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Accelerated Variational Quantum Eigensolver
Daochen Wang, Oscar Higgott, Stephen Brierley
TL;DR
Finding chemical Hamiltonian ground-state energies requires balancing QPE’s coherence demands against VQE’s sampling cost on near-term devices. The paper proposes α-VQE, which tunes this trade-off through a free parameter and recovers VQE and QPE scaling at its endpoints.
Problem
Ground-state energy estimation must balance QPE’s long coherent circuits against VQE’s O(1/ϵ^2) samples per expectation-estimation subroutine on near-term noisy devices.
Method
α-VQE introduces α ∈ [0, 1] to exploit available coherence, using a tunable measurement–depth trade-off and an expectation-estimation routine cast as amplitude estimation.
Results
α-VQE reduces total runtime by O(1/ϵ^α) for α > 0, while α = 0 and α = 1 recover VQE and QPE scaling, respectively.
Takeaways & Limitations
The algorithm provides a continuum between VQE and QPE, allowing the coherence time available on a quantum computer to determine the measurement cost and circuit depth.
Abstract
from arXiv · showhide
The problem of finding the ground state energy of a Hamiltonian using a quantum computer is currently solved using either the quantum phase estimation (QPE) or variational quantum eigensolver (VQE) algorithms. For precision $ε$, QPE requires $O(1)$ repetitions of circuits with depth $O(1/ε)$, whereas each expectation estimation subroutine within VQE requires $O(1/ε^{2})$ samples from circuits with depth $O(1)$. We propose a generalised VQE algorithm that interpolates between these two regimes via a free parameter $α\in[0,1]$ which can exploit quantum coherence over a circuit depth of $O(1/ε^α)$ to reduce the number of samples to $O(1/ε^{2(1-α)})$. Along the way, we give a new routine for expectation estimation under limited quantum resources that is of independent interest.
I. INTRODUCTION
Quantum chemistry ground-state estimation uses QPE or VQE, but VQE’s sampling cost can become impractical. The proposed α-VQE algorithm tunes coherence depth against measurement count, recovering both methods’ scaling at its endpoints.
- Motivation: VQE estimates ground-state energy with quantum expectation estimation and classical optimisation, but requires N = O(1/ε^2) samples per expectation.It is designed for near-term noisy devices with limited coherence time.
- Proposed approach: α-VQE introduces a free parameter α ∈ [0, 1] to exploit available quantum coherence and reduce the samples required for a target precision.Its coherent circuit depth is chosen as D(α) = O(1/ε^α), subject to the device’s maximum low-error depth Dmax.
- Performance: For every α > 0, α-VQE out-performs VQE in sample count and reduces total runtime O(N × D) by a factor O(1/ε^α).The comparison is stated in terms of measurement count and total runtime scaling.
- Performance: At α = 0, α-VQE recovers VQE scaling, while at α = 1 it recovers QPE scaling.The measurement limits are f(ε, 0) = O(1/ε^2) and f(ε, 1) = O(log(1/ε)), up to further log factors for iterative QPE.
- Proposed approach: The paper constructs α-VQE by replacing VQE’s expectation-estimation subroutine with a tunable Bayesian phase-estimation routine called α-QPE.The construction introduces α into Bayesian QPE and modifies a prior result to support expectation estimation.
II. GENERALISING VQE TO α-VQE
Standard VQE combines a parameterised quantum state, quantum expectation measurements, and classical optimisation to minimise an energy estimate. The Hamiltonian is decomposed into Pauli products so the energy can be assembled from individual expectations.
- VQE procedure: VQE prepares a variational state |ψ(λ)⟩ and uses a classical optimiser to minimise its estimated energy E(λ).The preparation circuit R(λ) maps a fixed starting state |0⟩ to |ψ(λ)⟩.
- Hamiltonian decomposition: The Hamiltonian is written as a finite sum of Pauli-product terms with real coefficients, typically with polynomially many terms in system size.This decomposition supports estimating each term separately and combining them into the total energy.
- Energy estimation: Quantum measurements estimate individual expectation values, while a classical computer forms their weighted sum.The resulting energy function is then supplied to the classical optimiser.
- Variational principle: The variational principle guarantees E(λ) ≥ Emin, with equality if and only if |ψ(λ)⟩ is the Hamiltonian’s ground state.Thus minimising the variational energy targets the ground-state eigenvalue.
- Sampling regime: Each expectation value is estimated by statistical sampling with circuit depth D = O(1) beyond state preparation and N = O(1/ε^2) repetitions for precision ε.This is the statistical sampling regime used by standard VQE.
