## Introduction

Quantum Amplitude Estimation (QAE)1 is a fundamental quantum algorithm with the potential to achieve a quadratic speedup for many applications that are classically solved through Monte Carlo (MC) simulation. It has been shown that we can leverage QAE in the financial service sector, e.g., for risk analysis2,3 or option pricing4,5,6, and also for generic tasks such as numerical integration7. While the estimation error bound of classical MC simulation scales as $${\mathcal{O}}(1/\sqrt{M})$$, where M denotes the number of (classical) samples, QAE achieves a scaling of $${\mathcal{O}}(1/M)$$ for M (quantum) samples, indicating the aforementioned quadratic speedup.

The canonical version of QAE is a combination of Quantum Phase Estimation (QPE)8 and Grover’s Algorithm. Since other QPE-based algorithms are believed to achieve exponential speedup, most prominently Shor’s Algorithm for factoring9, it has been speculated as to whether QAE can be simplified such that it uses only Grover iterations without a QPE-dependency. Removing the QPE-dependency would help to reduce the resource requirements of QAE in terms of qubits and circuit depth and lower the bar for practical applications of QAE.

Recently, several approaches have been proposed in this direction. In ref. 10 the authors show how to replace QPE with a set of Grover iterations combined with a Maximum Likelihood Estimation (MLE), in the following called Maximum Likelihood Amplitude Estimation (MLAE). In ref. 11, QPE is replaced by the Hadamard test, analog to Kitaev’s Iterative QPE12,13 and similar approaches14,15.

Both in refs. 10 and 11 propose potential simplifications of QAE but do not provide rigorous proofs of the correctness of the proposed algorithms. In ref. 11, it is not even clear how to control the accuracy of the algorithm other than possibly increasing the number of measurements of the evolving quantum circuits. Thus, the potential quantum advantage is difficult to compare and we will not discuss it in the remainder of this paper.

In ref. 16, another variant of QAE was proposed. There, for the first time, it was rigorously proven that QAE without QPE can achieve a quadratic speedup over classical MC simulation. Following16, we call this algorithm QAE, Simplified (QAES). Although this algorithm achieves the desired asymptotic complexity exactly (i.e., without logarithmic factors), the involved constants are very large, and likely to render this algorithm impractical unless further optimized—as shown later in this paper.

In the following, we propose a new version of QAE—called Iterative QAE (IQAE)—that achieves better results than all other tested algorithms. It provably has the desired asymptotic behavior up to a multiplicative $$\mathrm{log}\,(2/\alpha {\mathrm{log}\,}_{2}(\pi /4\epsilon ))$$ factor, where ϵ > 0 denotes the target accuracy, and 1 − α the resulting confidence level.

Like in ref. 16, our algorithm requires iterative queries to the quantum computer to achieve the quadratic speedup and cannot be parallelized. Only MLAE allows the parallel execution of the different queries as the estimate is derived via classical MLE applied to the results of all queries. Although parallelization is a nice feature, the potential speedup is limited. Assuming the length of the queries is doubled in each iteration (like for canonical QAE and MLAE) the speedup is at most a factor of two since the computationally most expensive query dominates all the others.

With MLAE, QAES, and IQAE we have three promising variants of QAE that do not require QPE and it is of general interest to empirically compare their performance. Of similar interest is the question of whether the canonical QAE with QPE—while being (quantum) computationally more expensive—might lead to some performance benefits. To be able to better compare the performance of canonical QAE with MLAE, QAES, and IQAE, we extend QAE by a classical MLE postprocessing based on the observed results. This improves the results without additional queries to the quantum computer and allows us to derive proper confidence intervals.

QAE was first introduced in ref. 1 and assumes the problem of interest is given by an operator $${\mathcal{A}}$$ acting on n + 1 qubits such that

$${\mathcal{A}}{\left|0\right\rangle }_{n}\left|0\right\rangle =\sqrt{1-a}{\left|{\psi }_{0}\right\rangle }_{n}\left|0\right\rangle +\sqrt{a}{\left|{\psi }_{1}\right\rangle }_{n}\left|1\right\rangle ,$$
(1)

