Filtered Spectral Projection for Quantum Principal Component Analysis

Sk Mujaffar Hossain; Satadeep Bhattacharjee

stat.MLMar 13, 2026

Filtered Spectral Projection for Quantum Principal Component Analysis

Authors: Sk Mujaffar Hossain, Satadeep Bhattacharjee

Organizations: Indo-Korea Science and Technology Center (IKST), Bengaluru 560064, India

Abstract

Quantum principal component analysis (qPCA) is commonly formulated as the extraction of eigenvalues and eigenvectors of a covariance-encoded density operator. Yet in many qPCA settings the practical goal is simpler: projection onto the dominant spectral subspace. Here we introduce a projection-first framework, the Filtered Spectral Projection Algorithm (FSPA), which bypasses explicit eigenvalue estimation while preserving the relevant spectral structure. FSPA amplifies any nonzero warm-start overlap with the leading subspace and remains robust in small-gap and near-degenerate regimes, without artificial symmetry breaking in the absence of bias. We show that FSPA achieves an oracle complexity $\mathcal{O}((\log(1/ε)+\log(1/|a_1|^2))/\log(λ_1/λ_2))$ ,which is tight by a matching lower bound, establishing it as anoptimal projection primitive. We derive a convergence rate for degenerate spectra, give a circuit resource analysis with $n+\mathcal{O}(1)$ qubit overhead independent of system dimension, and extend the method to threshold spectral projection, Threshold-FSPA, which converges in $\mathcal{O}(\log(1/ε))$ calls when the threshold lies between eigenvalues. In the density matrix exponentiation access model, FSPA gives an exponential copy-complexity advantage over classical methods. For classical datasets, we show that for amplitude-encoded centered data the ensemble density matrix $ρ=\sum_i p_i|ψ_i\rangle\langleψ_i|$ equals the covariance matrix. Numerical tests on chemistry density matrices, noisy circuit outputs, Breast Cancer Wisconsin, handwritten Digits, and 1--4-qubit scalability confirm the theory. A minimal Qiskit implementation validates magnitude invariance, signal amplification, and no spurious symmetry breaking. These results establish FSPA as an optimal and deployable quantum spectral projection primitive.

Figures & tables

Figure 1: Empirical validation of the gap-dependent oracle complexity of FSPA. The total number of oracle applications required to reach 99.99% fidelity is plotted against the theoretical scaling variable

1/\log(\lambda_1/\lambda_2)

. Linear regression confirms proportional scaling, consistent with the predicted complexity

\mathcal{O}(\log(1/\epsilon)/\log(\lambda_1/\lambda_2))

. Classical power iteration is shown for comparison.

Result	Type	Key statement	Section
Magnitude invariance	Proposition 1	Iterates of $\rho$ and $c\rho$ are identical	3
Gap-dependent amplification	Proposition 2	Overlap monotone; rate $\propto\lambda_{1}/\lambda_{2}$	3
Subspace convergence (qualitative)	Proposition 3	$\\|P_{\mathcal{S}_{R}}\|\phi_{T}\rangle\\|^{2}\to 1$ as $T\to\infty$	3
Oracle complexity (upper bound)	Theorem 1	$\mathcal{O}((\log(1/\epsilon)+\log(1/\|a_1\|^2))/\log(1/r))$	3
Circuit resources	Proposition 4	$n{+}\kappa{+}\mathcal{O}(1)$ qubits; gate count $\propto\mathcal{G}(U_{\rho})/\log(1/r)$	3
Oracle complexity (lower bound)	Proposition 5	$\Omega((\log(1/\epsilon)+\log(1/\|a_1\|^2))/\log(1/r))$ (optimal)	3

Table 1: Summary of all theoretical results established in this work.

r=\lambda_{2}/\lambda_{1}

;

\epsilon

= target infidelity;

\alpha

= initial subspace overlap;

r_{S}=\lambda_{R+1}/\lambda_{1}

;

\Delta_{\tau}

= normalised eigenvalue-threshold gap;

\kappa

= block-encoding ancilla overhead;

d=2^{n}

= system dimension.

Method	Access	Mag.	Gap	Complexity
Lloyd qPCA Lloyd et al. (2014)	DME	Yes	Yes	$\mathcal{O}(1/\epsilon^{2}\gamma)$
Filtered QPE Gilyén (2019)	BE	Yes	Yes	$\mathcal{O}(1/\gamma)$
Tang Tang (2021)	Samp.	No	Yes	Classical
FSPA (this work)	DME/BE	No	Yes	$\mathcal{O}(\log(1/\epsilon)/\log(1/r))$
Thresh.-FSPA	DME/BE	No	Part.	$\mathcal{O}(\log(1/\epsilon)/\Delta_{\tau})$

Table 2: Comparison of quantum spectral algorithms. DME = density matrix exponentiation; BE = block-encoding. Mag. = fails under uniform eigenvalue rescaling at fixed gap.

r=\lambda_{2}/\lambda_{1}

;

\gamma=\lambda_{1}-\lambda_{2}

;

\Delta_{\tau}

= eigenvalue-threshold gap.

Figure 2: Eigenvector instability versus subspace stability on the Breast Cancer Wisconsin dataset. Pedregosa et al. (2011) The real-data covariance matrix is constructed from standardized diagnostic features. Small perturbations strongly rotate individual leading eigenvectors, while dominant-subspace fidelity remains stable. This illustrates why subspace-level metrics are the appropriate object of study in near-degenerate regimes.

