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Minimizing estimation runtime on noisy quantum computers

Guoming Wang, Dax Enshan Koh, Peter D. Johnson, Yudong Cao

arXiv:2006.09350v4quant-ph

TL;DR

Expectation-value estimation is costly for VQE and other quantum algorithms, while methods with stronger scaling advantages require substantial coherence. The paper engineers likelihood functions to increase information gained per sample, and models runtime scaling across noise levels. The model interpolates between O(1/ε) and O(1/ε2) as noise varies.

  • Problem

    Expectation-value estimation is costly for VQE and other quantum algorithms, while phase-estimation methods with quadratic runtime speedups require resources beyond near-term devices for moderately large instances.

    Source: §I · p. 1, §II · p. 4

  • Method

    The paper develops tunable engineered likelihood functions that use signal processing to boost information gain per sample during Bayesian estimation.

    Source: §I · p. 2, B. Main results · p. 3

  • Results

    The runtime model interpolates between O(1/ε) and O(1/ε2) as a function of noise degree λ.

    Source: B. Main results · p. 4

  • Takeaways & Limitations

    The framework targets expectation-value and amplitude-estimation subroutines, including those used by VQE.

    Source: §I · p. 2

    Improving estimation alone does not resolve other VQE challenges, including optimization of parameterized quantum circuits.

    Source: §II · p. 4

Abstract

from arXiv · show

The number of measurements demanded by hybrid quantum-classical algorithms such as the variational quantum eigensolver (VQE) is prohibitively high for many problems of practical value. For such problems, realizing quantum advantage will require methods which dramatically reduce this cost. Previous quantum algorithms that reduce the measurement cost (e.g. quantum amplitude and phase estimation) require error rates that are too low for near-term implementation. Here we propose methods that take advantage of the available quantum coherence to maximally enhance the power of sampling on noisy quantum devices, reducing measurement number and runtime compared to the standard sampling method of the variational quantum eigensolver (VQE). Our scheme derives inspiration from quantum metrology, phase estimation, and the more recent "alpha-VQE" proposal, arriving at a general formulation that is robust to error and does not require ancilla qubits. The central object of this method is what we call the "engineered likelihood function" (ELF), used for carrying out Bayesian inference. We show how the ELF formalism enhances the rate of information gain in sampling as the physical hardware transitions from the regime of noisy intermediate-scale quantum computers into that of quantum error corrected ones. This technique speeds up a central component of many quantum algorithms, with applications including chemistry, materials, finance, and beyond. Similar to VQE, we expect small-scale implementations to be realizable on today's quantum devices.

I. INTRODUCTION

Near-term quantum algorithms face costly expectation-value estimation, while methods with stronger asymptotic speedups demand coherence beyond practical noisy devices. The paper develops engineered likelihood functions (ELFs) to use available coherence for more informative sampling, then examines noise, tunability, runtime, and a simple enhanced-sampling example.

I. INTRODUCTION

  • Expectation-value estimation is a costly bottleneck in VQE and other algorithms, while phase-estimation methods with quadratic runtime speedups require resources beyond near-term devices for moderately large instances.The introduction also identifies Monte Carlo estimation and linear-system algorithms as applications constrained in part by coherence requirements.Source: §I · p. 1

A. Prior work

  • α-VQE [20] interpolates between VQE sampling and phase estimation through a sample-count/coherence tradeoff, but lower asymptotic exponents require deeper circuits.Its scaling ranges from O(1/ε2) at α = 2 to O(1/ε) at α = 1.Source: §I · p. 1, A. Prior work · p. 3

