Parallelized Amplitude Estimation (PAE)

• PAE is a quantum amplitude estimation technique that gathers oscillatory data from independent circuits at varying Grover depths, enabling parallel execution.
• It utilizes classical post-processing such as maximum likelihood, ESPRIT, and Bayesian inference to fuse results from modular subproblems.
• PAE architectures offer trade-offs among depth, width, and classical computation, achieving near-Heisenberg query scaling and enhanced noise resilience.

Searching arXiv for recent and foundational papers on parallelized amplitude estimation. arxiv_search(query="parallelized amplitude estimation quantum", max_results=10) arxiv_search(query="Parallelized Amplitude Estimation", max_results=10) Parallelized amplitude estimation (PAE) is a family of quantum amplitude-estimation methods in which information about an unknown amplitude is gathered from multiple independent quantum circuits—typically at different Grover depths, phase-estimation windows, or low-depth signal-processing schedules—and then fused by classical post-processing. In contrast to canonical quantum amplitude estimation based on quantum phase estimation (QPE), which uses controlled powers of the Grover operator and an inverse quantum Fourier transform in a single long coherent circuit, PAE reorganizes the task into batchable or modular subproblems that are amenable to parallel execution across shots, circuit families, phase windows, or distributed branches (Suzuki et al., 2019, Braun et al., 2022, Labib et al., 2024, Oshio et al., 8 Aug 2025). The literature includes nonadaptive multi-depth maximum-likelihood methods, explicit parallel QPE/QAE constructions, signal-processing-based noniterative schemes, modular windowed variants, and low-depth batchable algorithms based on QSP, Bayesian inference, or statistical eigengap estimation (Tanaka et al., 2020, Rall et al., 2022, Shukla et al., 7 Aug 2025, Ramôa et al., 2024, Huang et al., 5 Mar 2026).

1. Formal setting and signal model

The underlying estimation problem is the standard amplitude-estimation task. A unitary $\mathcal A$ prepares a state of the form

$$\ket{\Psi}=\mathcal A\ket{0}_{n+1} =\sqrt{p}\,\ket{\Psi_1}\ket{1}+\sqrt{1-p}\,\ket{\Psi_0}\ket{0},$$

where the last qubit indicates whether a basis state is “good” or “bad,” and the target quantity is the success probability $p$ or, equivalently, the good-state amplitude $\sqrt p$. The conventional angular parameterization is

$$\theta=\arcsin\sqrt p,\qquad p=\sin^2\theta,\qquad \theta\in[0,\pi/2].$$

With Grover operator

$$\mathcal Q=-\mathcal A\,\mathcal S_0\,\mathcal A^{-1}\,\mathcal S_\chi,$$

one has the usual amplitude-amplification rotation

$$\mathcal Q^m\ket{\Psi} =\sin((2m+1)\theta)\ket{\Psi_1}\ket{1} +\cos((2m+1)\theta)\ket{\Psi_0}\ket{0},$$

so that the probability of measuring “good” after depth $m$ is $\sin^2((2m+1)\theta)$ (Zhao et al., 2022).

This sinusoidal dependence on Grover depth is the common signal model exploited across PAE variants. Some works parameterize the unknown directly as a success probability, while others parameterize the amplitude itself. In particular, the signal-processing-based csAE formulation writes

$$U\lvert 0^l\rangle=\cos\theta\,\lvert x,0\rangle+\sin\theta\,\lvert x',1\rangle$$

and sets $$\ket{\Psi}=\mathcal A\ket{0}_{n+1} =\sqrt{p}\,\ket{\Psi_1}\ket{1}+\sqrt{1-p}\,\ket{\Psi_0}\ket{0},$$0, so that the depth-$$\ket{\Psi}=\mathcal A\ket{0}_{n+1} =\sqrt{p}\,\ket{\Psi_1}\ket{1}+\sqrt{1-p}\,\ket{\Psi_0}\ket{0},$$1 success probability is $$\ket{\Psi}=\mathcal A\ket{0}_{n+1} =\sqrt{p}\,\ket{\Psi_1}\ket{1}+\sqrt{1-p}\,\ket{\Psi_0}\ket{0},$$2 rather than $$\ket{\Psi}=\mathcal A\ket{0}_{n+1} =\sqrt{p}\,\ket{\Psi_1}\ket{1}+\sqrt{1-p}\,\ket{\Psi_0}\ket{0},$$3 (Labib et al., 2024). Despite such notational differences, the core estimation object remains the same: an angle or phase governing oscillatory statistics across amplified experiments.

