Data informed Ansatz Selection for Variational Quantum Circuits

Manav Seksaria; Advith Desu; Anil Prabhakar

quant-phNov 3, 2026

Data informed Ansatz Selection for Variational Quantum Circuits

Authors: Manav Seksaria, Advith Desu, Anil Prabhakar

Organizations: Center for Quantum Information, Communication and Computing, Indian Institute of Technology Madras, Chennai, India · Engineering Systems and Design, Singapore University of Technology and Design, Singapore · Department of Electrical Engineering, Indian Institute of Technology Madras, Chennai, India

Abstract

Choosing a variational quantum circuit (VQC) usually means searching over ansatze and depths, and training each candidate. We give three rules that fix these choices from the data and the gate set alone, before any training. First, a circuit without entangling gates has a rank-one Fourier coefficient tensor on every subset of qubits. A singular value decomposition of a dataset's Fourier coefficients therefore indicates whether entanglement is needed, which qubits it connects, and a lower bound on the error of omitting it. Second, the rank of the quantum Fisher information matrix grows by $p_0$ per layer, after a first-layer deficit $d_0$ , until it reaches a ceiling set by the dynamical Lie algebra. This law fixes the depth. Third, reading out all $2^N - 1$ Pauli-Z products from one batch of shots resolves half of the state's directions, the most any single-basis readout can. A single-qubit readout resolves an exponentially small fraction. For the Madelon dataset, the rules prescribe 7 blocks of 3 qubits at depth 3: a 239-parameter classifier with 78.5% test accuracy, found without a grid search. Ablations confirm three predicted effects: shallower circuits, smaller blocks and dropping reuploading all lower accuracy.

Figures & tables

Panel of Figure 1 — Figure 1: One consideration applies at each stage of the circuit’s information path: representability at the encoding, QFIM rank at the state, extraction at the readout.

Figure 2: The rank grows at slope $p_{0}$ and turns a sharp corner at the orbit-stabiliser ceiling. Markers: measured QFIM rank of HEA at $N=5$ . Red line: the predictor $\min(p_{0}L-d_{0},\,C_{\mathrm{rank}})$ .

Ansatz	Gate set	Coupling	DLA $\mathfrak{g}$	$\dim\mathfrak{g}$	$p_{0}$	$d_{0}$	$C_{\mathrm{rank}}$	$\kappa$
HEA [ kandala2017 ]	$R_{y}R_{z}$	chain	$\mathfrak{su}(2^{N})$	$4^{N}{-}1$	$2N$	$0$	$2(2^{N}{-}1)$	$0$
EfficientSU2 [ qiskitlib ]	$R_{y}R_{z}$	complete	$\mathfrak{su}(2^{N})$	$4^{N}{-}1$	$2N$	$0$	$2(2^{N}{-}1)$	$0$
Product	$R_{y}R_{z}$	none	$\mathfrak{su}(2)^{N}$	$3N$	$2N$	$0$	$2N$	$0$
N-Local [ qiskitlib ]	$R_{y}R_{z}R_{x}$	ring	$\mathfrak{su}(2^{N})$	$4^{N}{-}1$	$3N$	$N$	$2(2^{N}{-}1)$	$0$
HVA chain [ wiersema2024 ]	$R_{ZZ},R_{x}$	chain	$\mathfrak{so}(2N)$	$N(2N{-}1)$	$2N{-}1$	$N{-}1$	$N^{2}{+}N{-}1$	$4(N{-}L)$
HVA ring [ wiersema2024 ]	$R_{ZZ},R_{x}$	ring	$\mathfrak{so}(2N)^{\oplus 2}$	$2N(2N{-}1)$	$2N$	$N$	$2N^{2}{-}2N{+}1$	$0$

Table 1: The rank law taken with equality,

\operatorname{rank}F(L)=\min(p_{0}L-d_{0},\,C_{\mathrm{rank}}-\kappa(L))

, with

C_{\mathrm{rank}}=\dim\mathfrak{g}-\dim\Stab_{\mathfrak{g}}(\ket{0})

and the codimension

\kappa(L)

of Sec. 3.3 (

0

for

L>N

), predicts the measured rank exactly for every family at

N\in\{3,\ldots,6\}

up to

L=12

, except at the two small-

N

exceptions explained in Appendix . Appendix works through the Brickwork row step by step.

Variational Quantum Circuits

Variational quantum circuits are trainable quantum circuits used for optimization, simulation, and hybrid quantum machine learning.

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The Average Relative Entropy and Transpilation Depth determines the noise robustness in Variational Quantum Classifiers

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Quantum Machine Learning Variational Quantum Algorithms

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Variational Quantum Algorithms (VQAs) have been extensively researched for applications in Quantum Machine Learning (QML), Optimization, and Molecular simulations. Although designed for Noisy Intermediate-Scale Quantum (NISQ) devices, VQAs are predominantly evaluated classically due to uncertain results on noisy devices and limited resource availability. Raising concern over the reproducibility of simulated VQAs on noisy hardware. While prior studies indicate that VQAs may exhibit noise resilience in specific parameterized shallow quantum circuits, there are no definitive measures to establish what defines a shallow circuit or the optimal circuit depth for VQAs on a noisy platform. These challenges extend naturally to Variational Quantum Classification (VQC) algorithms, a subclass of VQAs for supervised learning. In this article, we propose a relative entropy-based metric to verify whether a VQC model would perform similarly on a noisy device as it does on simulations. We establish a strong correlation between the average relative entropy difference in classes, transpilation circuit depth, and their performance difference on a noisy quantum device. Our results further indicate that circuit depth alone is insufficient to characterize shallow circuits. We present empirical evidence to support these assertions across a diverse array of techniques for implementing VQC, datasets, and multiple noisy quantum devices.

Abstract

Variational Quantum Circuits

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