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Quantum equation of motion for computing molecular excitation energies on a noisy quantum processor
Pauline J Ollitrault, Abhinav Kandala, Chun-Fu Chen, Panagiotis Kl Barkoutsos, Antonio Mezzacapo, Marco Pistoia, Sarah Sheldon, Stefan Woerner, Jay Gambetta, Ivano Tavernelli
TL;DR
Molecular excitation energies are important but difficult to compute because classical resources scale poorly and near-term quantum methods for correlated systems remain limited. The paper adapts the classical equation-of-motion method into qEOM, using quantum measurements with classical eigenvalue solution. Simulations achieve excitation-energy errors of ≤1.5 mH for H2, LiH, and H2O, while the study also evaluates noisy LiH hardware calculations.
Problem
Molecular excitation energies are important for predicting photo-induced reactions, but classical resource requirements scale poorly and near-term quantum methods for correlated molecular systems remain limited.
Method
The paper adapts the classical equation-of-motion approach into qEOM, measuring EOM matrix elements from a quantum-prepared ground state and solving the resulting eigenvalue problem classically.
Results
Errors ≤1.5 mH were achieved in simulations of excitation energies for H2, LiH, and H2O.
Takeaways & Limitations
qEOM combines variational ground-state preparation with EOM excited-state calculations and was demonstrated in noisy-quantum-computation studies of molecular excitation energies.
Abstract
from arXiv · showhide
The computation of molecular excitation energies is essential for predicting photo-induced reactions of chemical and technological interest. While the classical computing resources needed for this task scale poorly, quantum algorithms emerge as promising alternatives. In particular, the extension of the variational quantum eigensolver algorithm to the computation of the excitation energies is an attractive option. However, there is currently a lack of such algorithms for correlated molecular systems that is amenable to near-term, noisy hardware. In this work, we propose an extension of the well-established classical equation of motion approach to a quantum algorithm for the calculation of molecular excitation energies on noisy quantum computers. In particular, we demonstrate the efficiency of this approach in the calculation of the excitation energies of the LiH molecule on an IBM Quantum computer.
I. INTRODUCTION
Molecular excited-state calculations are difficult because classical resources scale poorly, while near-term quantum methods have mainly addressed ground states. The paper motivates adapting VQE-based strategies, especially EOM, to calculate excitation energies on noisy hardware.
- Motivation: Classical quantum-chemistry calculations face rapidly growing resource requirements as molecular degrees of freedom increase.Full CI has factorial scaling, while CCSD(T) scales as O(N^7).
- Related quantum methods: VQE uses parametrized quantum states and classical optimization to approximate molecular ground-state energies on near-term quantum computers.The Hamiltonian expectation value is evaluated for trial states whose gate parameters are optimized self-consistently.
- Excited-state challenge: Excited-state calculations require methods beyond ground-state optimization because higher-energy states are generally inaccessible through direct trial-state minimization.Classical approaches include linear-response and equation-of-motion formulations built from an optimized ground state.
- Related quantum methods: Existing VQE-based excitation algorithms include QSE, which extends VQE through additional measurements without modifying the quantum circuit.QSE was demonstrated for H2 using two qubits.
- Contribution: The paper adapts EOM into a quantum algorithm, tests H2, LiH, and H2O, and demonstrates LiH excited-state experiments on a 20-qubit IBM processor.The hardware study varies noise and adapts error mitigation to assess robustness.
II. THEORETICAL FOUNDATION
qEOM generates molecular excited states by applying excitation operators to a ground state and determining excitation energies from a variational EOM formulation. Quantum hardware measures the required matrix elements, while the resulting eigenvalue problem is solved classically.
- EOM formulation: EOM represents an excited state as an excitation operator applied to the electronic ground state.The corresponding de-excitation operator maps the excited state back to the ground state.
- EOM formulation: The Hamiltonian–excitation-operator commutator yields the excitation energy E₀ⁿ = Eₙ − E₀.The formulation starts from the energy difference between an excited state and the ground state.
- Relation to QSE: Unlike QSE, EOM uses Hermitian commutator expressions and directly returns excitation energies, making the approach size-intensive.The quantum implementation can still differ numerically from classical EOM because of ansatz, circuit, measurement, and hardware-noise effects.