B. Main results

  • ELFs tune likelihood functions to maximize information gain, using circuits whose likelihoods can have increasing polynomial degree as repetitions of the ansatz increase.The approach frames expectation estimation as inference on Π = ⟨A|O|A⟩ and optimizes circuit parameters for information gain.Source: §I · p. 2, B. Main results · p. 3
  • Noisy devices can suppress estimation bias only with noise tailoring and calibration, while noise and error still increase runtime.The authors report that noise does not necessarily introduce systematic bias, and that active noise-tailoring techniques and noise calibration can suppress bias.Source: B. Main results · p. 3
  • With noise degree λ, the runtime model interpolates between O(1/ε) and O(1/ε2), allowing runtime estimates from hardware specifications.The model is presented for target accuracy ε and devices with λ ∈ [0, ∞).Source: B. Main results · p. 4

II. A FIRST EXAMPLE

  • At Π = 0, single-sample Fisher information is 9 for enhanced sampling versus 1 for standard sampling, implying at least ninefold fewer measurements for the target error in this example.The enhanced-sampling likelihood also has a steeper slope at Π = 0; Figure 1 depicts the compared circuits and likelihoods.Source: §II · p. 5, Figure 1 · p. 6
  • Enhanced sampling’s greater layer depth raises time per sample, and accumulated circuit error eventually yields diminishing Fisher-information returns, so performance requires balancing the two.The example motivates Bayesian inference for processing outcomes from circuits with varying layer counts rather than relying on a sample mean.Source: §II · pp. 6–7

B. Main results

  • Noisy Chebyshev likelihood functions have estimation dead spots, which tunable generalized-reflection angles can mitigate by boosting information gain around flat regions.The authors say engineering likelihood functions can mitigate dead spots even in deeper enhanced-sampling circuits.Source: B. Main results · p. 4, §II · p. 7

III. ENGINEERED LIKELIHOOD FUNCTIONS

Engineered likelihood functions (ELFs) use tunable generalized reflections to shape measurement probabilities for estimating an expectation value. The section describes their circuit construction and how noise affects the resulting likelihood.

A. Quantum circuits for engineered likelihood functions

  • The ancilla-free circuit prepares ansatz state |A⟩=A|0n⟩, applies 2L generalized reflections with rotation angles xj, then measures outcome d∈{0,1}, defining an ELF probability distribution given unknown θ.Source: A. Quantum circuits for engineered… · p. 7
  • For L = 1, tuning generalized reflection angles can boost Fisher information in noisy estimation dead spots where the enhanced-sampling likelihood is flat.Figure 2 uses (x1, x2) = (−0.6847, 0.6847) for the engineered likelihood; even small circuit error can make Fisher information zero in flat regions.Source: Figure 2 · p. 8
  • Setting every reflection angle to π/2 yields the Chebyshev likelihood function, whose bias is a degree-(2L + 1) Chebyshev polynomial of Π.General ELFs allow the tunable parameters to produce likelihood functions beyond this fixed-angle case.Source: A. Quantum circuits for engineered… · p. 9
  • Noise rescales the ideal likelihood bias by the process fidelity f, making the resulting ELF less steep as errors increase.The model incorporates layer fidelity p and SPAM factor p̄, with f = p̄p^L.Source: A. Quantum circuits for engineered… · p. 9

B. Bayesian inference with engineered likelihood functions

The inference procedure tracks a Gaussian belief over θ, tunes circuit parameters to gain information, and updates that belief after each measurement. It converts between θ and Π to report the final estimate.

B. Bayesian inference with engineered likelihood functions

  • At each round, the algorithm selects circuit parameters, measures an outcome, and applies Bayes’ rule to update its distribution over θ.The final distribution over θ is converted to a distribution over Π, whose mean is the estimate.Source: B. Bayesian inference with engineered… · pp. 9–10, Figure 4 · p. 10
  • The method approximates each posterior over θ with a Gaussian of the same mean and variance, then uses it as the next round’s prior.The true posterior need not be Gaussian; the algorithm repeats measurement and update until the distribution is sufficiently concentrated.Source: B. Bayesian inference with engineered… · p. 10
  • For sufficiently small prior variance, the algorithm uses Fisher information at the prior mean as a proxy for variance reduction when tuning circuit parameters.When both prior variance and process fidelity are small, the variance reduction factor is approximately f^2(Δ′(µ;x))^2.Source: B. Bayesian inference with engineered… · p. 11
  • With fixed circuit depth, the section predicts MSE(ˆµk) = Θ(1/k) as the number of inference rounds k grows.The predicted asymptotic inverse-MSE growth rate per time step depends on Fisher information at the true value, maximized over circuit parameters.Source: B. Bayesian inference with engineered… · p. 11