where a [0, 1] is the unknown, and $${\left|{\psi }_{0}\right\rangle }_{n}$$ and $${\left|{\psi }_{1}\right\rangle }_{n}$$ are two normalized states, not necessarily orthogonal. QAE allows estimating a with high probability such that the estimation error scales as $${\mathcal{O}}(1/M)$$, where M corresponds to the number of applications of $${\mathcal{A}}$$. To this extent, an operator $${\mathcal{Q}}={\mathcal{A}}{{\mathcal{S}}}_{0}{{\mathcal{A}}}^{\dagger }{{\mathcal{S}}}_{{\psi }_{0}}$$ is defined where $${{\mathcal{S}}}_{{\psi }_{0}}={\mathbb{I}}-2\left|{\psi }_{0}\right\rangle_n \left\langle {\psi }_{0}\right|_n\otimes \left|0\right\rangle \left\langle 0\right|$$ and $${{\mathcal{S}}}_{0}={\mathbb{I}}-2{\left|0\right\rangle }_{n+1}{\left\langle 0\right|}_{n+1}$$ as introduced in1. In the following, we denote applications of $${\mathcal{Q}}$$ as quantum samples or oracle queries.

The canonical QAE follows the form of QPE: it uses m ancilla qubits—initialized in equal superposition—to represent the final result, it defines the number of quantum samples as M = 2m and applies geometrically increasing powers of $${\mathcal{Q}}$$ controlled by the ancillas. Eventually, it performs an inverse QFT on the ancilla qubits before they are measured, as illustrated in Fig. 1. Subsequently, the measured integer y {0, …, M − 1} is mapped to an angle $${\tilde{\theta }}_{a}=y\pi /M$$. Thereafter, the resulting estimate of a is defined as $$\tilde{a}={\sin }^{2}({\tilde{\theta }}_{a})$$. Then, with a probability of at least 8/π2 ≈ 81%, the estimate $$\tilde{a}$$ satisfies

$$| a-\tilde{a}| \le \frac{2\pi \sqrt{a(1-a)}}{M}+\frac{{\pi }^{2}}{{M}^{2}},$$
(2)

which implies the quadratic speedup over a classical MC simulation, i.e., the estimation error $$\epsilon ={\mathcal{O}}(1/M)$$. The success probability can quickly be boosted to close to 100% by repeating this multiple times and using the median estimate2. These estimates $$\tilde{a}$$ are restricted to the grid $$\left\{{\sin }^{2}\left(y\pi /M\right):y=0,\ldots ,M/2\right\}$$ through the possible measurement outcomes of y.

Alternatively, and similarly to MLAE, it is possible to apply MLE to the observations for y. For a given θa, the probability of observing $$\left|y\right\rangle$$ when measuring the ancilla qubits is derived in1 and given by

$${\mathbb{P}}[\left|y\right\rangle ]=\frac{{\sin }^{2}(M{{\Delta }}\pi )}{{M}^{2}{\sin }^{2}({{\Delta }}\pi )},$$
(3)

where Δ is the minimal distance on the unit circle between the angles θa and $$\pi \tilde{y}/M$$, and $$\tilde{y}=y$$ if y ≤ M/2 and $$\tilde{y}=M/2-y$$ otherwise. Given a set of y-measurements, this can be leveraged in an MLE to get an estimate of θa that is not restricted to grid points. Furthermore, it allows using the likelihood ratio to derive confidence intervals17. This is discussed in more detail in Supplementary Section 1. In our tests, the likelihood ratio confidence intervals were always more reliable than other possible approaches, such as the (observed) Fisher information. Thus, in the following, we will use the term QAE for the canonical QAE with the application of MLE to the y measurements to derive an improved estimate and confidence intervals based on the likelihood ratio.

All variants of QAE without QPE—including ours—are based on the fact that

$${{\mathcal{Q}}}^{k}{\mathcal{A}}{\left|0\right\rangle }_{n}\left|0\right\rangle =\cos ((2k+1){\theta }_{a}){\left|{\psi }_{0}\right\rangle }_{n}\left|0\right\rangle +\sin ((2k+1){\theta }_{a}){\left|{\psi }_{1}\right\rangle }_{n}\left|1\right\rangle ,$$
(4)

where θa is defined as $$a={\sin }^{2}({\theta }_{a})$$. In other words, the probability of measuring $$\left|1\right\rangle$$ in the last qubit is given by

$${\mathbb{P}}[\left|1\right\rangle ]={\sin }^{2}((2k+1){\theta }_{a}).$$
(5)

The algorithms mainly differ in how they derive the different values for the powers k of $${\mathcal{Q}}$$ and how they combine the results into a final estimate of a.

