ANASTAARS: Noise-Aware Quantum Optimizer

• ANASTAARS is a noise-aware optimizer that employs adaptive random subspace methods to efficiently handle the high-dimensional parameter spaces in variational quantum algorithms like QAOA.
• The algorithm reduces shot noise effects by reusing previous measurements and constructing surrogate models in low-dimensional affine spaces, thereby improving shot efficiency.
• Adaptive subspace enlargement after unsuccessful iterations contributes to robust and scalable optimization by dynamically balancing model complexity and measurement cost.

Searching arXiv for the ANASTAARS paper and closely related acronyms to ground the article in current arXiv records. ANASTAARS, expanded as A Noise-Aware Stochastic Trust-region Algorithm using Adaptive Random Subspaces, is a derivative-free classical optimizer for variational quantum algorithms, especially the quantum approximate optimization algorithm (QAOA), in regimes where the ansatz depth is large and the parameter dimension $d=2p$ becomes difficult to manage. It is designed for shot-noisy objective functions and addresses the simultaneous problems of scaling, stochastic measurement error, and the cost of repeated circuit evaluations by combining stochastic trust-region methodology, adaptive random subspace models, selective reuse of previously acquired measurements, and an explicitly noise-aware acceptance test (Dzahini et al., 15 Jul 2025).

1. Definition and optimization setting

ANASTAARS is formulated for the QAOA setting in which the $p$-layer state is

$$\bm{\psi}(\bm{\gamma},\bm{\beta}) = e^{-i\beta_p\bm{H}_M}e^{-i\gamma_p\bm{H}_P}\cdots e^{-i\beta_1\bm{H}_M}e^{-i\gamma_1\bm{H}_P}\bm{\psi}_0,$$

with parameter vectors

$$\bm{\gamma}=(\gamma_1,\dots,\gamma_p)^\top,\qquad \bm{\beta}=(\beta_1,\dots,\beta_p)^\top.$$

The associated classical optimization problem is written as minimizing the expected problem-Hamiltonian energy,

$$\underset{\bm{\gamma},\bm{\beta}}{\min}\; \bm{\psi}(\bm{\gamma},\bm{\beta})^\top \bm{H}_P \bm{\psi}(\bm{\gamma},\bm{\beta}),$$

or, after defining

$$x=(\gamma_1,\dots,\gamma_p,\beta_1,\dots,\beta_p)^\top \in \mathbb{R}^d,\qquad d=2p,$$

as the stochastic program

$$\underset{x\in\mathbb{R}^d}{\min}\; f(x), \qquad f(x)=\mathbb{E}[f_\theta(x)].$$

Here $f_\theta(x)$ is a noisy shot-based realization of the cost function (Dzahini et al., 15 Jul 2025).

The optimizer is motivated by a specific failure mode of high-depth QAOA workflows. As $p$ grows, the dimension $d=2p$ grows correspondingly; full-space model-based derivative-free optimization becomes expensive because model construction scales polynomially with $p$0; gradient-based schemes are often shot-expensive because gradients must be estimated; and deterministic trust-region methods can stagnate in the presence of shot noise. ANASTAARS is intended to mitigate all of these effects in a single framework. This suggests a design objective that is not merely asymptotic dimensionality reduction, but practical shot-efficiency under NISQ-era measurement constraints (Dzahini et al., 15 Jul 2025).

2. Core algorithmic principle: adaptive random subspace trust regions

The defining mechanism of ANASTAARS is optimization in low-dimensional random affine subspaces rather than in the full parameter space. At iteration $p$1, with incumbent $p$2, the method chooses a matrix

$p$3

whose columns span a $p$4-dimensional random subspace. The corresponding affine search space is

$p$5

A local model $p$6 approximates the objective restricted to this affine space,

$p$7

and, for quadratic models, takes the form

$p$8

The trust region is defined in subspace coordinates by

$p$9

and the trial step satisfies

$$\bm{\psi}(\bm{\gamma},\bm{\beta}) = e^{-i\beta_p\bm{H}_M}e^{-i\gamma_p\bm{H}_P}\cdots e^{-i\beta_1\bm{H}_M}e^{-i\gamma_1\bm{H}_P}\bm{\psi}_0,$$0

