A second eigendecomposition can reduce the number of terms to O(MN), with typical systems retaining M ∼ N log N eigenvectors.The remaining structure is expressed through sums of squares of one-body Hamiltonians in a Majorana representation.
  • Cost reduction: The double-factorized normalizing factor benefits from Schatten norms that can be up to a factor of N smaller than the entry-wise norms used previously.The approach also provides a factor-of-eight reduction in the prefactor of the two-electron norms.
  • Cost reduction: With empirical scaling for M and R, the algorithm uses c_W = Õ(N) Toffoli gates, improving on the Õ(N^3/2) cost of the single-factorized approach.The comparison concerns encoding electronic spectra for atom-centered basis sets into the walk operator.
  • Truncation: Incoherent eigenvalue truncation better matches benchmarked errors than coherent truncation but does not rigorously bound total error and still overestimates observed error.The incoherent scheme assumes truncated errors combine by a sum of squares rather than linearly.

VI. RESULTS

The study evaluates density-functional and wave-function calculations for catalytic intermediates and transition states, then estimates quantum phase-estimation costs for chemical-accuracy energies. Both small and larger active spaces are considered.

  • Computational analysis: The study computes DFT data and wave-function results for catalytic intermediates and transition states selected from the reported ruthenium-catalyst mechanism.The selected structures comprise eight intermediates and transition-state structures shown in Figure 1.
  • Quantum-resource estimates: Quantum phase-estimation costs are evaluated to chemical accuracy for active spaces of 52–65 orbitals and for larger active spaces.The analysis also discusses the corresponding runtimes and qubit requirements.

A. Selection of active orbitals

The study selects active spaces containing strongly correlated orbitals, then expands them with weakly correlated orbitals to examine quantum-algorithm performance across increasing orbital-space sizes.

  • The electronic Hamiltonian is parameterized by one- and two-electron integrals over a restricted molecular-orbital subspace called the active space.
  • The selected active spaces contain five to sixteen orbitals, with all strongly correlated orbitals identified for the carbon-dioxide-fixation process.
  • Three active-space sizes—small, intermediate, and large—are chosen for each molecular structure in the catalytic cycle.
  • Beyond the classical limit, weakly correlated orbitals are added, making active-space selection from orbital entanglement alone increasingly difficult.The study therefore uses reproducible criteria that are also chemically reasonable.

B. Resource estimates

The paper estimates phase-estimation resources for catalytic-cycle electronic energies using truncated double-factorized Hamiltonians, evaluating accuracy, scaling, and hardware-aware costs. The estimates show strong algorithmic improvements while exposing trade-offs between truncation error, active-space size, qubit count, and Toffoli cost.

  • 52–65 orbitals are used for the catalytic-cycle active spaces, while supplementary estimates extend to 2–250 orbitals.
  • M ∼N^2.5 describes eigenvector scaling with active-space size when the number of atoms is fixed, differing from M ∼N log N when increasing atom number.
  • 1mHartree is chosen as the truncation threshold because simulations indicate it closely reflects chemical accuracy, whereas 100mHartree can exceed chemical-accuracy energy shifts.
  • ∼N^3.25 is the observed phase-estimation Toffoli-cost scaling across active spaces ranging from 2 to 250 orbitals.
  • 1.22×10^10 Toffoli gates using 3600 qubits are estimated for the 54-orbital FeMoco representation.This improves on the earlier 6.0 × 10^14 T-gate estimate and the 1.2 × 10^12 Toffoli single-factorized approach.
  • 33 to 120 times smaller αDF values account for much of the double-factorized improvement because the approaches have roughly equal term counts at the comparison threshold.
  • The resource estimates target 1 mHartree energy accuracy and allow a trade-off between required logical qubits and Toffoli count.
  • Figure 5 relates Toffoli cost and eigenvector count to truncation threshold for phase estimation at 1mHartree precision, assuming 10µs between Toffoli gates.

C. Runtimes and qubit counts

Runtime estimates depend strongly on fault-tolerant gate times and hardware scale. The study also reports substantial qubit requirements and examines cost scaling with active-space size and truncation.

  • Runtime assumptions: 10^10 Toffoli gates correspond to about 28 hours at 10µs per gate, but several years at 10ms per gate.Fast logical gate times are therefore essential for realistic quantum computation of chemical catalysis.
  • Hardware assumptions: Layout overhead is assumed negligible because the dominant Fanout-based table-lookup subroutine maps well to a nearest-neighbor planar topology.The assumed gate time also includes layout and Clifford-gate overhead.
  • Qubit requirements: 4000 logical qubits would require millions of physical qubits, assuming hundreds to thousands of physical qubits per logical qubit.This implies a scalable architecture reaching millions of physical qubits.

D. State preparation

The state-preparation analysis uses DMRG to construct an approximate ground state and evaluates a Hartree–Fock trial state for phase estimation. The selected ruthenium complexes have sufficiently large overlaps for this single-determinant choice.

  • Trial-state construction: DMRG calculations provide approximate ground states used to assess trial-state overlap with the true ground state.The approximate states are obtained for each molecular system.
  • Trial-state construction: The quantum computer prepares a Hartree–Fock determinant, with overlap given by the squared Hartree–Fock coefficient in the configuration-interaction expansion.The determinant occupies the first N/2 orbitals for N electrons.
  • Overlap and applicability: Large overlaps were found for all systems, and the authors judge the dominant single-determinant state sufficient for the studied molecular structures.The selected Ru complexes do not exhibit strong multireference character.

VII. CONCLUSIONS

The paper applies quantum computing to catalytic energy estimation using a double-factorized qubitization algorithm and resource estimates for carbon-dioxide-to-methanol chemistry. It identifies active-space growth and dynamical correlation as central challenges for achieving practical advantage.