MLAE first approximates $${\mathbb{P}}[\left|1\right\rangle ]$$ for k = 2j and j = 0, 1, 2, …, m − 1, for a given m, using Nshots measurements from a quantum computer for each j, i.e., in total, $${\mathcal{Q}}$$ is applied Nshots(M − 1) times, where M = 2m. It has been shown in ref. 10 that the corresponding Fisher information scales as $${\mathcal{O}}({N}_{\text{shots}}{M}^{2})$$, which implies a lower bound of the estimation error scaling as $${{\Omega }}(1/(\sqrt{{N}_{\text{shots}}}M))$$. Crucially10, does not provide an upper bound for the estimation error. Confidence intervals can be derived from the measurements using, e.g., the likelihood ratio approach, see Supplementary Section 1.

In contrast to MLAE, QAES requires the different powers of $${\mathcal{Q}}$$ to be evaluated iteratively and cannot be parallelized. It iteratively adapts the powers of $${\mathcal{Q}}$$ to successively improve the estimate and carefully determines the next power of $${\mathcal{Q}}$$. However, instead of a lower bound, a rigorous error upper bound is provided. QAES achieves the optimal asymptotic query complexity $${\mathcal{O}}(\mathrm{log}\,(1/\alpha )/\epsilon )$$, where α > 0 denotes the probability of failure. In contrast to the other algorithms considered, QAES provides a bound on the relative estimation error. Although the algorithm achieves the desired asymptotic scaling exactly, the constants involved are very large—likely too large for practical applications unless they can be further reduced.

In the following, we introduce a new variant of QAE without QPE. As for QAES, we provide rigorous performance proof. Although our algorithm only achieves the quadratic speedup up to a multiplicative factor $$\mathrm{log}\,(2/\alpha {\mathrm{log}\,}_{2}(\pi /4\epsilon ))$$, the constants involved are orders of magnitude smaller than for QAES. Moreover, in practice, this doubly logarithmic factor is small for any reasonable target accuracy ϵ and confidence level 1 − α, as we will show in the following.

## Results

### Algorithm description

IQAE leverages similar ideas as refs. 10,11,16 but combines them in a different way, which results in a more efficient algorithm while still allowing for a rigorous upper bound on the estimation error and computational complexity. As mentioned before, we use the quantum computer to approximate $${\mathbb{P}}[\left|1\right\rangle ]={\sin }^{2}((2k+1){\theta }_{a})$$ for the last qubit in $${{\mathcal{Q}}}^{k}{\mathcal{A}}{\left|0\right\rangle }_{n}\left|0\right\rangle$$ for different powers k. In the following, we outline the rationale behind IQAE, which is formally given in Algorithm 1. The main sub-routine FINDNEXTK is outlined in Algorithm 2.

Suppose a confidence interval [θl, θu]  [0, π/2] for θa and a power k of $${\mathcal{Q}}$$, as well as an estimate for $${\sin }^{2}((2k+1){\theta }_{a})$$. We can translate our estimates for $${\sin }^{2}((2k+1){\theta }_{a})$$ into estimates for $$\cos ((4k+2){\theta }_{a})$$. However, unlike in Kitaev’s Iterative QPE, we cannot estimate $$\sin ((4k+2){\theta }_{a})$$, and the cosine alone is only invertible without ambiguity if we know that the argument is restricted to either [0, π] or [π, 2π], i.e., the upper or lower half-plane. Thus, we want to find the largest k such that the scaled interval $${[(4k+2){\theta }_{l},(4k+2){\theta }_{u}]}_{\text{mod}2\pi }$$ is fully contained either in [0, π] or [π, 2π]. If this is given, we can invert $$\cos ((4k+2){\theta }_{a})$$ and improve our estimate for θa with high confidence. This implies an upper bound of k, and the heart of the algorithm is the procedure used to find the next k given [θl, θu], which is formally introduced in Algorithm 2 and illustrated in Fig. 2. In the following theorem, we provide convergence results for IQAE that imply the aforementioned quadratic speedup. The respective proof is given in Supplementary Section 2.