This structure replaces a full-space local model with a model in dimension $$\bm{\psi}(\bm{\gamma},\bm{\beta}) = e^{-i\beta_p\bm{H}_M}e^{-i\gamma_p\bm{H}_P}\cdots e^{-i\beta_1\bm{H}_M}e^{-i\gamma_1\bm{H}_P}\bm{\psi}_0,$$1 whenever possible (Dzahini et al., 15 Jul 2025).

The method’s distinctive novelty is not only the use of random subspaces, but the adaptive enlargement of those subspaces after unsuccessful iterations. If the previous iteration was successful, or if further enlargement would exceed $$\bm{\psi}(\bm{\gamma},\bm{\beta}) = e^{-i\beta_p\bm{H}_M}e^{-i\gamma_p\bm{H}_P}\cdots e^{-i\beta_1\bm{H}_M}e^{-i\gamma_1\bm{H}_P}\bm{\psi}_0,$$2, the algorithm resets to $$\bm{\psi}(\bm{\gamma},\bm{\beta}) = e^{-i\beta_p\bm{H}_M}e^{-i\gamma_p\bm{H}_P}\cdots e^{-i\beta_1\bm{H}_M}e^{-i\gamma_1\bm{H}_P}\bm{\psi}_0,$$3, draws a fresh random subspace, and builds a new $$\bm{\psi}(\bm{\gamma},\bm{\beta}) = e^{-i\beta_p\bm{H}_M}e^{-i\gamma_p\bm{H}_P}\cdots e^{-i\beta_1\bm{H}_M}e^{-i\gamma_1\bm{H}_P}\bm{\psi}_0,$$4-dimensional model. If the previous iteration was unsuccessful, it augments the previous subspace by one orthogonal direction instead of discarding it. If the previous basis is represented by $$\bm{\psi}(\bm{\gamma},\bm{\beta}) = e^{-i\beta_p\bm{H}_M}e^{-i\gamma_p\bm{H}_P}\cdots e^{-i\beta_1\bm{H}_M}e^{-i\gamma_1\bm{H}_P}\bm{\psi}_0,$$5, the new direction is sampled as

$$\bm{\psi}(\bm{\gamma},\bm{\beta}) = e^{-i\beta_p\bm{H}_M}e^{-i\gamma_p\bm{H}_P}\cdots e^{-i\beta_1\bm{H}_M}e^{-i\gamma_1\bm{H}_P}\bm{\psi}_0,$$6

and the augmented subspace is defined by

$$\bm{\psi}(\bm{\gamma},\bm{\beta}) = e^{-i\beta_p\bm{H}_M}e^{-i\gamma_p\bm{H}_P}\cdots e^{-i\beta_1\bm{H}_M}e^{-i\gamma_1\bm{H}_P}\bm{\psi}_0,$$7

Operationally, this means failure triggers a richer local model in a strictly larger subspace rather than repeated optimization within the same inadequate slice of parameter space (Dzahini et al., 15 Jul 2025).

3. Random embeddings, interpolation models, and sample reuse

The probabilistic justification for ANASTAARS rests on the requirement that the random subspace retain enough alignment with a useful descent direction. The desired condition is

$$\bm{\psi}(\bm{\gamma},\bm{\beta}) = e^{-i\beta_p\bm{H}_M}e^{-i\gamma_p\bm{H}_P}\cdots e^{-i\beta_1\bm{H}_M}e^{-i\gamma_1\bm{H}_P}\bm{\psi}_0,$$8