IV. EFFICIENT HEURISTIC ALGORITHMS FOR CIRCUIT PARAMETER TUNING AND BAYESIAN INFERENCE

The section presents heuristic methods for tuning ELF circuit parameters by optimizing proxies for variance reduction. It uses coefficient calculations and iterative optimization to make that tuning efficient.

IV. EFFICIENT HEURISTIC ALGORITHMS FOR CIRCUIT PARAMETER TUNING AND BAYESIAN INFERENCE

  • The tuning algorithms maximize Fisher information or likelihood slope as proxies for the variance reduction factor.For sufficiently small prior variance, Fisher information approximates the variance reduction factor; at low fidelity, the slope proxy is also used.Source: §IV · p. 12, A. Efficient maximization of proxies… · p. 12

A. Efficient maximization of proxies of the variance reduction factor

  • Given θ and all parameters except xj, the coefficient functions used in the tuning procedures can be computed in O(L) time.The functions are the cosine, sine, and bias decomposition coefficients and their θ-derivatives.Source: A. Efficient maximization of proxies… · p. 12

1. Maximizing the Fisher information of the likelihood function

  • Gradient ascent updates all parameters using the objective gradient, while coordinate ascent successively optimizes one parameter and increases Fisher information monotonically.Each gradient-ascent iteration and coordinate-ascent round takes O(L^2) time; coordinate ascent avoids step sizes and may converge faster.Source: §1 · pp. 12–13
  • ELF Fisher information exceeds CLF Fisher information for most θ values, and is less sensitive to θ in the L = 6, p = 0.9, no-SPAM comparison.The figure reports f = 0.531441; CLF Fisher information is exactly zero at θ = jπ/(2L + 1).Source: §1 · p. 13, Figure 5 · p. 13

2. Maximizing the slope of the likelihood function

The section describes slope-based optimization of engineered likelihood functions (ELFs) and an approximate Bayesian update that avoids numerical integration. Near the prior mean, a sinusoidal fit supplies analytic approximations to the posterior mean and variance.

2. Maximizing the slope of the likelihood function

  • Gradient-ascent and coordinate-ascent algorithms tune circuit parameters to maximize the likelihood slope at the prior mean.The optimized parameters are evaluated across θ to compare ELF and CLF slopes.Source: §2 · p. 13
  • Slope-based ELFs are steeper than CLFs for most θ and are less sensitive to θ, giving them more statistical power in the low-fidelity setting.The comparison uses L = 6; the CLF slope is exactly zero at θ = jπ/(2L + 1).Source: §2 · p. 13, Figure 6 · p. 13

B. Approximate Bayesian inference with engineered likelihood functions

  • Approximating the likelihood near the prior mean with a sinusoid enables analytic posterior updates instead of time-consuming numerical integration.The fit is constructed from points around µ, and its parameters determine approximations to posterior mean and variance.Source: B. Approximate Bayesian inference with… · pp. 13–14
  • The sinusoidal fit can be inaccurate far from µ, but those θ values have little influence on posterior mean and variance when the prior is narrow.For θ far from µ, the prior contribution decays exponentially with distance.Source: B. Approximate Bayesian inference with… · pp. 13, 15

V. SIMULATION RESULTS

Simulations compare engineered and Chebyshev likelihood functions (ELFs and CLFs) in ancilla-free and ancilla-based schemes. ELFs perform better or as well as CLFs, and the ancilla-free ELF performs best among the compared schemes; their inverse-MSE growth rates are roughly, but not reliably, predicted by ˆR0.