From the standpoint of architecture, the central departure from canonical QAE is that the oscillatory information is extracted from many independent executions rather than from one monolithic phase-estimation circuit. In PAE-like schemes, the computational burden shifts from long coherent control to classical inference over measurements gathered at multiple depths, windows, or branches.

2. Emergence from non-QPE multi-depth estimation

A major precursor to PAE is the multi-depth maximum-likelihood approach of Suzuki et al., which estimates an unknown amplitude by running a family of circuits $$\ket{\Psi}=\mathcal A\ket{0}_{n+1} =\sqrt{p}\,\ket{\Psi_1}\ket{1}+\sqrt{1-p}\,\ket{\Psi_0}\ket{0},$$4 at predetermined depths $$\ket{\Psi}=\mathcal A\ket{0}_{n+1} =\sqrt{p}\,\ket{\Psi_1}\ket{1}+\sqrt{1-p}\,\ket{\Psi_0}\ket{0},$$5, collecting counts $$\ket{\Psi}=\mathcal A\ket{0}_{n+1} =\sqrt{p}\,\ket{\Psi_1}\ket{1}+\sqrt{1-p}\,\ket{\Psi_0}\ket{0},$$6, and maximizing the joint likelihood over $$\ket{\Psi}=\mathcal A\ket{0}_{n+1} =\sqrt{p}\,\ket{\Psi_1}\ket{1}+\sqrt{1-p}\,\ket{\Psi_0}\ket{0},$$7 (Suzuki et al., 2019). The depth schedules studied explicitly are the linearly incremental sequence

$$\ket{\Psi}=\mathcal A\ket{0}_{n+1} =\sqrt{p}\,\ket{\Psi_1}\ket{1}+\sqrt{1-p}\,\ket{\Psi_0}\ket{0},$$8

and the exponentially incremental sequence

$$\ket{\Psi}=\mathcal A\ket{0}_{n+1} =\sqrt{p}\,\ket{\Psi_1}\ket{1}+\sqrt{1-p}\,\ket{\Psi_0}\ket{0},$$9

Because these schedules are fixed in advance, every depth circuit can be executed independently. The paper explicitly contrasts this with adaptive simplified-counting approaches and states that the method “can be run in parallel on multiple quantum devices” (Suzuki et al., 2019).

The statistical mechanism is classical maximum likelihood. If depth $p$0 is repeated $p$1 times and produces $p$2 good outcomes, then the likelihood contribution is

$p$3

and the estimator maximizes the product over depths. The asymptotic analysis distinguishes two regimes: LIS yields $p$4, whereas EIS yields $p$5, matching Heisenberg-like scaling in total query count (Suzuki et al., 2019). This fixed-schedule, multi-depth, classically fused architecture is one of the clearest antecedents of later PAE terminology.

The noise-aware extension formulates the same multi-depth architecture as a two-parameter estimation problem for amplitude and noise. Under a depolarizing-noise model with parameter $p$6, the success probability becomes

$p$7

and the estimator jointly maximizes the corresponding likelihood over $p$8 (Tanaka et al., 2020). This work shows that noisy PAE can retain quantum query advantage for a range of budgets, but also that the estimation error saturates and that the Fisher information matrix can become nearly singular for anomalous target amplitudes such as $p$9 (Tanaka et al., 2020).

3. Principal parallelized architectures

The literature now contains several distinct architectural interpretations of PAE. Some parallelize the phase-kickback mechanism itself, some parallelize predetermined Grover-depth experiments, and some modularize QPE into independently executable windows.