The paper uses the concept of a $$\bm{\psi}(\bm{\gamma},\bm{\beta}) = e^{-i\beta_p\bm{H}_M}e^{-i\gamma_p\bm{H}_P}\cdots e^{-i\beta_1\bm{H}_M}e^{-i\gamma_1\bm{H}_P}\bm{\psi}_0,$$9 random matrix sequence satisfying

$$\bm{\gamma}=(\gamma_1,\dots,\gamma_p)^\top,\qquad \bm{\beta}=(\beta_1,\dots,\beta_p)^\top.$$0

This is supported through Johnson–Lindenstrauss transforms. Specifically, for suitable $$\bm{\gamma}=(\gamma_1,\dots,\gamma_p)^\top,\qquad \bm{\beta}=(\beta_1,\dots,\beta_p)^\top.$$1,

$$\bm{\gamma}=(\gamma_1,\dots,\gamma_p)^\top,\qquad \bm{\beta}=(\beta_1,\dots,\beta_p)^\top.$$2

and a Haar-based construction is used: if $$\bm{\gamma}=(\gamma_1,\dots,\gamma_p)^\top,\qquad \bm{\beta}=(\beta_1,\dots,\beta_p)^\top.$$3 is Haar distributed and $$\bm{\gamma}=(\gamma_1,\dots,\gamma_p)^\top,\qquad \bm{\beta}=(\beta_1,\dots,\beta_p)^\top.$$4 is its first $$\bm{\gamma}=(\gamma_1,\dots,\gamma_p)^\top,\qquad \bm{\beta}=(\beta_1,\dots,\beta_p)^\top.$$5 rows, then

$$\bm{\gamma}=(\gamma_1,\dots,\gamma_p)^\top,\qquad \bm{\beta}=(\beta_1,\dots,\beta_p)^\top.$$6

has the desired JL-style property when

$$\bm{\gamma}=(\gamma_1,\dots,\gamma_p)^\top,\qquad \bm{\beta}=(\beta_1,\dots,\beta_p)^\top.$$7

for an absolute constant $$\bm{\gamma}=(\gamma_1,\dots,\gamma_p)^\top,\qquad \bm{\beta}=(\beta_1,\dots,\beta_p)^\top.$$8 (Dzahini et al., 15 Jul 2025).

Within each subspace, ANASTAARS builds interpolation or surrogate models. The paper presents three model families.

For linear subspace interpolation, given a poised set

$$\bm{\gamma}=(\gamma_1,\dots,\gamma_p)^\top,\qquad \bm{\beta}=(\beta_1,\dots,\beta_p)^\top.$$9

the shot-averaged estimates are

$$\underset{\bm{\gamma},\bm{\beta}}{\min}\; \bm{\psi}(\bm{\gamma},\bm{\beta})^\top \bm{H}_P \bm{\psi}(\bm{\gamma},\bm{\beta}),$$0

and the linear model is

$$\underset{\bm{\gamma},\bm{\beta}}{\min}\; \bm{\psi}(\bm{\gamma},\bm{\beta})^\top \bm{H}_P \bm{\psi}(\bm{\gamma},\bm{\beta}),$$1

The coefficient vector is obtained through the interpolation system

$$\underset{\bm{\gamma},\bm{\beta}}{\min}\; \bm{\psi}(\bm{\gamma},\bm{\beta})^\top \bm{H}_P \bm{\psi}(\bm{\gamma},\bm{\beta}),$$2

After a failed iteration, the old interpolation points are embedded in the expanded subspace so that old physical points in $$\underset{\bm{\gamma},\bm{\beta}}{\min}\; \bm{\psi}(\bm{\gamma},\bm{\beta})^\top \bm{H}_P \bm{\psi}(\bm{\gamma},\bm{\beta}),$$3 are preserved: $$\underset{\bm{\gamma},\bm{\beta}}{\min}\; \bm{\psi}(\bm{\gamma},\bm{\beta})^\top \bm{H}_P \bm{\psi}(\bm{\gamma},\bm{\beta}),$$4 The new set is constructed as

$$\underset{\bm{\gamma},\bm{\beta}}{\min}\; \bm{\psi}(\bm{\gamma},\bm{\beta})^\top \bm{H}_P \bm{\psi}(\bm{\gamma},\bm{\beta}),$$5