A. Experimental details

  • The simulations estimate RMSE across 300 inference runs and use pseudo-randomly generated measurement outcomes rather than directly simulating quantum circuits.The experiments assume no SPAM error and approximate each round’s time cost as 2L+1 units of ansatz-circuit duration.Source: A. Experimental details · pp. 15–16
  • A lookup table of optimized parameters across 4,001 Π values accelerates repeated parameter tuning, with little performance deterioration reported.The simulations use 11 uniformly spaced points across µ±σ for sinusoidal fitting of the likelihood.Source: A. Experimental details · p. 16

B. Comparing the performance of various schemes

  • The reported scheme comparisons use L = 6 and layer fidelity p = 0.9 across varied true Π values.The figure captions also show individual settings where ELF outperforms CLF and AF ELF achieves the best performance.Source: B. Comparing the performance of… · p. 17, Figure 8 · p. 18, Figure 9 · p. 18
  • ELF performs better than or as well as CLF in both ancilla settings, while AF ELF always outperforms AB ELF and AF CLF outperforms AB CLF on average.ELF estimator RMSE tends to zero with increasing time, unlike CLF RMSE in all cases; CLF performance depends more strongly on the true Π.Source: B. Comparing the performance of… · p. 17

C. Understanding the performance of Bayesian inference with ELFs

1. Analyzing the impact of layer fidelity on the performance of estimation

The simulations show that higher layer fidelity improves estimation, while increasing circuit depth does not guarantee better performance. The runtime model connects gate fidelity and circuit depth to estimation time, with practical gains depending on balancing fidelity improvements against gate-time overhead.

1. Analyzing the impact of layer fidelity on the performance of estimation

  • Higher layer fidelity makes the estimator’s RMSE decrease faster, consistent with monotonically increasing AB ELF and AF ELF variance-reduction factors as f = p^L increases.Figures 14 and 15 vary layer fidelity from 0.75 to 0.95 at fixed depth L = 6.Source: §1 · p. 19, Figure 14 · p. 21, Figure 15 · p. 21

2. Analyzing the impact of circuit depth on the performance of estimation

  • Choosing depth remains unresolved: larger L need not increase the variance-reduction factor, and an efficient optimizer must account for layer fidelity and the prior on Π.Evaluating the factor across candidate depths can identify a satisfactory, though not necessarily optimal, choice.Source: §2 · pp. 19–20, Figure 19 · p. 22

VI. A MODEL FOR NOISY ALGORITHM PERFORMANCE

  • The runtime model treats inverse-MSE growth as half the inverse-variance rate and uses posterior variance as a proxy for estimator variance; its ELF upper-bound extension is conjectured.The analysis uses Chebyshev likelihood bounds and assumes a sufficiently peaked phase-estimator distribution when converting phase MSE to amplitude MSE.Source: §VI · pp. 20, 24

2. Analyzing the impact of circuit depth on the performance of estimation

  • At p = 0.9, both AB ELF and AF ELF achieve their best performance at L = 4 rather than L = 5.As depth increases, likelihood steepness can improve statistical power, but circuit fidelity decreases exponentially and implementation cost grows linearly.Source: §2 · p. 19, Figure 17 · p. 22, Figure 18 · p. 22

VI. A MODEL FOR NOISY ALGORITHM PERFORMANCE

  • For F ≪ 1/λ^2, inverse squared error grows quadratically as F(t) ∼ t^2; for F ≫ 1/λ^2, it grows linearly as F(t) ∼ t.The paper identifies these respectively as Heisenberg-limit and shot-noise-limit regimes.Source: §VI · p. 23
  • As two-qubit gate fidelities improve, deeper enhanced-sampling circuits warrant implementation and yield shorter estimated runtimes; error-correction overhead must not erase this gain.The model suggests the product of gate infidelity and error-corrected gate time should decrease as useful error correction is incorporated.Source: Figure 20 · p. 24, §VI · p. 24
  • The simplified cost analysis omits gate-to-gate fidelity variation and differing gate durations, and the model is expected to break down for λ ≥ 1.The authors describe the analysis as a useful design tool despite these practical subtleties.Source: §VI · pp. 23, 25

VII. OUTLOOK

The outlook presents enhanced sampling and engineered likelihood functions as a route to faster estimation across noisy and error-corrected devices. It highlights near-term performance, resource trade-offs, and unresolved questions about calibration and measurement grouping.