### Theorem 1

(Correctness of IQAE). Suppose a confidence level 1 − α (0, 1), a target accuracy ϵ > 0, and a number of shots Nshots {1, ..., Nmax(ϵ, α)}, where

$${N}_{\text{max}}(\epsilon ,\alpha )=\frac{32}{{(1-2\sin (\pi /14))}^{2}}\mathrm{log}\,\left(\frac{2}{\alpha }{\mathrm{log}\,}_{2}\left(\frac{\pi }{4\epsilon }\right)\right).$$
(6)

In this case, IQAE (Algorithm 1) terminates after a maximum number of $${\mathrm{log}\,}_{2}(\pi /8\epsilon )$$ rounds, where we define one round as a set of iterations with the same ki, and each round consists of at most Nmax(ϵ, α)/Nshots iterations. IQAE returns [al, au] with au − al ≤ 2ϵ and

$${\mathbb{P}}[{a}\,\notin \,[{a}_{l},{a}_{u}]]\le \alpha.$$
(7)

Thus, $$\tilde{a}=({a}_{l}+{a}_{u})/2$$ leads to an estimate for a with $$| a-\tilde{a}| \le \epsilon$$ with a confidence of 1 − α.

Furthermore, for the total number of $${\mathcal{Q}}$$-applications, Noracle, it holds that

$${N}_{\text{oracle}}<\frac{50}{\epsilon }\mathrm{log}\,\left(\frac{2}{\alpha }{\mathrm{log}\,}_{2}\left(\frac{\pi }{4\epsilon }\right)\right).$$
(8)

### Algorithm 1

Iterative Quantum Amplitude Estimation

FunctionIQAE (ϵ, α, Nshots, ci):

// ci is a chosen confidence interval method, which can be either Clopper-Pearson18,19or Chernoff-Hoeffding20

i = 0 // initialize iteration count

ki = 0 // initialize power of$${\mathcal{Q}}$$

upi = True // keeps track of the half-plane

[θl, θu] = [0, π/2] // initialize conf. interval

T = $${\mathrm{log}\,}_{2}(\pi /8\epsilon )$$ // max. number of rounds

calculate Lmax according to Eqs. (9) and (10) // max. error on every iteration

whileθu − θl > 2ϵdo

i = i + 1

ki, upi = FindNextK (ki−1, θl, θu, upi−1)

set Ki = 4ki + 2

ifKi > $${L}_{\max }/\epsilon$$then

N = NshotsLmax/ϵ/Ki/10 // no-overshooting condition

else

N = Nshots

approximate $${a}_{i}={\mathbb{P}}[\left|1\right\rangle ]$$ for the last qubit of $${{\mathcal{Q}}}^{{k}_{i}}{\mathcal{A}}{\left|0\right\rangle }_{n}\left|0\right\rangle$$ by measuring N times

ifki = ki−1then

combine the results of all iterations j ≤ i with kj = ki into a single result, effectively increasing the number of shots

if ci = "Chernoff-Hoeffding" then

$${\epsilon }_{{a}_{i}}=\sqrt{\frac{1}{2N}\mathrm{log}\,(\frac{2{\it{T}}}{\alpha })}$$

$${a}_{i}^{\,\text{max}\,}=\min (1,{a}_{i}+{\epsilon }_{{a}_{i}})$$

$${a}_{i}^{\,\text{min}\,}=\max (0,{a}_{i}-{\epsilon }_{{a}_{i}})$$

if ci = "Clopper-Pearson" then

$${a}_{i}^{\,\text{max}\,}={I}^{-1}(\frac{\alpha }{2T};N{a}_{i},N(1-{a}_{i})+1)$$

$${a}_{i}^{\,\text{min}\,}={I}^{-1}(1-\frac{\alpha }{2T};N{a}_{i}+1,N(1-{a}_{i}))$$ // see Supplementary Eqs. (33)–(36)

calculate the confidence interval $$[{\theta }_{i}^{\,\text{min}},{\theta }_{i}^{\text{max}\,}]$$ for $${\{{K}_{i}{\theta }_{a}\}}_{\text{mod}2\pi }$$ from $$[{a}_{i}^{\,\text{min}},{a}_{i}^{\text{max}\,}]$$ and boolean flag upi by inverting $$a=(1-\cos ({K}_{i}\theta ))/2$$7D2

$${\theta }_{l}=\frac{{\left\lfloor {K}_{i}{\theta }_{l}\right\rfloor }_{\text{mod}2\pi }+{\theta }_{i}^{\text{min}\,}}{{K}_{i}}$$

$${\theta }_{u}=\frac{{\left\lfloor {K}_{i}{\theta }_{u}\right\rfloor }_{\text{mod}2\pi }+{\theta }_{i}^{\text{max}\,}}{{K}_{i}}$$

$$[{a}_{l},{a}_{u}]=[{\sin }^{2}({\theta }_{l}),{\sin }^{2}({\theta }_{u})]$$

return [al, au]