  • Scope and contributions: The study estimates accurate electronic energies for intermediates and transition states in a ruthenium catalytic cycle converting carbon dioxide to methanol.It also validates truncation schemes against DMRG and evaluates single-determinant state preparation.
  • Algorithmic contribution: Double-factorized integral representations reduce qubitization runtime by minimizing coefficient loading and the Hamiltonian spectral-norm bound αDF.The approach targets the rapid growth of four-index two-electron integrals.
  • Limitations and outlook: Competitive electronic-structure calculations require faster algorithms capable of treating much larger active spaces.The paper identifies larger active spaces as necessary for addressing dynamical correlation without reduced-dimensional corrections.
  • Limitations and outlook: An active space an order of magnitude larger than those considered would increase quantum computational requirements by three orders of magnitude.Further algorithmic improvements are therefore needed before quantum computers become superior to traditional approaches in real-world chemistry applications.

VIII. COMMENTS ON THE INTEGRAL FILES

The integral files used in the study are available upon request from the corresponding author.

  • The study’s integral files can be obtained by requesting them from the corresponding author.

Supplementary Material for Quantum Computing Enhanced

The supplementary material describes computational setups for molecular structures, reaction energies, basis sets, and electronic-structure calculations. It reports DFT procedures and defines the basis and energy tables used for the catalyst study.

  • Structures and reaction energies: Seven catalyst intermediates and transition states were taken from prior work and labeled using roman numerals.Their structures were previously optimized with M06-L/def2-SVP, with single-point M06-L/def2-TZVP energies used for the main-text diagram.
  • Structures and reaction energies: Relative reaction energies were computed from electronic energies of complexes and small molecules using the formula specified in Table II.The calculations included CO2, H2, H2O, THF, and methanol, with DFT calculations performed using Gaussian09 and Turbomole.
  • Structures and reaction energies: PBE/def2-TZVP and PBE0/def2-TZVP single-point calculations provided electronic and relative reaction energies for the complexes and remaining compounds.The study also optimized structures with PBE/def2-TZVP and used frequency calculations to distinguish intermediates from transition states.
  • Basis sets and orbitals: The study used minimal and full atomic-orbital bases, with the latter combining ANO-RCC-VTZP for light elements and ANO-RCC-VQZP for ruthenium.The number of molecular orbitals equals the number of atomic-orbital basis functions.

A. HF, CASSCF, and DMRG-CI electronic energies

This section evaluates HF, CASSCF, and DMRG-CI electronic energies and examines wave-function and integral-truncation accuracy. It also identifies dynamic correlation as a challenge for selecting larger active spaces.

  • Active spaces and electronic energies: CASSCF orbitals were constructed from localized HF orbitals by selecting chemically relevant ruthenium, carbon, oxygen, hydrogen, and carbon-dioxide orbitals.The active-space construction targeted bonding and antibonding orbitals and other orbitals associated with the catalyst and reactants.
  • Active spaces and electronic energies: HF, CASSCF, and DMRG-CI electronic energies were reported for the catalyst complexes using different Cholesky-decomposition thresholds.The DMRG-CI results are given in the supplementary energy tables, while tight-threshold HF calculations were not tabulated when initialized from CASSCF orbitals.
  • DMRG convergence: Increasing the DMRG bond dimension from 1000 to 2048 changed the energy of catalyst structure IX by -1.7 mHartree and the overlap by 0.004.The authors conclude that further convergence has negligible effect on the qualitative wave-function structure.
  • DMRG convergence: Fiedler orbital ordering improved the energy by about 0.6 mHartree to 0.45 Hartree, while increasing the bond dimension from 500 to 1000 lowered catalyst energies by 1-2 mHartree.These comparisons were made between calculations with otherwise matched settings.
  • Correlation and active-space limits: The intermediates and transition states were found to be mainly dynamically correlated, making larger active-space selection ambiguous.Small active spaces could be identified, but dominant dynamic correlation complicated extension to larger spaces.

B. Results

The results compare quantum-resource estimates and develop circuit primitives for double-factorized Hamiltonians. Double-factorized qubitization gives the lowest reported T-counts at reasonable qubit numbers, while data lookup dominates several block-encoding costs.

  • Resource comparison: Double-factorized qubitization offers the lowest T-counts at reasonable qubit numbers compared with Trotter and other qubitization representations.The comparison includes Trotter, unfactorized qubitization, and single-factorized qubitization resource estimates.
  • Resource comparison: The resource estimates cover carbon-dioxide-fixation steps across active spaces from 52–65 orbitals and from 2–250 orbitals.The 2–250-orbital tables mark examples with 20 or fewer orbitals as classically tractable by full configuration-interaction methods.
  • Quantum circuit primitives: The data-lookup oracle uses d−1 Toffoli gates, Θ(db) Clifford gates, and ⌈log2(d)⌉ clean ancillary qubits.Additional clean qubits can reduce the Toffoli cost, while measurement-based uncomputation can reduce the additive λb term to λ.
  • Quantum circuit primitives: Multiplexed state preparation replaces a non-power-of-two uniform superposition with a power-of-two superposition prepared by Hadamard gates.The modification is intended to simplify multiplexed state preparation while keeping approximation errors bounded.
  • Block encoding: The one-electron and two-electron block-encoding costs are dominated by data lookup for multiplexing basis-transformation rotations.The two-electron construction uses asymptotic estimates such as R = O(N^1.5), M(r) = O(N), and M = O(N^2.5) for the studied active-space regime.
  • Matrix norm analysis: A matrix inequality used in the analysis is shown to be tight, and the derivation is presented as potentially independently interesting.The tightness result is supported by a diagonal single-entry choice in the associated bound.
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