Approach	Parallel element	Characteristic claim
Parallel QPE/QAE (Braun et al., 2022)	Parallel kickbacks, duplicated work registers, entangled controls, or reinitialization	Parallelization can reduce depth and improve resilience for specific error models
csAE (Labib et al., 2024)	Predetermined Grover-depth set $\sqrt p$0 with independent execution and ESPRIT post-processing	Reported $\sqrt p$1 sequential and $\sqrt p$2 parallel query complexity at $\sqrt p$3 confidence
AWQAE (Shukla et al., 7 Aug 2025)	Independent phase windows/chunks with blockwise QPE-like circuits	Precision $\sqrt p$4 with per-block coherent qubit cost $\sqrt p$5
GHZ/QSP PAE (Oshio et al., 8 Aug 2025)	Global GHZ state followed by separated low-depth branches	$\sqrt p$6 and depth $\sqrt p$7

The explicit parallel QPE/QAE construction of Imhof et al. reorganizes amplitude estimation by parallelizing the phase-kickback generation that underlies QPE. The work describes three styles: simple parallelization with many copies of the work register, indirect parallelization using entangled control qubits so that many single kickbacks reproduce higher powers, and reinitialization-based recycling when width is limited (Braun et al., 2022). For amplitude estimation, the key caveat is that each branch must be initialized in a single Grover eigenstate rather than in the usual superposition used by canonical QAE.

The csAE method recasts amplitude estimation as a classical direction-of-arrival problem. The amplified state

$\sqrt p$8

induces the signal

$\sqrt p$9

so $$\theta=\arcsin\sqrt p,\qquad p=\sin^2\theta,\qquad \theta\in[0,\pi/2].$$0 appears as a frequency that can be extracted by ESPRIT from measurement data obtained at sparse, predetermined depths (Labib et al., 2024). The method is explicitly nonadaptive, phase-estimation-free, and designed so that all depth experiments can be run independently in parallel.

AWQAE preserves the QPE/QAE viewpoint but partitions the phase register into windows of sizes $$\theta=\arcsin\sqrt p,\qquad p=\sin^2\theta,\qquad \theta\in[0,\pi/2].$$1 and reconstructs the full phase classically from small QPE-like blocks (Shukla et al., 7 Aug 2025). Each block uses a 2-qubit phase-resolution register, an $$\theta=\arcsin\sqrt p,\qquad p=\sin^2\theta,\qquad \theta\in[0,\pi/2].$$2-qubit counting register, the target register, and one ancilla. The architecture is described as modular, scalable, and adaptive, with blocks “independent” and “amenable to parallel processing,” although the final estimate still depends on an ambiguity-aware LSB-to-MSB correction routine (Shukla et al., 7 Aug 2025).

The most explicit recent use of the term PAE is the GHZ/QSP framework of Yang et al. It prepares a $$\theta=\arcsin\sqrt p,\qquad p=\sin^2\theta,\qquad \theta\in[0,\pi/2].$$3-qubit GHZ state,

$$\theta=\arcsin\sqrt p,\qquad p=\sin^2\theta,\qquad \theta\in[0,\pi/2].$$4

then applies separated QSP-constructed phase-shift circuits $$\theta=\arcsin\sqrt p,\qquad p=\sin^2\theta,\qquad \theta\in[0,\pi/2].$$5 on each branch, so that the effective signal amplification is proportional to $$\theta=\arcsin\sqrt p,\qquad p=\sin^2\theta,\qquad \theta\in[0,\pi/2].$$6 (Oshio et al., 8 Aug 2025). This yields a tunable depth-width-query tradeoff and, in the fully parallel regime $$\theta=\arcsin\sqrt p,\qquad p=\sin^2\theta,\qquad \theta\in[0,\pi/2].$$7, a logarithmic-depth algorithm with near-Heisenberg-limited total query scaling.

4. Statistical post-processing paradigms

Although the quantum front ends differ, PAE methods are equally distinguished by their classical inference layers. Maximum-likelihood estimation is the foundational paradigm: all observed counts from different depths are combined into a single one-parameter or two-parameter likelihood and optimized globally (Suzuki et al., 2019, Tanaka et al., 2020). In this formulation, the nonuniqueness of $$\theta=\arcsin\sqrt p,\qquad p=\sin^2\theta,\qquad \theta\in[0,\pi/2].$$8 at a single depth is resolved by cross-consistency across several depths.