### Algorithm 2

Procedure for finding ki+1

FunctionFindNextK (ki, θl, θu, upi, r = 2):

Ki = 4ki + 2 // currentθ-factor

$${\theta }_{i}^{\,\text{min}\,}={K}_{i}{\theta }_{l}$$ // lower bound for scaled θ

$${\theta }_{i}^{\,\text{max}\,}={K}_{i}{\theta }_{u}$$ // upper bound for scaledθ

$${K}_{\text{max}}=\left\lfloor \frac{\pi }{{\theta }_{u}-{\theta }_{l}}\right\rfloor$$ // set an upper bound forθ-factor

$$K={K}_{\text{max}}-{({K}_{\text{max}}-2)}_{\text{mod} \, 4}$$ // largest potential candidate of the form 4k + 2

whileK ≥ rKido

q = K/Ki // factor to scale$$[{\theta }_{i}^{\,\text{min}},{\theta }_{i}^{\text{max}\,}]$$

if$${\{q\cdot {\theta }_{i}^{\mathrm{max}}\}}_{{{\mathrm{mod}}\, \, 2\pi }}\le \pi \ \,{\mathbf{and}}\ {\{q\cdot {\theta }_{i}^{\mathrm{min}}\}}_{{{\mathrm{mod}}\,\, 2\pi }}\le \pi$$then

// $$[{\theta }_{i+1}^{\,\text{min}},{\theta }_{i+1}^{\text{max}\,}]$$ is in upper half-plane

Ki+1 = K

upi+1 = True

ki+1 = (Ki+1 − 2)/4

return (ki+1, upi+1)

if$${\{q\cdot {\theta }_{i}^{\mathrm{max}}\}}_{{{\mathrm{mod}}\,2\pi }}\ge \pi \,{\mathrm{and}}\,{\{q\cdot {\theta }_{i}^{\mathrm{min}}\}}_{{{\mathrm{mod}}\,2\pi }}\ge \pi$$then

// $$[{\theta }_{i+1}^{\,\text{min}},{\theta }_{i+1}^{\text{max}\,}]$$ is in lower half-plane

Ki+1 = K

upi+1 = False

ki+1 = (Ki+1 − 2)/4

return (ki+1, upi+1)

K = K − 4

return (ki, upi) // return old value

More intuitively, the heart of our algorithm—the sub-routine FINDNEXTK—allows us to maximize Fisher Information $${\mathcal{I}}$$ on a given iteration in a greedy fashion. The way to see this is to notice, that $${\mathcal{I}}$$ is proportional to NshotsK2, where K := 4k + 210.

Note that the maximum number of applications of $${\mathcal{Q}}$$ given in Theorem 1 is a loose upper bound since the proof uses Chernoff-Hoeffding bound to estimate sufficiently narrow intermediate confidence intervals in Algorithm 1. Using more accurate techniques instead, such as Clopper-Pearson’s confidence interval for Bernoulli distributions18, can lower the constant overhead in Noracle by a factor of 3 (see Supplementary Section 3) but is more complex to analyze analytically.

In Algorithm 2, we require that Ki+1/Ki≥ r = 2, otherwise we continue with Ki. The choice of the lower bound r is optimal in the proof, i.e., it gives us the lowest coefficient for the upper bound (see Supplementary Section 2). Moreover, the chosen lower bound was working very well in practice.

In Algorithm 1 we imposed the "no-overshooting" condition in order to ensure, that we do not make unnecessary measurement shots at the last iterations of the algorithm. This condition also allows us to keep constants small in the proof (see Supplementary Eq. (11)). It utilizes a quantity Lmax—the maximum possible error, which could be returned on a given iteration using Nshots measurements. It is calculated before the start of the algorithm for chosen ϵ, α and number of shots Nshots. It also depends on the type of chosen confidence interval. For Chernoff-Hoeffding one can write a direct analytical expression:

$${L}_{\text{max}}({N}_{\text{shots}},\epsilon ,\alpha )\!\!:\,=\arcsin {\left(\frac{2}{{N}_{\text{shots}}}\mathrm{log}\,\left(\frac{2T(\epsilon )}{\alpha }\right)\right)}^{1/4},$$
(9)

which is derived from Supplementary Eqs. (15) and (21). For Clopper-Pearson, one can only calculate it numerically:

$${L}_{\text{max}}({N}_{\text{shots}},\epsilon ,\alpha )\!\!:\,=\mathop{\max }\limits_{\theta }{h}_{{N}_{\text{shots}},\epsilon ,\alpha }(\theta ),$$
(10)

where function h is defined in Supplementary Section 3. It is derived by analogy with Supplementary Eq. (39), where instead of Nmax(ϵ, α) one should use Nshots.