A. Tunable Bayesian QPE (α-QPE)
α-QPE modifies Bayesian RFPE with a tunable α to trade measurement count against coherent circuit depth, interpolating between statistical sampling and phase estimation.
- Motivation: Kitaev-type QPE uses logarithmically many measurements but coherent depth proportional to 1/ϵ, while RFPE provides a Bayesian framework for alternative measurement–depth trade-offs.The phase-estimation regime is N = ˜O(log(1/ϵ)) and D = ˜O(1/ϵ), neglecting further logarithmic factors.
- α-QPE construction: α ∈ [0, 1] parameterises modified RFPE sequences that achieve precision ϵ with different measurement counts and coherent depths.The modification includes re-preparing the eigenstate at each iteration and defines precision as expected posterior standard deviation.
- Performance: N = f(ϵ, α) measurements and D = O(1/ϵ^α) coherent depth are required by α-QPE for precision ϵ.This is the stated measurement–depth trade-off in Proposition 1.
- Depth constraint: Choosing the largest α compatible with Dmax minimises measurements, and when Dmax < 1/ϵ the resulting measurement scaling is inversely quadratic in Dmax.The optimisation follows because f(ϵ, α) decreases with α while D = 1/ϵ^α is bounded by Dmax.
B. Casting expectation estimation as α-QPE
The paper casts VQE expectation estimation as α-QPE by using amplitude estimation for Pauli expectations and addressing eigenstate-preparation complications.
- Amplitude estimation: Amplitude estimation encodes |⟨ψ|P|ψ⟩| as an eigenphase of a constructed rotation, allowing QPE to estimate the Pauli expectation.The rotation has eigenphases ±φ, with |⟨ψ|P|ψ⟩| = cos(±φ/2).
- α-QPE adaptation: α-QPE replaces QPE in this amplitude-estimation construction to obtain the measurement–depth trade-off of Proposition 1.The replacement requires re-preparing one of the relevant eigenstates at each iteration.
- State preparation: The method assumes |A| stays bounded away from 0 and 1 by a constant δ > 0 so the superposition can be collapsed efficiently before each α-QPE iteration.This assumption addresses the equal superposition of the ±φ eigenstates.
- State preparation: A two-stage procedure first estimates A a constant number of times, then uses α-QPE when the δ condition holds and statistical sampling otherwise.The first stage also determines the sign of A.
- Implementation: The controlled-unitary implementation has overhead from Pauli gates, a controlled sign flip, ancillas, and repeated state-preparation circuits, without ϵ dependence.The depth of state preparation R is noted as potentially the most significant overhead.
C. Generalised α-VQE
Generalised α-VQE replaces VQE’s expectation-estimation routine with α-QPE, allowing tunable use of circuit coherence while retaining core variational features.
- Construction: α-VQE is defined by replacing VQE expectation estimation with the α-QPE routine.Figure 2 schematically illustrates this construction.
- Resource scaling: Each measurement uses a circuit of depth O((CR + log n)/ϵ^α), where CR is the depth of the state-preparation circuit R(λ).The total measurement count also scales with Hamiltonian terms and classical-optimiser iterations.
- Preserved features: Because only expectation estimation is modified, α-VQE preserves parallelisation across Pauli terms, variational self-correction, and classical storage of λ for state re-preparation.These are identified as three key advantages inherited from standard VQE.
- Endpoint regimes: α = 0 corresponds to statistical sampling or standard VQE, while α = 1 corresponds to the phase-estimation regime.These endpoint regimes are shown in the α-VQE schematic.
III. α-VQE AS ACCELARATED VQE
α-VQE accelerates VQE by using bounded coherence to access intermediate resource regimes between statistical sampling and phase estimation.
- Resource trade-off: Standard VQE requires N = O(1/ϵ^2) samples, whereas phase estimation requires D = O(1/ϵ) coherence time, motivating intermediate regimes.The paper frames α-VQE as a continuous interpolation between these two extremes.
- Comparison: For α > 0, α-VQE’s expectation-estimation method is always superior to statistical sampling in the compared resource accounting.The comparison neglects small overheads needed to cast expectation estimation as α-QPE.
- Acceleration: Given Dmax, selecting α to exploit the available depth provides the mechanism for α-VQE’s acceleration, quantified by Eqn. 7.Figure 3 plots this acceleration for different maximum circuit depths.
- Broader scope: The restricted-depth perspective also applies beyond VQE to expectation estimation and phase estimation under limited circuit depth.The paper presents this as a broader theoretical interpretation of its results.
- Acceleration: Figure 3 compares statistical sampling, phase estimation, and α-VQE regimes, with α-VQE occupying the shaded region between the two extremes.The plots show how increasing Dmax can reduce measurements up to exponentially.
