Metric Learning with Quantum Feature Spaces

Vinayak Sharma; Lord Sen; Ashish Padhy; Sourav Behera; Gennaro De Luca; Shyamapada Mukherjee; Aviral Shrivastava

quant-phDec 4, 2023

Metric Learning with Quantum Feature Spaces

Authors: Vinayak Sharma, Lord Sen, Ashish Padhy, Sourav Behera, Gennaro De Luca, Shyamapada Mukherjee, Aviral Shrivastava

Organizations: School of Computing and Augmented Intelligence Arizona State University, Tempe, AZ 85281 · National Institute of Technology, Rourkela Sector 1, Rourkela, Odisha - 769008

Abstract

The $2^N$ dimensions of Hilbert space give quantum metric learning exponentially many mutually orthogonal states with which to separate classes, but the same exponential dimension also degrades its optimization landscape and the tractability of evaluating its metric. By letting a classical network predict all gate parameters of a fixed ansatz, we isolate the reachable state manifold and its fidelity metric. We prove that this separability is already present on the unentangled manifold $\mathbb{M}_{\text{unentangled}} = (\mathbb{CP}^1)^N$ , which saturates the $2^N$ maximal orthogonal capacity needed for classification, while entanglement adds states only at intermediate separations, which are costly to resolve under finite sampling. The same size carries costs. Global fidelity causes exponential gradient collapse across both unentangled ( $Θ(c^{-N})$ ) and entangled ( $Θ(4^{-N})$ ) manifolds; as fidelities concentrate, the fidelity signal-to-noise ratio at a fixed shot budget also decays exponentially in $N$ ; and because $\mathbb{M}_{\text{unentangled}}$ already holds $2^N$ orthogonal states, expressive, unconstrained encoders under contrastive losses reach trivial minima by geometric shattering even without entanglement. We introduce Quantum Projective Metric Learning (QPMeL), an exact $O(N)$ closed-form kernel on $\mathbb{M}_{\text{unentangled}}$ that circumvents circuit simulation. QPMeL resolves gradient collapse via a curriculum annealing from a $1$ -local observable regime (restoring $Θ(1/N^2)$ polynomial variance) to the global product kernel. Empirically, QPMeL as a complete design exceeds eight and matches two of 11 published quantum few-shot designs.

Figures & tables

Figure 1: QPMeL architecture. A domain metric encoder maps the input to a latent vector, and an angle head projects it through bounded activations to $N$ coordinate pairs $(\theta_{q},\gamma_{q})$ , one per qubit.

Figure 2: QPMeL training flow, shown for an MS-COCO image and caption. The PMeF (shaded) projects the encoder’s angles onto real coordinate vectors $C_{q}$ and computes each qubit’s fidelity with the CKF, and the metric loss on the resulting kernel sends gradients back to the encoder. Every block is classical, and no circuit runs during training.

MNIST
Model	Circuit	Protocol	Baseline	QPMeL
Thumwanit et al. (2021)	E-VQC	Binary $(3,6)$	$91.7$ (9)	$\mathbf{99.80\pm 0.20}$ (5)
NQE ( Hur et al., 2023 )	CPQK	Binary $(0,1)$	$99.50$ (12)	$99.60\pm 1.71$ (5)
Huang et al. (2025)	E-VQC	2w-10s $(3,5)$	$89.69$ (9)	$\mathbf{99.00\pm 1.01}$ (5)
Wang et al. (2024)	E-VQC	3w-1s $(0,1,2)$	$50.5$ (6)	$\mathbf{98.27\pm 1.86}$ (5)
Wang et al. (2024)	E-VQC	3w-10s $(0,1,2)$	$82.5$ (6)	$\mathbf{99.73\pm 0.37}$ (5)

Table 1: Few-shot classification against published quantum models.