Theorem 1 provides a bound on the query complexity, i.e., the total number of oracle calls with respect to the target accuracy. However, it is important to note that the computational complexity, i.e., the overall number of operations, including classical steps such as all applications of FINDNEXTK and computing the intermediate confidence intervals, scales in exactly the same way.

### Numerical experiments

Next, we empirically compare IQAE, MLAE, QAES, QAE, and classical MC with each other and determine the total number of oracle queries necessary to achieve a particular accuracy. We are only interested in measuring the last qubit of $${{\mathcal{Q}}}^{k}{\mathcal{A}}{\left|0\right\rangle }_{n}\left|0\right\rangle$$ for different powers k, and we know that $${\mathbb{P}}[\left|1\right\rangle ]={\sin }^{2}((2k+1){\theta }_{a})$$. Thus, for a given θa and k, we can consider a Bernoulli distribution with corresponding success probability or a single-qubit Ry-rotation with angle 2(2k + 1)θa to generate the required samples. All algorithms mentioned in this paper are implemented and tested using Qiskit21 in order to be run on simulators or real quantum hardware, e.g., as provided via the IBM Quantum Experience.

For IQAE and MC, we compute the (intermediate) confidence intervals based both on Chernoff-Hoeffding20 and on Clopper-Pearson18. For QAE and MLAE, we use the likelihood ratio17, see Supplementary Section 1. For QAES, we report the outputted accuracy of the algorithm.

To compare all algorithms we estimate a = 1/2 with a 1 − α = 95% confidence interval. For IQAE, MLAE, and QAE, we set Nshots = 100. As shown in Fig. 3, IQAE outperforms all other algorithms. QAES, even though achieving the best asymptotical behavior, performs worst in practice. On average, QAES requires about 108 times more oracle queries than IQAE which is even more than for classical MC simulation with the tested target accuracies. MLAE performs comparable to IQAE, however, the exact MLE becomes numerically challenging with increasing m. In order to observe the scaling of the quantum part of the algorithm, we collect more data points via the usage of a geometrically smaller search domain around estimated θ with each new round instead of brute force search on the whole initial domain for θ. Lastly, QAE with MLE-postprocessing performs a bit worse than IQAE and MLAE, which answers the question raised at the beginning: Applying QPE in the QAE setting does not lead to any advantage but only increases the complexity, even with an MLE-postprocessing. Thus, using IQAE instead does not only reduce the required number of qubits and gates, but it also improves the performance. Note that the MLE problem resulting from canonical QAE is significantly easier to solve than the problem arising in MLAE since the solution can be efficiently computed with a bisection search, see Supplementary Section 1. However, to evaluate QAE we need to simulate an increasing number of (ancilla) qubits, even for the very simple problem considered here, which makes the simulation of the quantum circuits more costly.

In the remainder of this section, we analyze the performance of IQAE in more detail. In particular, we empirically analyze the total number of oracle queries when using both Chernoff-Hoeffding the Clopper-Pearson confidence intervals, as well as the resulting k-schedules.

More precisely, we run IQAE for all a {i/100i = 0, …, 100} discretizing [0, 1], for all ϵ {10ii = 3, …, 6}, and for all α {1%, 5%, 10%}. We choose Nshots = 100 for all experiments. For each combination of parameters, we evaluate the resulting number of total oracle calls Noracle and compute

$$\frac{{N}_{\text{oracle}}}{\mathrm{log}\,(2/\alpha {\mathrm{log}\,}_{2}(\pi /4\epsilon ))/\epsilon },$$
(11)

i.e., the constant factor of the scaling with respect to ϵ and α. We evaluate the average, as well as the worst-case overall considered values for a. The results are illustrated in Figs. 4 and 5. The empirical complexity analysis of Chernoff-Hoeffding IQAE leads to:

