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Variational Quantum Computation of Excited States
Oscar Higgott, Daochen Wang, Stephen Brierley
TL;DR
Excited-state calculations matter for molecular processes, but existing approaches can require high-depth controlled-unitaries or many additional samples. VQD extends VQE by adding overlap penalties for previously found eigenstates. It uses VQE-scale qubits and at most twice the circuit depth, and accurately recovered all six H2 eigenstates including a degenerate subspace in simulation.
Problem
Excited-state energies are needed for applications including optical, charge-transfer, and reaction-rate calculations, while existing methods can impose substantial quantum-resource overhead.
Method
VQD extends VQE by adding overlap terms that deflate previously found eigenstates and enable iterative excited-state optimisation.
Results
VQD requires the same number of qubits as VQE and at most twice the maximum circuit depth; on H2, its median error was less than 4×10^-6 Ha and it found all six eigenstates.
Takeaways & Limitations
VQD is presented as a low-resource approach for calculating low-lying excited-state energies and degeneracies on near-term quantum computers.
Takeaways & Limitations
The inverse-circuit overlap implementation requires accurate knowledge of each previous preparation circuit, which device errors may compromise; higher-state calculations can also accumulate errors.
Abstract
from arXiv · showhide
The calculation of excited state energies of electronic structure Hamiltonians has many important applications, such as the calculation of optical spectra and reaction rates. While low-depth quantum algorithms, such as the variational quantum eigenvalue solver (VQE), have been used to determine ground state energies, methods for calculating excited states currently involve the implementation of high-depth controlled-unitaries or a large number of additional samples. Here we show how overlap estimation can be used to deflate eigenstates once they are found, enabling the calculation of excited state energies and their degeneracies. We propose an implementation that requires the same number of qubits as VQE and at most twice the circuit depth. Our method is robust to control errors, is compatible with error-mitigation strategies and can be implemented on near-term quantum computers.
1 Introduction
Excited-state energies are important for predicting molecular processes, but classical methods may fail where ground-state calculations succeed. VQD extends low-depth VQE to systematically target excited states using overlap information.
- Excited-state energies support predictions of charge and energy transfer in photovoltaics and photodissociation reactions.
- Density functional theory can fail to determine excited states even for materials whose ground-state energies it can calculate.
- VQE uses parameterised circuits and classical optimisation to find Hamiltonian ground states on near-term quantum computers.
- Existing excited-state extensions can require quadratically more terms, non-systematic symmetry choices, high-depth controlled-unitaries, or many additional samples.
- VQD adds overlap terms to the VQE objective, exploiting orthogonal eigenvectors and reusable ansatz parameters to target excited states.
2 Variational quantum deflation algorithm
VQD finds excited states by augmenting VQE energy minimisation with penalties for overlap with previously found states. It proceeds iteratively, using quantum measurements and classical parameter updates.
- VQD optimises an excited-state ansatz by adding overlap penalties to the VQE energy objective.
- Sufficiently large β_i values make the objective minimum equal the k-th state energy when the ansatz is sufficiently expressive.
- The energy term uses VQE circuits, while overlap terms with earlier states are computed using dedicated quantum circuits.
- The algorithm first finds λ_0 with VQE, then determines each successive λ_k using previously obtained parameters.
- A classical optimiser updates λ_k from measured Hamiltonian expectations and overlaps until a stopping criterion is reached.
- The required sample count M^(k) is bounded for measuring the VQD objective to precision ε, with the additional cost expected to be very small for well-chosen β_i.
3 Overlap weighting
The penalty weights must exceed energy differences from previously found states to guarantee the desired excited-state minimum. A spectral-width bound provides a practical way to choose them, while iterative enlargement can self-correct undersizing.
- The effective-Hamiltonian viewpoint expresses VQD as ground-state optimisation at stage k over previously found eigenstates.
- The Hamiltonian expectation decomposes into weighted contributions from previously found states and higher-energy eigenstates.
- β_i > E_k−E_i guarantees a minimum at E_k for a sufficiently powerful ansatz.
- For Pauli-expanded Hamiltonians, choosing β_i from the bound Δ≤2∥H∥≤2Σ|c_j| can guarantee validity.
- If weights are too small, repeatedly increasing γ until finding an energy below γ can identify a sufficiently large penalty.
4 Low-depth implementations
Low-depth overlap estimation makes VQD practical with either VQE-sized qubit counts or nearly VQE-sized circuit depth. Simulations compare its energy levels with exact diagonalisation across molecular separations.
- Overlap estimation applies the trial circuit followed by the inverse preparation circuit and counts all-zero outcomes to estimate squared overlap.The method uses O(1/ε^2) computational-basis samples for precision ε.
- The inverse preparation circuit must be known, and device errors can make its practical implementation inaccurate.
- VQD can retain VQE-like robustness to control errors by optimising parameters for the inverse preparation circuit.
- The inverse-circuit implementation uses the same number of qubits as VQE and around twice its circuit depth.
- An alternative destructive SWAP-test implementation uses almost the same circuit depth as VQE but twice as many qubits.
- Figure 2 compares exact-diagonalisation and VQD energy levels for H2 in STO-3G across internuclear separations.
5 Numerical simulation: H2
Simulations of VQD on H2 using a UCCGSD ansatz achieved highly accurate energies across all six eigenstates, including a three-dimensional degenerate excited-state subspace.
- VQD systematically recovered all six eigenstates, including the three degenerate states spanning the first, second, and third excited levels.The authors identify degenerate-state recovery as a key advantage over folded-spectrum and WAVES approaches.
