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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