Source-linked AI summary
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.
Source: §I · p. 1Method
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.
Source: §I · p. 1, A. Tunable Bayesian QPE (α-QPE) · p. 3, B. Casting expectation estimation as… · p. 3Results
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.
Source: §I · p. 1Takeaways & 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.
Source: §I · p. 1, A. Tunable Bayesian QPE (α-QPE) · p. 3
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.Source: §I · p. 1
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.Source: §I · p. 1
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.Source: §I · p. 1
- 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.Source: §I · p. 1
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.Source: §I · p. 1
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)⟩.Source: §II · pp. 1–2
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.Source: §II · p. 2
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.Source: §II · p. 2
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.Source: §II · p. 2
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.Source: §II · p. 2
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.Source: A. Tunable Bayesian QPE (α-QPE) · p. 2
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.Source: A. Tunable Bayesian QPE (α-QPE) · p. 3
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.Source: A. Tunable Bayesian QPE (α-QPE) · p. 3
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.Source: A. Tunable Bayesian QPE (α-QPE) · p. 3
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).Source: B. Casting expectation estimation as… · p. 3
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.Source: B. Casting expectation estimation as… · p. 3
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.Source: B. Casting expectation estimation as… · p. 3
- 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.Source: B. Casting expectation estimation as… · p. 3
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.Source: B. Casting expectation estimation as… · p. 3
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.Source: C. Generalised α-VQE · p. 3, Figure 2 · p. 4
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.Source: C. Generalised α-VQE · p. 3
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.Source: C. Generalised α-VQE · pp. 3–4
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.Source: Figure 2 · p. 4, C. Generalised α-VQE · p. 4
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.Source: §III · p. 4
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.Source: §III · p. 4
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.Source: §III · p. 4, Figure 3 · p. 4
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.Source: §III · p. 4
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.Source: Figure 3 · p. 4
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.Source: Appendix A · p. 6
- 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.Source: Appendix A · p. 6
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.Source: Appendix A · pp. 6–7
- 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.Source: Appendix A · p. 7
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.Source: Appendix A · p. 8, Figure 5 · p. 9
- 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.Source: Appendix A · pp. 8–9, Figure 6 · p. 9
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.Source: Appendix B · p. 10
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.Source: Appendix B · p. 10
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.Source: Appendix B · p. 10
- 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.Source: Appendix B · p. 10
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.Source: Appendix C · p. 10
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)].Source: Appendix C · p. 10
- 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.Source: Appendix C · p. 11
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.Source: Appendix C · pp. 10–11
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.Source: Appendix C · p. 11
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.Source: Appendix C · p. 11