VII. OUTLOOK

  • Enhanced sampling uses available quantum coherence along a continuum between standard sampling and amplitude or phase estimation, without requiring ancilla qubits.The authors expect estimation-based quantum advantage to occur within this continuum rather than at either extreme.Source: §VII · p. 25
  • The proposed estimation workflow uses adaptive approximate Bayesian inference and tunes likelihood functions to improve statistical power, account for calibrated noise, balance circuit depth against error, and mitigate dead spots.The work also proposes a model for predicting enhanced-sampling performance as device quality improves.Source: §VII · p. 25
  • At near-term two-qubit gate fidelities of ∼99.92%, enhanced sampling with ELFs outperforms standard VQE sampling; optimal circuit fidelities are 0.5−0.7, allowing 50-fold greater depth than the 0.99 logical fidelities cited in [16].The authors obtain this balance by minimizing estimation runtime.Source: §VII · p. 25
  • The runtime model predicts that a 1,000-fold increase in gate time requires three additional nines of gate fidelity for equal estimation runtimes, while seven nines suffice in the 100-qubit example.For error correction to help estimation, its runtime overhead must be smaller than the reduction factor in logical gate error rates.Source: §VII · p. 25
  • Whether implementing generalized reflections for grouped measurements adds worthwhile circuit depth relative to the resulting variance reduction remains an open question.The authors also anticipate opportunities to parallelize measurements dedicated to enhanced sampling.Source: §VII · p. 25
  • Accurate algorithm-level error models and likelihood calibration remain important open problems because miscalibration can introduce systematic bias in parameter estimates.The authors leave for future work determining the precision needed to make that bias negligible.Source: §VII · pp. 25–26

Appendix A: A sufficient model of circuit noise

Appendix A presents a sufficient depolarizing-noise model for the likelihood generated by the circuit, including state-preparation and measurement errors. Appendix B proves that specified coefficient functions and their derivatives can be evaluated in time linear in circuit layers.

Appendix A: A sufficient model of circuit noise

  • The proposed circuit-noise model mixes each target layer with a completely depolarizing channel and models state-preparation and measurement errors as SPAM noise.The authors describe this as a sufficient, but not necessary, model for the likelihood noise model used in the work.Source: Appendix A · p. 26
  • The combined likelihood-generation fidelity is f = p̄p^L, and the noisy likelihood is expressed using this fidelity and the ideal likelihood's bias.The model combines state-preparation and measurement error parameters into p̄.Source: Appendix A · p. 26

Appendix B: Proof of Lemma 1

  • Lemma 1 establishes that the CSBD coefficient functions of ∆ and ∆′ with respect to any x_j can be evaluated in O(L) time.The proof handles even and odd indices separately, expressing ∆ through cosine, sine, and bias coefficients.Source: Appendix B · pp. 26, 27, 29
  • For both index cases, the proof computes the relevant matrix products and coefficient functions in O(L) time, then uses dynamic programming to obtain the derivative coefficients in O(L) time.The appendix gives this procedure for even and odd j separately.Source: Appendix B · pp. 28, 29, 31

Appendix C: Algorithms for maximizing the slope of the ancilla-free likelihood function

The appendix gives gradient-ascent and coordinate-ascent algorithms for maximizing the slope of the ancilla-free likelihood function at the prior mean. Both use coefficient-derivative procedures from Lemma 1.