The resulting closed form shows that only one new point is needed to move from dimension $$\underset{\bm{\gamma},\bm{\beta}}{\min}\; \bm{\psi}(\bm{\gamma},\bm{\beta})^\top \bm{H}_P \bm{\psi}(\bm{\gamma},\bm{\beta}),$$6 to $$\underset{\bm{\gamma},\bm{\beta}}{\min}\; \bm{\psi}(\bm{\gamma},\bm{\beta})^\top \bm{H}_P \bm{\psi}(\bm{\gamma},\bm{\beta}),$$7: $$\underset{\bm{\gamma},\bm{\beta}}{\min}\; \bm{\psi}(\bm{\gamma},\bm{\beta})^\top \bm{H}_P \bm{\psi}(\bm{\gamma},\bm{\beta}),$$8 This measurement reuse is one of the method’s central practical contributions (Dzahini et al., 15 Jul 2025).

For minimum-Frobenius-norm quadratic models, which are the main nonlinear models used in experiments, the paper defines $$\underset{\bm{\gamma},\bm{\beta}}{\min}\; \bm{\psi}(\bm{\gamma},\bm{\beta})^\top \bm{H}_P \bm{\psi}(\bm{\gamma},\bm{\beta}),$$9 and the polynomial bases

$$x=(\gamma_1,\dots,\gamma_p,\beta_1,\dots,\beta_p)^\top \in \mathbb{R}^d,\qquad d=2p,$$0

The model is

$$x=(\gamma_1,\dots,\gamma_p,\beta_1,\dots,\beta_p)^\top \in \mathbb{R}^d,\qquad d=2p,$$1

where $$x=(\gamma_1,\dots,\gamma_p,\beta_1,\dots,\beta_p)^\top \in \mathbb{R}^d,\qquad d=2p,$$2 solve

$$x=(\gamma_1,\dots,\gamma_p,\beta_1,\dots,\beta_p)^\top \in \mathbb{R}^d,\qquad d=2p,$$3

subject to interpolation constraints. After a failed iteration and dimension increase, the $$x=(\gamma_1,\dots,\gamma_p,\beta_1,\dots,\beta_p)^\top \in \mathbb{R}^d,\qquad d=2p,$$4-dimensional MFN model is formed analogously, again adding only one new function value while reusing the previous interpolation data (Dzahini et al., 15 Jul 2025).

The paper also presents a diagonal-Hessian quadratic model with $$x=(\gamma_1,\dots,\gamma_p,\beta_1,\dots,\beta_p)^\top \in \mathbb{R}^d,\qquad d=2p,$$5 parameters, based on the set

$$x=(\gamma_1,\dots,\gamma_p,\beta_1,\dots,\beta_p)^\top \in \mathbb{R}^d,\qquad d=2p,$$6

and basis

$$x=(\gamma_1,\dots,\gamma_p,\beta_1,\dots,\beta_p)^\top \in \mathbb{R}^d,\qquad d=2p,$$7

This cheaper construction yields a quadratic model with diagonal Hessian (Dzahini et al., 15 Jul 2025).

4. Noise awareness and trust-region acceptance

ANASTAARS is explicitly noise-aware. The objective is available only through shot-based realizations $$x=(\gamma_1,\dots,\gamma_p,\beta_1,\dots,\beta_p)^\top \in \mathbb{R}^d,\qquad d=2p,$$8, and the function estimates at the current and trial points are written as

$$x=(\gamma_1,\dots,\gamma_p,\beta_1,\dots,\beta_p)^\top \in \mathbb{R}^d,\qquad d=2p,$$9

These are required to be $$\underset{x\in\mathbb{R}^d}{\min}\; f(x), \qquad f(x)=\mathbb{E}[f_\theta(x)].$$0-accurate in the stochastic trust-region sense,

$$\underset{x\in\mathbb{R}^d}{\min}\; f(x), \qquad f(x)=\mathbb{E}[f_\theta(x)].$$1

though the paper emphasizes that such bounds hold only probabilistically because $$\underset{x\in\mathbb{R}^d}{\min}\; f(x), \qquad f(x)=\mathbb{E}[f_\theta(x)].$$2 is not directly observed (Dzahini et al., 15 Jul 2025).