- Operator basis: The excitation operator is expanded as a linear combination of single- and double-excitation operators with variable coefficients.Singles move one electron from an occupied to a virtual orbital, while doubles excite an occupied electron pair.
- Generalized eigenvalue problem: Variational minimization in the coefficient space converts the EOM expression into a generalized eigenvalue problem for excitation energies.The operator basis includes de-excitation terms, which are important for many-determinant quantum ground states.
- Quantum implementation: qEOM measures EOM matrix elements on a quantum-prepared ground state, then classically diagonalizes the resulting 2n-dimensional problem to obtain excitation energies.The matrix rank is set by the included single and double excitations, while quantum hardware supplies expectation values.
III. SIMULATIONS OF THE QEOM ALGORITHM
Statevector simulations test qEOM for H2, LiH, and H2O using VQE-prepared ground states and UCCSD wave functions, while examining error propagation and subspace selection. H2 and LiH achieve chemical accuracy across the dissociation curves, whereas H2O is slightly less accurate.
- Simulation setup: qEOM is tested on H2, LiH, and H2O using Hartree-Fock/STO-3G Hamiltonians and VQE-prepared UCCSD ground states.The simulations use statevector-type circuits without sampling or hardware noise.
- Simulation setup: Selecting excitation operators allows qEOM to target specific particle- and spin-number sectors rather than the globally lowest states.The reported excited states are the lowest within the selected subspace.
- Simulation results: H2 and LiH produce excited-state energies within chemical accuracy, with errors < 0.015 Hartree across all dissociation geometries.For H2O, less accurate VQE ground states lead to excited states slightly above chemical accuracy.
- Error analysis: Excitation-energy errors grow more slowly than ground-state-energy errors as errors are added to the ground-state parameters.Error propagation proceeds through the M, V, Q, and W matrices in the qEOM equations.
- Error analysis: Near-degeneracy between the ground and first excited states worsens matrix conditioning and can reduce excitation-energy quality.The issue occurs when the lowest generalized-eigenvalue solution approaches zero.
IV. HARDWARE CALCULATION OF THE EXCITATION ENERGIES OF LIH
The LiH hardware experiment reduces the circuit to a four-orbital active space and evaluates qEOM on IBM Q Poughkeepsie under several noise levels. Noise amplification, randomized gate characterization, and zero-noise extrapolation are used to compare measured excitation energies with simulations.
- Circuit and active space: The original LiH UCCSD circuit has over 12000 CNOT gates and 92 variational parameters, motivating reduction from 10 to 4 active orbitals.The reduced active space is selected using orbital contributions to the CI expansion.
- Hardware results: Figure 3 compares ground-state correlation energies, energy-gap errors, and five LiH transition-energy profiles across stretch factors, mitigation, and qEOM simulations.The transition-energy panel uses mitigated experimental markers and statevector qEOM dashed lines.
- Scope boundary: Discrepancies are expected to increase beyond 2.5 Å, where stronger correlation requires a larger active space.Within the studied dissociation range, the reported energy error is ≤7 mHa.
- Hardware implementation: The experiment uses four superconducting qubits of the 20-qubit IBM Q Poughkeepsie processor, with each trial circuit containing 6 CNOT gates.The selected qubits are Q0, Q1, Q2, and Q5.
- Hardware characterization: The hardware study characterizes single- and two-qubit gate fidelities for each stretch factor using randomized benchmarking.These fidelities are reported in Table I.
- Error mitigation: Noise amplification uses stretch factors c = 1, 1.25, 1.5, followed by linear extrapolation to estimate the zero-noise limit.Expectation values are re-measured at amplified noise strengths, with a four-pulse echo sequence used for the ZX90 gate.
V. CONCLUSIONS
The paper introduces qEOM, a quantum adaptation of the classical equation-of-motion method, and tests it on small molecules and noisy hardware. Simulations reach chemical accuracy, while LiH excitation energies are computed on IBM Q Poughkeepsie after error mitigation.
- Contribution: qEOM adapts the classical equation-of-motion approach to calculate electronic excited-state energies on quantum computers.The method combines variational ground-state calculations with quantum expectation-value measurements.