Appendix C: Algorithms for maximizing the slope of the ancilla-free likelihood function

  • The algorithms seek parameters x that maximize the magnitude of the ancilla-free likelihood slope at the prior mean, |P′(µ; f, x)| = f|∆′(µ; x)|/2.The appendix presents gradient-ascent and coordinate-ascent approaches.Source: Appendix C · p. 31
  • The gradient-ascent algorithm uses coefficient derivatives to compute the partial derivative of (∆′(µ; x))^2 with respect to each parameter x_j.It calls Lemma 1 to evaluate C′, S′, and B′ for the other fixed parameters.Source: Appendix C · p. 31
  • The coordinate-ascent algorithm instead uses those coefficient derivatives to update each x_j directly.Both algorithms rely on the procedures in Lemma 1.Source: Appendix C · p. 31

Appendix D: Ancilla-based scheme

The appendix describes an ancilla-based circuit for generating engineered likelihood functions, with tunable parameters and a corresponding Bayesian-inference procedure. It also defines a special Chebyshev likelihood case and identifies proxies for optimizing variance reduction.

Appendix D: Ancilla-based scheme

  • The ancilla-based scheme generates its engineered likelihood function with a circuit using the same U, V, and Q as the ancilla-free scheme, with tunable parameters x ∈ R^2L.The circuit is shown in Figure 21.Source: Appendix D · p. 31, Figure 21 · p. 33
  • In the noiseless case, the ancilla-based likelihood is expressed using Λ; under noise, its likelihood model uses the process fidelity f.Unlike ancilla-free ∆, Λ is trigonometric-multilinear in x rather than trigonometric-multiquadratic.Source: Appendix D · pp. 31–33
  • Setting every x parameter to π/2 gives Λ(θ; x) = (−1)^L cos(Lθ), the ancilla-based Chebyshev likelihood function.Source: Appendix D · p. 32
  • The ancilla-based scheme tunes parameters and performs Bayesian inference as in the ancilla-free case, using the same variance-reduction-factor definition with Λ substituted for ∆.Fisher information and likelihood slope at the prior mean are proxies for the variance-reduction factor under reasonable assumptions.Source: Appendix D · p. 33

1. Efficient maximization of proxies of the variance reduction factor

The section presents efficient procedures for evaluating the bias’s CSD coefficient functions and their θ-derivatives, which support the proxy-maximization algorithms. Each function can be computed in O(L) time.

a. Evaluating the CSD coefficient functions of the bias and its derivative

  • Because the bias is trigono-multilinear in the circuit parameters, each coordinate can be separated into coefficient functions Cj and Sj of the remaining parameters.Their θ-derivatives are denoted C′j and S′j.Source: a. Evaluating the CSD coefficient… · p. 33
  • Each CSD coefficient function and its θ-derivative can be computed in O(L) time for given θ and the other circuit parameters.The procedures use standard dynamic programming techniques.Source: a. Evaluating the CSD coefficient… · pp. 33–34
  • The evaluation procedure treats even and odd parameter indices separately, computing the corresponding coefficient functions and derivatives in O(L) time.For both cases, it computes intermediate matrices and then forms the coefficient functions and their θ-derivatives.Source: a. Evaluating the CSD coefficient… · pp. 34–35

b. Maximizing the Fisher information of the likelihood function

The section optimizes Fisher-information and likelihood-slope proxies by tuning circuit parameters, then uses the resulting engineered likelihood functions for Bayesian inference. The optimized ancilla-based ELF proxies outperform the corresponding CLF proxies for most θ values, and the likelihood is fitted with a sinusoidal form for inference.

b. Maximizing the Fisher information of the likelihood function

c. Maximizing the slope of the likelihood function

2. Approximate Bayesian inference with engineered likelihood functions

  • Bayesian inference fits the likelihood with a sinusoidal function and uses the fitted parameters to approximate posterior mean and variance at each round.The authors report that the approximation errors have negligible impact on overall algorithm performance.Source: §2 · pp. 36, 38, 39, Figure 24 · p. 39

Appendix E: Bias and variance of the ELF-based

The appendix characterizes the ELF-based estimator’s large-sample behavior: its squared bias is below its variance, and its perceived variance tracks the estimator’s variance with high probability. Together, these properties bound its mean squared error by twice the perceived variance with high probability.