Appendix A: Derivation of Proposition 1
Appendix A derives an analytical Bayes-risk solution for the tunable RFPE scheme and tests its approximations against numerical simulations. The analysis connects the parameter α to interpolation between phase-estimation and statistical-sampling behavior.
- Analytical setup: The RFPE expected posterior variance is expressed as r²(M, θ), with M controlling circuit depth and θ controlling oscillations in the update.The analysis uses the expected posterior variance as its Bayes-risk measure and studies its dependence on the controllable parameters M and θ.
- Analytical setup: Choosing θ = μ ± σ and trial M = a/σ is justified because it makes the measurement probability vary across the posterior’s uncertainty domain.The uncertainty domain is D = (μ − σ, μ + σ), and the choice aims to map M(D − θ) across a region where cosine varies substantially.
- RFPE convergence: The function g has maximum value approximately 0.307 at x = ±1.154, yielding an optimized variance-reduction factor L² approximately 0.693 per RFPE iteration.The resulting expected standard deviation decreases exponentially with the number of RFPE iterations.
- RFPE convergence: Because RFPE uses M = O(1/σ_k) while σ_k decreases exponentially, its required coherent depth grows exponentially with the requested precision.The appendix addresses this phase-estimation limitation by modifying the dependence of M on σ at each iteration.
- Validation: The derived Eqn. A16 agrees well with numerical simulations across α values, although its derivation relies on multiple assumptions and approximations.Agreement improves for later initial conditions, where r_k is smaller; simulations also compare Bayes risk with median estimation error.
- Validation: The appendix’s main interpolation result follows by replacing (k, D_k, r_k) with (N, D, ϵ), under the assumption that the eigenstate can be prepared initially and after each measurement.Figure 6 reports good agreement for standard deviations but only qualitative, not quantitative, agreement for median error.
Appendix B: RFPE-with-restarts
Appendix B analyzes RFPE-with-restarts under a maximum coherent depth and compares it with α-QPE. The restart strategy preserves the interpolation scaling while offering an advantage under Bayes-risk minimization.
- Restart strategy: RFPE-with-restarts switches from phase estimation to statistical sampling when decoherence is detected, with the sampling depth capped at Dmax.The analysis assumes immediate decoherence detection and holds M constant at Dmax after the phase-estimation phase.
- Scaling: The first regime exhibits inverse-quadratic scaling with Dmax, reducing the required iterations as the allowed coherent depth increases.The appendix explicitly identifies this inverse-quadratic dependence in the relevant depth range.
- Comparison: RFPE-with-restarts is always advantageous over α-QPE with respect to minimizing Bayes risk.Introducing β through Dmax = 1/ϵ^β allows the restart method to be viewed as β-QPE, which satisfies the same proposition with α replaced by β.
- Comparison: Exploratory simulations find that α-QPE can achieve better mean accuracy than β-QPE for a fixed number of iterations and constant Dmax.If β-QPE performs better under a desired metric, the authors note that β-VQE can be used instead.
Appendix C: δ bound and state collapse
The appendix presents a two-stage procedure that first statistically tests whether |A| is bounded away from 0 and 1, then uses α-QPE when successful and statistical sampling otherwise. Successful state collapse produces a state suitable for α-QPE with probability at least 0.25, while expanding the interval requires stronger Stage 2 resolution.
- Overview: A two-stage method removes the δ bound assumption on |A| and details collapse into |±φ⟩.The procedure combines an initial statistical test with a conditional state-collapse step.
- Stage 1: Stage 1 statistically samples A a constant number of times to test whether |A| lies in I and simultaneously determine its sign.The explicit interval is I := [cos(5π/12), cos(π/12)].
- Stage 1: With n = 1000 and t = 0.1, an estimate satisfying |Â| ∈ Î := [0.36, 0.85] marks Stage I successful.The appendix specifies this numerical interval as the success condition for the explicit specialisation.
- Stage 2: If Stage I succeeds, the correct sign of A is inferred with probability greater than 0.99, while unsuccessful cases continue with statistical sampling.Successful cases proceed to state collapse before α-QPE; unsuccessful cases do not use α-QPE.
- State collapse: After collapse, φ ∈ [π/6, 5π/6], and with probability at least 0.25 the resulting state gives either |φ⟩ or |−φ⟩ probability greater than 0.75.α-QPE is then performed on the collapsed state.
- Scope and trade-off: Expanding Î can improve Stage 1 success likelihood but requires more Stage 2 measurements m to resolve |φ⟩ from |−φ⟩.The need for larger m arises because the two states can become closer when the interval is expanded.