- Results: Simulations of H2, LiH, and H2O produce excitation energies within chemical accuracy, with errors ≤1.5 mH.
- Results: Error-mitigated qEOM computes LiH excitation energies on the IBM Q Poughkeepsie processor and shows robustness against hardware noise.
Supplementary Materials for: Quantum equation of motion for computing molecular
The supplied passage identifies the work as addressing excitation energies on a noisy quantum processor but provides no supplementary-materials findings.
- Scope: The supplementary-materials passage supplies only the phrase “excitation energies on a noisy quantum processor.”
I. ERROR PROPAGATION
The analysis examines how errors in the VQE ground-state parameters and finite-shot sampling propagate into qEOM excitation energies. It also evaluates noisy simulations for H2 and LiH across variational parameters and molecular conditions.
- Parameter-error propagation: Adding errors to optimized ground-state parameters changes both ground- and excited-state energies, with the excited-state errors compared against the ground-state error.Figure S1 reports absolute energy differences and the ratio of each excited-state error to the ground-state error.
- Statistical-error propagation: Shot noise is evaluated through absolute errors in the first three excitation energies and in the norms of the qEOM matrices M, V, and Q.The sampling analysis varies the number of shots across 8192, 4096, 2048, and 1024 and averages 100 computations for each bond length.
- Statistical-error propagation: The excitation-energy error follows the error in the matrix norms, with the V matrix identified as mostly affected.This links the observed energy sensitivity to errors in the measured qEOM matrices.
- Molecular and numerical conditions: Noisy simulations compare ground-state and qEOM excitation energies with noise-free references for H2 and LiH as functions of the variational parameter θ.For H2, three excitation energies are shown; for LiH, the five lowest nondegenerate states are displayed.
- Molecular and numerical conditions: qEOM is expected to be less accurate in strong-correlation regimes, such as large bond lengths, where obtaining the correct ground state is difficult.Near-degeneracy of the ground and first excited states produces an ill-conditioned problem and numerical instabilities.
II. COMPARISON WITH QSE
The paper compares qEOM with QSE for noisy H2 excitation-energy calculations. It emphasizes qEOM’s treatment of de-excitation operators, direct access to excitation energies, and greater robustness in the reported noise tests.
- Methodological differences: QSE neglects de-excitation operators and includes the identity operator, limiting its applicability and making it not size-intensive.The paper connects this limitation to the Tamm-Dancoff approximation and to the calculation of energy differences.
- Accuracy requirement: The accuracy of the ground-state calculation is crucial for recovering excitation energies within chemical accuracy, but current statistical and hardware noise makes that precision challenging.This requirement motivates the paper’s focus on noisy-hardware performance and error mitigation.
- H2 comparison: For H2, qEOM is reported to be more robust than QSE against noise in preparation of the ground-state wave function.The comparison considers noise sources including gate noise and decoherence effects.
B. Lithium Hydride
For LiH, the paper reduces the UCCSD representation and active space to make qEOM feasible on quantum hardware. The reduced-circuit simulations target excitation energies with expected errors between 1 and 10 mHa.
- Circuit and active-space reduction: The full LiH UCCSD circuit requires over 12000 CNOT gates and 92 optimized parameters.The circuit is reduced by selecting one excitation using the largest MP2 coefficient and restricting the active space.
- Circuit and active-space reduction: The LiH active space is reduced from 10 qubits to a 4-qubit register, while six inert qubits remain uncorrelated with the active register.Because the inert qubits are in |0⟩, only the active register needs to be modeled in quantum hardware.
- Circuit and active-space reduction: The modified UCC circuit reduces the double-excitation construction from eight entangling blocks to a single block.The regular construction uses eight blocks with fixed pre- and post-rotations and a shared parameter θ.
- qEOM calculation: The LiH calculation evolves the ground state in the reduced active space, measures qEOM operators, and reconstructs the full pseudo-eigenvalue problem.The target system is LiH in the STO-3G 10-qubit basis, with the ground-state wave function prepared in the reduced space.
- Accuracy: 1–10 mHa is the expected accuracy range for excitation energies obtained with the reduced LiH circuit.The estimate is based on the dissociation-profile results shown for the reduced circuit.