The local noise magnitude is estimated from repeated samples at the current point using the sample standard deviation

$$\underset{x\in\mathbb{R}^d}{\min}\; f(x), \qquad f(x)=\mathbb{E}[f_\theta(x)].$$3

The algorithm explicitly allows reuse of samples from previous iterations when estimating this noise level. Rather than using the standard ratio

$$\underset{x\in\mathbb{R}^d}{\min}\; f(x), \qquad f(x)=\mathbb{E}[f_\theta(x)].$$4

ANASTAARS employs the modified, noise-aware ratio

$$\underset{x\in\mathbb{R}^d}{\min}\; f(x), \qquad f(x)=\mathbb{E}[f_\theta(x)].$$5

The acceptance rule requires both

$$\underset{x\in\mathbb{R}^d}{\min}\; f(x), \qquad f(x)=\mathbb{E}[f_\theta(x)].$$6

If the step is accepted,

$$\underset{x\in\mathbb{R}^d}{\min}\; f(x), \qquad f(x)=\mathbb{E}[f_\theta(x)].$$7

otherwise,

$$\underset{x\in\mathbb{R}^d}{\min}\; f(x), \qquad f(x)=\mathbb{E}[f_\theta(x)].$$8

and a failure flag is set so that the next iteration attempts a larger subspace (Dzahini et al., 15 Jul 2025).

This coupling of trust-region logic to an estimated noise magnitude is the method’s main robustness mechanism against shot noise. A plausible implication is that the method is designed to avoid two opposite pathologies: accepting illusory improvements caused by stochastic fluctuation, and rejecting genuine progress because a noisy estimate makes the apparent decrease too small.

5. Scalability, shot efficiency, and computational rationale

ANASTAARS is presented as a response to the cost of building full-space stochastic interpolation models in large parameter dimensions. The paper states that, in a simple noise framework, full-space stochastic model construction may require at least

$$\underset{x\in\mathbb{R}^d}{\min}\; f(x), \qquad f(x)=\mathbb{E}[f_\theta(x)].$$9

function evaluations, whereas a $f_\theta(x)$0-dimensional subspace construction replaces $f_\theta(x)$1 by $f_\theta(x)$2 (Dzahini et al., 15 Jul 2025).

The practical shot-saving mechanisms are described explicitly. First, the algorithm works in dimension $f_\theta(x)$3. Second, the subspace model has far fewer parameters than a full-space model. Third, after a failed iteration it does not discard previous interpolation data. Fourth, in the linear and MFN constructions, increasing the model dimension from $f_\theta(x)$4 to $f_\theta(x)$5 can require only one new function evaluation. Since every function value is itself obtained through repeated circuit executions, these design choices directly target shot cost rather than only floating-point cost (Dzahini et al., 15 Jul 2025).

The paper situates ANASTAARS relative to three optimizer families. It lies between classical model-based derivative-free trust-region methods such as BOBYQA and NEWUOA, stochastic or noise-aware derivative-free methods such as ANATRA, and random subspace methods such as STARS. Its distinguishing combination is: stochastic trust-region methodology, adaptive random subspace models, sample reuse across iterations, and explicit noise awareness (Dzahini et al., 15 Jul 2025).

The algorithmic pseudocode reflects this hybrid position. The required parameters are

$f_\theta(x)$6

and the method cycles through subspace construction, approximate trust-region minimization, estimation of $f_\theta(x)$7, $f_\theta(x)$8, and $f_\theta(x)$9, then success or failure updates. The paper does not state a separate formal stopping rule in the excerpted pseudocode, and it does not present a new standalone convergence theorem for ANASTAARS itself. Its theoretical support is instead inherited primarily from the STARS/random-subspace framework and the JL-style probabilistic embedding guarantees (Dzahini et al., 15 Jul 2025).