$${N}_{\,\text{oracle}}^{\text{avg}\,}\le \frac{2}{\epsilon }\mathrm{log}\,\left(\frac{2}{\alpha }{\mathrm{log}\,}_{2}\left(\frac{\pi }{4\epsilon }\right)\right)\,\text{, and}\,$$
(12)
$${N}_{\,\text{oracle}}^{\text{wc}\,}\le \frac{6}{\epsilon }\mathrm{log}\,\left(\frac{2}{\alpha }{\mathrm{log}\,}_{2}\left(\frac{\pi }{4\epsilon }\right)\right),$$
(13)

where $${N}_{\,\text{oracle}}^{\text{avg}\,}$$ denotes the average and $${N}_{\,\text{oracle}}^{\text{wc}\,}$$ the worst-case complexity, respectively. Furthermore, the analysis of Clopper-Pearson IQAE leads to:

$${N}_{\,\text{oracle}}^{\text{avg}\,}\le \frac{0.8}{\epsilon }\mathrm{log}\,\left(\frac{2}{\alpha }{\mathrm{log}\,}_{2}\left(\frac{\pi }{4\epsilon }\right)\right)\,\text{, and}\,$$
(14)
$${N}_{\,\text{oracle}}^{\text{wc}\,}\le \frac{1.4}{\epsilon }\mathrm{log}\,\left(\frac{2}{\alpha }{\mathrm{log}\,}_{2}\left(\frac{\pi }{4\epsilon }\right)\right),$$
(15)

To analyze the k-schedule, we set a = 1/2, ϵ = 10−6, α = 5%, and again Nshots = 100. Figure 6 shows for each iteration the resulting average, standard deviation, minimum, and maximum of Ki+1/Ki, over 1000 repetitions of the algorithm, for the Ki defined in Algorithm 2. As explained before, we want to achieve as high a possible value of Ki+1/Ki for each iteration. Therefore, it can be seen that Nshots = 100 seems to be too small for the first round, i.e., another iteration with the same Ki is necessary before approaching an average growth rate slightly larger than four.

## Discussion

We introduced Iterative Quantum Amplitude Estimation, a new variant of QAE that realizes a quadratic speedup over classical MC simulation. Our algorithm does not require QPE, i.e., it is solely based on Grover iterations, and allows us to prove rigorous error and convergence bounds. We demonstrate empirically that our algorithm outperforms the other existing variants of QAE, some even by several orders of magnitude. This development is an important step towards applying QAE on quantum hardware to practically relevant problems and achieving a quantum advantage.

Our algorithm achieves the quadratic speedup up to a $$\mathrm{log}\,(2/\alpha {\mathrm{log}\,}_{2}(\pi /4\epsilon ))$$-factor. In contrast, QAES, the other known variant of QAE without QPE and with a rigorous convergence proof, achieves optimal asymptotic complexity at the cost of very large constants. It is an open question for future research whether there exists a variant of QAE without QPE that is practically competitive while having an asymptotically optimal performance bound. Another difference between IQAE and QAES is the type of error bound: IQAE provides an absolute and QAES a relative bound. Both types are relevant in practice, however, in the context of QAE, where problems often need to be normalized, a relative error bound is sometimes more appropriate. We leave the question of a relative error bound for IQAE open to future research.

Another research direction that seems of interest is the existence of parallel versions of QAE. More precisely, is it possible to realize the powers of the operator $${\mathcal{Q}}$$ distributed somehow in parallel over additional qubits, instead of sequential application on the quantum register? However, as shown in ref. 22, this does not seem to be possible.

Another open question for further investigation is the optimal choice of parameters for IQAE. We can set the required minimal growth rate for the oracle calls, as well as the number of classical shots per iteration, and both affect the performance of the algorithm. Determining the most efficient setting may further reduce the required number of oracle calls for particular target accuracy.

We also demonstrated that the gap between the bound on the total number of oracle calls provided in Theorem 1 and the actual performance is not too big. The proof technique for the upper bound almost achieves the actual performance. However, one may still ask whether an even tighter analytic bound is possible.

To summarize, we introduced and analyzed a new variant of QAE without QPE that outperforms the other known approaches. Moreover, we provide a rigorous convergence theory. This helps to reduce the requirements on quantum hardware and is an important step towards leveraging quantum computing for real-world applications.