			QSTConv		QNervFormer		QNervFormer (dropout)
Head	Similarity	$L$	Train	Held-out	Train	Held-out	Train	Held-out
Chen et al.	$\langle Z\rangle$ cosine	4	–	$3.2$	–	$4.3$	–	–
E-VQC	Fidelity	4	$1.5\pm 0.9$	$1.2\pm 0.4$	$10.1\pm 2.6$	$0.2\pm 0.3$	$2.8\pm 0.4$	$2.3\pm 1.4$
DR	Fidelity	1	$13.1\pm 6.5$	$12.3\pm 5.2$	$65.0\pm 17.9$	$1.0\pm 0.9$	$7.9\pm 6.2$	$6.6\pm 4.6$
DR	Fidelity	2	$2.9\pm 1.8$	$3.7\pm 1.2$	$43.0\pm 8.0$	$0.3\pm 0.6$	$2.9\pm 2.4$	$2.3\pm 2.0$
DR	Fidelity	4	$0.6\pm 0.7$	$0.7\pm 0.4$	$6.5\pm 1.7$	$0.5\pm 0.4$	$0.9\pm 1.0$	$0.9\pm 0.7$

Table 2: THINGS-EEG2 200-way top-1 accuracy (%) with only the head changed;

L

circuit layers.

Dataset	Task	SWAP 2q gates	Shots	Exact	IBM-Prague model	IBM Marrakesh
MNIST	10w-1s	73	128	$96.7\pm 0.6$	$96.3\pm 0.8$	$97.6\pm 3.4$
Fashion-MNIST	10w-1s	93	128	$85.03\pm 0.4$	$80.9\pm 1.5$	$74.8\pm 4.2$
Omniglot	5w-3s	184	128	$93.9\pm 0.7$	$93.4\pm 1.2$	$84.0\pm 6.0$

Table 3: Few-shot accuracy (%) on IBM hardware and under a device noise model.

Appendix figures & tables6 assets

Supplementary material from the paper’s appendix.

Appendix

	Top-1		Top-5		Final loss		Collapsed
Encoder	Curr.	Add.	Curr.	Add.	Curr.	Add.	Curr.	Add.
QSTConv	$\mathbf{23.96\pm 5.2}$	$19.11\pm 3.9$	$\mathbf{56.82\pm 7.6}$	$48.56\pm 5.5$	$4.81$	$5.38$	$0$	$0$
QNervFormer	$0.92\pm 0.8$	$0.73\pm 0.5$	$4.92\pm 2.0$	$3.93\pm 1.4$	$0.50$	$1.40$	$0$	$0$

Appendix

Table E.1: QPMeL kernel schedule on THINGS-EEG2, 10 subjects and 5 seeds: 200-way held-out accuracy (%), final training loss, and runs whose final loss stays above

6.9

50

. Here, Curr. is Curriculum and Add. is Additive.

		Kernel at initialization
Head	Similarity	Mean	Range (nats)	Structure $\rho$	Encoder gradient	Train (%)
E-VQC	$\langle Z\rangle$ cosine	$-3.9\times 10^{-3}$	$6.57$	$0.19$	$10.6$	$3.2$
E-VQC	Fidelity	$8.8\times 10^{-4}$	$0.09$	$0.79$	$0.010$	$10.1$
E-VQC	$\log$ fidelity	$8.8\times 10^{-4}$	–	$0.79$	$955$	$0.3$
DR	Fidelity	$1.0\times 10^{-3}$	$0.02$	$0.06$	$0.021$	$6.5$
QPMeL	PMeF, product	$9.9\times 10^{-4}$	$0.07$	n/m	$0.62$	$\mathbf{96.5}$

Appendix

Table E.2: Kernel and gradient diagnostics on THINGS-EEG2 with QNervFormer; four-layer circuits.