6 Error accumulation
Imperfect lower-state preparation can accumulate error in VQD, but simulations show negligible accumulation for low-lying H2 states and increased error for higher excitations.
- Imperfect preparation of previously found states means the deflated Hamiltonian no longer has the exact excited-state energy as its lowest eigenvalue.The analysis begins with an approximate ground state and tracks the resulting error in subsequent deflation stages.
- The accumulated energy error remains linear in the original ground-state error when the next-stage error is approximately comparable.The bound is given as ϵ′_1 = O(ϵ_0β_0 + 2ϵ_1||H||).
- Approximately 20% of 225 simulations produced eigenstates in the wrong order and were excluded from the sampling-error analysis.The remaining approximately 180 runs per sampling rate were used to compute median errors.
- For k < 4, median errors matched exact-deflation comparisons and remained below chemical accuracy, indicating negligible accumulation for these states.This pattern held across 10^6, 10^7, and 10^8 samples per Hamiltonian subterm and overlap term.
- For k = 4 and k = 5, accumulated error was substantially larger than with exact lower states, showing reduced effectiveness for higher-lying states.Chemical accuracy for k = 5 required 10^7 samples rather than 10^6 for 0 < k < 4.
- Alternative effective Hamiltonians or a hybrid VQD–WAVES approach are proposed to address error accumulation, although both alternatives require greater circuit depth.The WAVES eigenstate witness does not share the same error-accumulation problem and may refine energy estimates.
7 Choice of effective Hamiltonian
The paper places its effective Hamiltonian within a broader family of deflation methods and compares its error sensitivity with projection- and Schur-complement-based alternatives.
- The proposed effective Hamiltonian is one choice among deflation methods commonly used for positive semi-definite matrices such as covariance matrices.The paper relates this broader use to eigenvalue problems beginning with the largest eigenvalues.
- After transforming H to H′ = −H + E′, Hotelling’s deflation corresponds to the paper’s method with β_i = E′ − E_i.The transformation requires E′ ≥ E_d−1, with E′ = ||H|| given as an example.
- Projection and Schur-complement deflation ensure that the effective Hamiltonian’s true ground state does not overlap with the previous eigenstate estimate, regardless of its accuracy.These methods have empirically outperformed Hotelling’s method on some PCA datasets.
- Projection deflation introduces terms involving overlaps and Hamiltonian matrix elements between the trial state and previously found states.Without the orthogonality approximation, overlaps between previously found states must also be calculated.
- All additional projection-deflation terms can still be evaluated on a quantum computer.The paper points to an existing quantum-computation procedure for these measurements.
8 Discussion
VQD extends VQE to excited states with modest resource overhead, while overlap estimation supports implementation and sampling analysis. The method also remains compatible with alternative SWAP-test realizations and robustness to control errors.
- 8 Discussion: VQD uses the same number of qubits as VQE, at most twice the maximum circuit depth, and a negligible increase in measurements.These resources contrast with the larger overhead reported for existing quantum excited-state methods.
- 8 Discussion: The sampling overhead is expected to be very small because Hamiltonian coefficients typically dominate well-chosen deflation weights, while overlap variances tend to zero at convergence.This expectation assumes low-lying excited states and well-chosen β_i values.
- A Sampling cost: If β_i = 2 sum_j |c_j| is used throughout optimisation, the upper-bound estimate becomes M^(k) = (1 + k)^2 M.This choice guarantees sufficiently large deflation weights but can increase sampling costs with the excited-state index k.
- B Destructive SWAP test: Overlap estimation can use the destructive SWAP test, which removes the ancilla and controlled-SWAP gate through parallel Bell-basis measurements and classical logic.The original construction uses 2N + 1 qubits and depth O(N), whereas the destructive version attains the same outcome distribution without an ancilla.
- 8 Discussion: The standard VQD implementation uses the same qubit count as VQE and around twice its circuit depth; an alternative destructive-SWAP implementation nearly preserves depth but doubles qubits.Nearest-neighbour ansatz layouts can support the SWAP-test comparison, and parameter variation can preserve robustness to control errors.
C Methods for numerical simulation
The simulations use VQE-style ansätze and optimisation, while VQD deflates previously found states through overlap penalties. The method also addresses symmetry restrictions and error accumulation through numerical and analytical analyses.
- Ansatz and optimisation: UCCGSD replaces UCCSD by allowing orbital indices irrespective of reference-state occupation.The UCCGSD ansatz uses a Hartree–Fock reference and a generalized cluster operator T = T1 + T2.
- Ansatz and optimisation: Random initialisation was used, although informed parameter guesses can reduce optimiser iterations.MP2-based initialisation has been proposed for ground-state VQE, while other approaches apply to ground and excited states.
- Variational quantum deflation: Overlap terms can be cached across iterations to inform optimisation for later excited states and potentially reduce required steps and circuits.The cached quantities are squared overlaps between previously computed states and the current trial state.
- Symmetry constraints: Symmetry-conserving ansätze or penalty terms restrict VQD to the desired Hilbert-space sector.The UCC ansatz conserves electron number for the neutral Hydrogen example, while operators such as N_e, S^2, and S_z can impose constraints.
- Error mitigation: Overlap estimation can support error mitigation by discarding symmetry-violating bitstrings during classical post-processing.This is more efficient than the ancilla-based method when the symmetry operator is computational-basis diagonal and commutes with the ansatz.