QSP-based amplitude-estimation frameworks replace Grover-only oscillations by designed polynomial response functions. The sampling theorem of the QSP approach states that one can sample a bit $$\theta=\arcsin\sqrt p,\qquad p=\sin^2\theta,\qquad \theta\in[0,\pi/2].$$9 with

$$\mathcal Q=-\mathcal A\,\mathcal S_0\,\mathcal A^{-1}\,\mathcal S_\chi,$$0

for QSP-implementable polynomials $$\mathcal Q=-\mathcal A\,\mathcal S_0\,\mathcal A^{-1}\,\mathcal S_\chi,$$1, using $$\mathcal Q=-\mathcal A\,\mathcal S_0\,\mathcal A^{-1}\,\mathcal S_\chi,$$2 oracle queries (Rall et al., 2022). Its hybrid theorem makes the depth-for-repetitions tradeoff explicit: $$\mathcal Q=-\mathcal A\,\mathcal S_0\,\mathcal A^{-1}\,\mathcal S_\chi,$$3 where $$\mathcal Q=-\mathcal A\,\mathcal S_0\,\mathcal A^{-1}\,\mathcal S_\chi,$$4 is total degree and $$\mathcal Q=-\mathcal A\,\mathcal S_0\,\mathcal A^{-1}\,\mathcal S_\chi,$$5 is maximum degree used in any one circuit (Rall et al., 2022). The framework is adaptive across batches, but all repetitions within a batch are independent and parallelizable.

Bayesian amplitude estimation supplies a different inference backbone. BAE represents the posterior over $$\mathcal Q=-\mathcal A\,\mathcal S_0\,\mathcal A^{-1}\,\mathcal S_\chi,$$6 by sequential Monte Carlo particles, chooses the next Grover depth by expected utility, and updates after batches of repeated shots (Ramôa et al., 2024). The paper emphasizes that reweighting and resampling are parallelizable over particles, that utility evaluations are parallelizable over candidate controls, and that $$\mathcal Q=-\mathcal A\,\mathcal S_0\,\mathcal A^{-1}\,\mathcal S_\chi,$$7 shots at a chosen depth can be run in parallel on multiple devices (Ramôa et al., 2024). The annealed variant replaces variance-minimizing utility with an ESS-based design criterion, but the outer loop remains adaptive.

A further statistical reformulation treats amplitude estimation as eigengap estimation of an effective Hamiltonian generated by Grover dynamics. In this picture,

$$\mathcal Q=-\mathcal A\,\mathcal S_0\,\mathcal A^{-1}\,\mathcal S_\chi,$$8

and the estimation target is extracted from expectation values such as $$\mathcal Q=-\mathcal A\,\mathcal S_0\,\mathcal A^{-1}\,\mathcal S_\chi,$$9 under nonadaptive Gaussian schedules (Huang et al., 5 Mar 2026). GLSAE minimizes a least-squares objective

$$\mathcal Q^m\ket{\Psi} =\sin((2m+1)\theta)\ket{\Psi_1}\ket{1} +\cos((2m+1)\theta)\ket{\Psi_0}\ket{0},$$0

while GDMAE, in the flag-qubit setting, maximizes

$$\mathcal Q^m\ket{\Psi} =\sin((2m+1)\theta)\ket{\Psi_1}\ket{1} +\cos((2m+1)\theta)\ket{\Psi_0}\ket{0},$$1

Both algorithms are noniterative and highly parallelizable across sampled schedule points (Huang et al., 5 Mar 2026).

5. Resource tradeoffs, performance, and noise

PAE does not remove the fundamental query cost of amplitude estimation; rather, it redistributes that cost across depth, width, and classical computation. This is explicit in the GHZ/QSP PAE theorem,

$$\mathcal Q^m\ket{\Psi} =\sin((2m+1)\theta)\ket{\Psi_1}\ket{1} +\cos((2m+1)\theta)\ket{\Psi_0}\ket{0},$$2

which reduces to

$$\mathcal Q^m\ket{\Psi} =\sin((2m+1)\theta)\ket{\Psi_1}\ket{1} +\cos((2m+1)\theta)\ket{\Psi_0}\ket{0},$$3

for $$\mathcal Q^m\ket{\Psi} =\sin((2m+1)\theta)\ket{\Psi_1}\ket{1} +\cos((2m+1)\theta)\ket{\Psi_0}\ket{0},$$4 (Oshio et al., 8 Aug 2025). Here the