6. Numerical experiments, benchmark behavior, and limitations

The experimental variant studied is ANASTAARS-QD2, which uses MFN quadratic subspace models, adaptive dimensions $p$0, $p$1, and $p$2. The parameters used in the experiments are

$p$3

The test problems are MaxCut instances on a toy graph with MaxCut value $p$4 and on the Chvátal graph with MaxCut value $p$5 (Dzahini et al., 15 Jul 2025).

The initial experiments use $p$6 QAOA layers, so $p$7, on the Qiskit QASM simulator with per-evaluation shot counts

$p$8

and results are reported over 30 trials. Baselines include STARS-QD2, PyBOBYQA, ImFil, NOMAD, ANATRA, and NEWUOA; in the larger-scale experiments, the reported comparison is mainly against ANASTAARS-QD2, NEWUOA, and NOMAD (Dzahini et al., 15 Jul 2025).

For $p$9 ($d=2p$0), the paper reports that ANASTAARS-QD2 matches the best long-term performance achieved by PyBOBYQA on both the toy and Chvátal graphs, and that, unlike STARS-QD2, it does not easily get stuck. The paper attributes this difference to adaptive subspace growth after unsuccessful iterations. At $d=2p$1 and large shot budget, it is described as among the best optimizers in trial quantiles (Dzahini et al., 15 Jul 2025).

The scalability study extends to

$d=2p$2

Across shot counts $d=2p$3, the reported qualitative findings are that in the smallest-budget cases the optimizer’s lead increases as dimension increases, and in the largest-budget cases it maintains its lead as dimension grows. Performance is described as consistent across shot regimes, from low-shot to high-shot settings. The paper’s overall conclusion is that ANASTAARS-QD2 provides a path toward larger-scale QAOA optimization than is common in the cited literature (Dzahini et al., 15 Jul 2025).

The stated limitations are equally important. The experiments are performed on simulated MaxCut QAOA rather than on hardware. The tested implementation is specifically ANASTAARS-QD2, not every possible model variant. The method introduces several hyperparameters, including $d=2p$4. Subspace success remains probabilistic because it depends on alignment between the sampled subspace and useful descent directions. The paper also does not state a strong new convergence theorem for ANASTAARS in the provided text (Dzahini et al., 15 Jul 2025). These caveats suggest that the method’s current contribution is primarily algorithmic and empirical rather than a completed theoretical characterization.

7. Name disambiguation and related acronyms

The acronym ANASTAARS should be distinguished from several similarly named but unrelated arXiv topics. A-STAR, expanded as The All-Sky Transient Astrophysics Reporter, is a proposed high-cadence, wide-field X-ray transient survey mission for multimessenger astronomy, with instruments named Owl and Lobster and a survey strategy covering the available sky twice per 24 hours; it is not a quantum optimization method (Osborne et al., 2013). ANTARESS, expanded as Advanced and Neat Techniques for the Accurate Retrieval of Exoplanetary and Stellar Spectra, is a workflow for high-resolution exoplanet transit and occultation spectroscopy, optimized for extracted 2D echelle spectra and spatially resolved stellar-spectrum extraction; it is likewise unrelated to QAOA optimization (Bourrier et al., 2024). The Antarctic optical survey program AST3-2 concerns a robotic time-domain telescope at Dome A, Antarctica, with public release of processed images, catalogs, and light curves from its 2016 survey; this is an astronomical survey infrastructure result rather than an optimizer (Yang et al., 2023).

This distinction matters because acronym similarity could suggest a common research lineage where none exists. Within the supplied arXiv record, ANASTAARS refers specifically to a noise-aware adaptive random-subspace trust-region optimizer for variational quantum algorithms, and particularly for QAOA in shot-noisy, moderate-to-large parameter regimes (Dzahini et al., 15 Jul 2025).