Appendix E: Bias and variance of the ELF-based

  • For sufficiently large t, the ELF-based estimator’s squared bias is smaller than its variance in both ancilla-based and ancilla-free cases.The appendix states Bias(µ̂t)² < Var(µ̂t).Source: Appendix E · p. 39
  • For sufficiently large t, the perceived variance µ̂’s associated variance estimate is close to Var(µ̂t) with high probability.The appendix reports this behavior for both ancilla-based and ancilla-free cases.Source: Appendix E · p. 39
  • For sufficiently large t, MSE(µ̂t) is at most 2σ̂²t with high probability; similar statements hold for the ELF-based estimator of θ.This conclusion combines the squared-bias and perceived-variance properties.Source: Appendix E · p. 39

Appendix F: Comparison of exact optimization with optimization of proxies

The appendix studies how the variance-reduction factor behaves in limiting regimes and compares direct optimization with proxy objectives. It motivates the proxies through small-variance and low-fidelity conditions, then numerically examines the optimization for small L.

1. Limiting behavior of the variance reduction factor

  • The variance-reduction factor is analyzed as prior variance vanishes, fidelity approaches zero, and fidelity equals one, under the stated assumption for the f = 1 case.The results are stated for the ancilla-free bias and also apply to the ancilla-based bias after replacing ∆ with Λ.Source: Appendix F · p. 39, §1 · pp. 39–40
  • The appendix motivates proxy optimization from limiting-case expressions, with Fisher information corresponding to the noisy ELF in the vanishing-variance case.It expects the stated proxy objectives to approximate variance-reduction-factor optimization when σ is small, or when both σ and f are small.Source: §1 · p. 40, §2 · p. 41

2. Implementing the exact variance reduction factor optimization and comparison with proxies

  • The authors numerically maximize the expected posterior variance and proxy objectives for small L, presenting the optimization results in Figures 28–31.For L = 1, the ancilla-free-bias optimization problem (F9) has an analytical solution.Source: §2 · p. 41

3. Analytical expressions for L = 1 slope proxy

For L = 1, the appendix gives an analytical solution to the ancilla-free bias optimization and characterizes the maximum likelihood slope. The proxy’s accuracy depends on f, σ, circuit depth, and whether the scheme uses an ancilla.

3. Analytical expressions for L = 1 slope proxy

  • For the ancilla-free scheme, the L = 1 bias optimization is solved analytically, with its solution expressed through polynomial roots.The notation roota(p) denotes the ath smallest real root of p, starting at index 1.Source: §3 · pp. 41, 43, 46
  • For the ancilla-free scheme at L = 1 and L = 2, proxies work well when f ≤0.9 and σ ≤0.1, but fail as approximations when f = 0.999 and σ = 0.2.The plotted variance reduction factor V is shown against prior mean µ.Source: Figure 28 · p. 42, Figure 29 · p. 43
  • For the ancilla-based scheme, proxies match exact variance-reduction optimization at L = 1, while at L = 2 they work well only when f and σ are not large.At L = 2, they fail when f = 0.999 and σ = 0.2.Source: Figure 30 · p. 44, Figure 31 · p. 45
  • The L = 1 likelihood has a maximum slope magnitude achieved by a specified pair of angles, with the angle definitions given through Eqs. (F19)–(F22).The angle definitions involve roots of two octic polynomials.Source: §3 · pp. 44, 46
  • Figure 32 graphs the optimal slope using the angles defined by Eqs. (F19)–(F22).The text identifies the plotted quantity as Eq. (F16).Source: Figure 32 · p. 46, §3 · p. 46

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