		Layers $L$
Head		1	2	4	8	16
DR	Rings in the kernel	$0$	$1$	$3$	$7$	$15$
	Structure $\rho$	$0.64$	$0.46$	$0.06$	$0.00$	$0.00$
	Encoder gradient	$1.4\times 10^{-4}$	$8.3\times 10^{-3}$	$0.022$	$0.041$	$0.068$
	Train (%)	$\mathbf{65.0\pm 17.9}^{\ddagger}$	$43.0\pm 8.0$	$7.0\pm 3.1$	$1.0\pm 0.0$	$2.0\pm 0.9$
	Held-out, QSTConv	$\mathbf{12.3\pm 5.2}$	$3.7\pm 1.2$	$0.7\pm 0.4$	n/r	n/r

Appendix

Table E.3: Circuit depth under the fidelity readout on THINGS-EEG2. Structure is measured at initialization within one tower on inputs uniform in

[-\pi,\pi]

, and the encoder gradient is the RMS at the first epoch on QNervFormer. Train is the fit to 200 training images on QNervFormer after 200 epochs, three seeds ( ‡ ten). Held-out rows are on QSTConv and on QNervFormer with branch dropout, with the checkpoint selection of Table , five seeds ( † four); QPMeL reaches

19.9

and

11.8\%

there. n/r, not run.

		QSTConv			QNervFormer, branch dropout
Head	$L$	Matched	Unmatched	$<0.01$	Matched	Unmatched	$<0.01$
E-VQC, fidelity	4	$0.057$	$0.034$	$0.39$	$0.127$	$0.072$	$0.18$
DR	1	$0.282$	$0.161$	$0.02$	$0.214$	$0.121$	$0.22$
DR	2	$0.188$	$0.117$	$0.06$	$0.123$	$0.072$	$0.28$
DR	4	$1.3\times 10^{-3}$	$1.2\times 10^{-3}$	$0.99$	$0.047$	$0.029$	$0.60$
DR	8	n/r	n/r	n/r	$0.019$	$8.9\times 10^{-3}$	$0.87$

Appendix

Table E.4: Trained kernels on THINGS-EEG2: mean fidelity of matched and of unmatched pairs, and the fraction of unmatched pairs below

0.01

, on 1024 training pairs at the checkpoint of Table ; five seeds ( † four). ‡ QNervFormer without branch dropout at epoch 200, three seeds. Haar-random states give

9.8\times 10^{-4}

9.8\times 10^{-4}

and

1.00

. n/r, not run.

	Image $\rightarrow$ text		Text $\rightarrow$ image
Way	PMeF	Simulated	PMeF	Simulated
3	$99.6\pm 0.6$	$98.7\pm 1.9$	$98.9\pm 1.0$	$98.0\pm 2.3$
5	$98.5\pm 0.9$	$98.8\pm 1.4$	$98.5\pm 0.8$	$98.0\pm 1.7$
7	$97.8\pm 0.9$	$96.3\pm 2.1$	$97.5\pm 1.1$	$96.3\pm 2.4$
9	$98.1\pm 0.8$	$95.3\pm 2.1$	$97.0\pm 1.0$	$95.1\pm 2.4$
11	$96.5\pm 0.9$	$94.4\pm 2.1$	$96.8\pm 0.8$	$95.1\pm 1.7$

Appendix

Table E.5: MS-COCO cross-modal matching on training pairs (%), 20 qubits: closed-form PMeF and statevector simulation of the same circuit. Chance is

100/n

for

n

-way.

Primitive	Setting	Qubits	2-qubit gates	Depth
Inversion test	$N=5$	5	0	6
Inversion test	$N=6$	6	0	6
Inversion test	$N=10$	10	0	6
SWAP test	$N=5$	11	73	187
SWAP test	$N=6$	13	93	232
SWAP test	$N=10$	21	167	394

Appendix

Table E.6: Transpiled cost of the two fidelity primitives, targeting the ibm_marrakesh coupling map and basis at optimization level

1

. Two-qubit counts are native cz gates after routing; the pre-routing logical count is

8N

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