Figure 3: Uniform spectral rescaling at fixed gap. Lloyd-style qPCA collapses below a resolution threshold, while FSPA remains stable under global eigenvalue downscaling.

Figure 4: Algorithmic regime map versus spectral gap. Phase-estimation-based methods show a sharp threshold behavior tied to finite resolution; FSPA degrades smoothly as the gap shrinks, consistent with gap-limited amplification.

Figure 5: Minimal Qiskit demonstration of FSPA on a

4\times 4

PSD matrix (2-qubit system, eigenvalues

\{0.05,0.10,0.20,1.00\}

). Each oracle call is realised as a Qiskit AerSimulator statevector circuit that prepares the post-selected outcome of a block-encoded application of

\rho

, followed by renormalization. Left (Demo 1): Starting from a weak warm-start bias (

|\langle\psi_{1}|\phi_{0}\rangle|^{2}\approx 0.001

), FSPA amplifies the overlap with the dominant eigenvector

|\psi_{1}\rangle

to unit fidelity; the Qiskit circuit (orange) matches the classical reference (blue) exactly. Centre (Demo 2): Uniform spectral rescaling

\rho\to\alpha\rho

with

\alpha\in\{1,0.01,0.0001\}

leaves the overlap trajectory unchanged to within floating-point precision (

\lesssim 10^{-15}

), confirming Proposition 1 directly on a quantum circuit. Right (Demo 3): Initialising in

|\psi_{2}\rangle

(zero overlap with

|\psi_{1}\rangle

), FSPA converges to

|\psi_{2}\rangle

and the overlap with

|\psi_{1}\rangle

remains at machine zero (

\sim 10^{-23}

), confirming that FSPA amplifies existing spectral bias without creating an arbitrary preferred direction.

Figure 6: FSPA applied to one-body reduced density matrices (1-RDMs) of H 2 , LiH, and BeH 2 computed by FCI in the STO-3G basis. (A) Dominant natural orbital recovery for all three molecules; solid lines show the classical reference, dashed lines show the Qiskit circuit. (B) Occupation number spectra (eigenvalues of the 1-RDM), showing the contrast between H 2 (large gap) and LiH/BeH 2 (near-degenerate occupations). (C) Magnitude invariance on the LiH 1-RDM: rescaling all occupation numbers by

\alpha\in\{1,0.01,0.0001\}

leaves the convergence trajectory unchanged to within

\lesssim 10^{-15}

, confirming Proposition 1 . (D) LiH near-degenerate case (

\Delta=0.023

): eigenvector fidelity

|\langle\psi_{1}|\phi\rangle|^{2}

(red) reaches 0.83 while subspace fidelity

\|P_{\mathcal{S}}|\phi\rangle\|^{2}

(blue) converges to 1.000, validating Theorem 2 .

Figure 7: FSPA on density matrices from 2-qubit random circuits under depolarising noise with rate

p

. (A) Overlap trajectories at noise rates

p\in\{0,0.01,0.05,0.10\}

; FSPA degrades gracefully. (B) FSPA versus Lloyd-style qPCA: Lloyd-style qPCA collapses abruptly while FSPA degrades smoothly. (C) Mean fidelity

\pm 1\sigma

across five random circuit seeds; FSPA achieves

\geq 99\%

fidelity for

p\leq 0.10

Figure 8: Scalability analysis of FSPA from 1 to 4 qubits (

d=2

d=16

), target fidelity 99%. (A) Oracle calls versus qubit count

n

at fixed spectral gaps

\Delta\in\{0.50,0.20,0.10\}

; flat curves confirm

\mathcal{O}(d^{0})

scaling as predicted by Theorem 1 . (B) Empirical oracle counts versus theoretical prediction of Theorem 1 at

n=2

for six spectral gaps; excellent agreement validates the complexity characterisation. (C) Qubit overhead: FSPA uses

n+1

qubits total (

n

data qubits plus one renormalization ancilla), which is optimal.

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Principal component analysis (PCA) is traditionally implemented through a covariance or kernel matrix, leading-eigenvector extraction, and hard rank-

k

projection. These steps can be computationally costly in high-dimensional and quantum-data settings, sensitive to small eigengaps, and unnecessary when downstream tasks only require principal-subspace scores. Such score-based objectives are important in applications such as anomaly detection, spectral-energy profiling, and other postselection tasks. To address these needs, we introduce a measurement-based soft PCA framework replacing the hard top-

k

projector with an entropy-regularized Fermi--Dirac filter. This filter is the unique optimizer of an entropy-regularized variational formulation of PCA and converges to the classical PCA projector in the zero-temperature limit. This filter has a direct interpretation as a quantum measurement, which naturally suggests a quantum approach. For centered covariance operators represented by quantum feature states, a single fixed circuit, together with threshold calibration, accesses all optimal filters for different rank budgets or retained-variance levels without rank-dependent circuit updates or eigenvector recovery. For new inputs, the same calibrated quantum circuit yields soft principal subspace scores, spectral energy profiles, and postselected filtered states. The required centering of both training and test data is performed coherently inside the quantum protocol, which is particularly important for quantum data where no classical feature vectors or centered Gram matrix are directly available. By reframing PCA as a calibrated measurement task, this framework bypasses the need for iterative eigenvector extraction and achieves a dimension-independent sample complexity

O(η^{-2})

for normalized fractional-rank or retained variance scoring at additive accuracy

η

Abstract

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