Structured Unitary Tensor Network Representations for Circuit-Efficient Quantum Data Encoding

Guang Lin; Toshihisa Tanaka; Qibin Zhao

quant-phFeb 18, 2026

Structured Unitary Tensor Network Representations for Circuit-Efficient Quantum Data Encoding

Authors: Guang Lin, Toshihisa Tanaka, Qibin Zhao

Organizations: RIKEN Center for Advanced Intelligence Project · Tokyo University of Agriculture and Technology

Abstract

Encoding classical data into quantum states is a central bottleneck in quantum machine learning: many widely used encodings are circuit-inefficient, requiring deep circuits and substantial quantum resources, which limits scalability on quantum hardware. In this work, we propose TNQE, a circuit-efficient quantum data encoding framework built on structured unitary tensor network (TN) representations. TNQE first represents each classical input via a TN decomposition and then compiles the resulting tensor cores into an encoding circuit through two complementary core-to-circuit strategies. To make this compilation trainable while respecting the unitary nature of quantum operations, we introduce a unitary-aware constraint that parameterizes TN cores as learnable block unitaries, enabling them to be directly optimized and directly encoded as quantum operators. The proposed TNQE framework enables explicit control over circuit depth and qubit resources, allowing the construction of shallow, resource-efficient circuits. Across a range of benchmarks, TNQE achieves encoding circuits as shallow as $0.04\times$ the depth of amplitude encoding, while naturally scaling to high-resolution images ( $256 \times 256$ ) and demonstrating practical feasibility on real quantum hardware.

Figures & tables

Figure 1: Different-resolution visual comparisons in simulation are conducted using $32\times 32$ , $256\times 256$ , and $256\times 256\times 3$ images. The ordering of methods is consistent across resolutions, and the circuit structure is reported as (qubit, depth). Amplitude encoding enables accurate encoding in simulation at the cost of deep circuits, TNQEs provide different trade-offs between qubit and depth.

Figure 2: Illustration of the proposed TNQEs. First, a) classical data is represented using quantized tensor train decomposition. Then, b) full tensor network encoding and c) core-wise encoding are developed to convert the core tensors into quantum circuits.

Methods	Qubits	Depth	Image size	E2E
Basis encoding	$O(s^{2})$	$O(1)$	limited	✓
Amplitude encoding	$O(\log s^{2})$	$O(s^{2})$	limited	✓
MPS-based encoding	$O(p(\log s^{2}/p))$	$O(s^{2}/p)$	limited	✓
CNN-based hybrid	$O(d)$	$O(1)$	arbitrary	$\times$
TNQE-full	$O(\log s^{2}+\log r)$	$O(4r^{2}\log s)$	limited	✓
TNQE-core	$O(\log s\log r)$	$O(4r^{2})$	scalable	✓

Table 1: Comparison of quantum data encoding methods in terms of scalability and resource requirements. Here

s^{2}

denotes the image size (

s\times s

), and

r

and

N_{\ell}

are hyperparameters of TNQE.

Figure 3: Illustration of TNQE-unitary. Each core is parameterized by a learnable block unitary acting on the bond and physical qubits, with $N_{\ell}$ layers of single-qubit rotations and fixed entanglers.

Figure 4: Comparison of quantum circuit for (a) amplitude encoding (partially shown due to its depth), (b) TNQE-full encoding and TNQE-unitary encoding, and (c) TNQE-core encoding.

Methods	Qubits	Depth	Unitary	CNOT	Ops.
Baseline (Amplitude)	10	2028	1023	1013	2036
Automatic encoding	10	208	371	129	500
TNQE-full	12	945	900	475	1375
TNQE-core	26	116	219	193	412
TNQE-unitary	13	81	100	120	220

Table 2: Comparison of quantum circuits on

32\times 32

images.

Figure 5: Scalability of different quantum data encoding methods. (a) The number of qubits, (b) circuit depth, and (c) the number of quantum operators as image size increases from $4\times 4$ to $512\times 512$ . (d) Classification accuracy (%) on approximated MNIST images using CNN and MLP classifiers.

Results with original image
Methods	MSE	BCE	PSNR	SSIM
TN reconstruction	0.013135	0.113215	19.3358	0.924981
Automatic encoding	0.015656	0.147523	18.3669	0.882500
TNQE-full	0.015700	0.129583	18.6845	0.906974
TNQE-core	0.015639	0.129311	18.6903	0.907203
TNQE-unitary	0.011249	0.092333	20.5837	0.915903

Table 3: Quantitative comparisons of approximated image using quantum data encoding methods in simulation on MNIST dataset.

Methods	Rank	1	2	3	4	5	8	9	16
Full	MSE	0.088905	0.056156	0.032136	0.014735	0.008940	0.001833	0.001195	0.003042
Full	Structure	(10, 7)	(11, 195)	(12, 945)	(12, 945)	(13, 4185)	(13, 4185)	(14, 17625)	(14, 17625)
Core	MSE	0.089138	0.056174	0.032164	0.014731	0.008943	0.001852	0.001185	0.002972
Core	Structure	(10, 4)	(18, 24)	(26, 116)	(26, 116)	(31, 242)	(34, 496)	(39, 1006)	(42, 2028)
Unitary	MSE	0.162642	0.045030	0.018398	0.018493	0.011287	0.011249	0.008355	0.008307
Unitary	Structure	(10, 8)	(11, 56)	(12, 61)	(12, 61)	(13, 81)	(13, 81)	(14, 82)	(14, 82)

Table 4: Ablation study on different ranks using TNQE on MNIST. The circuit structure is reported as (qubit, depth).

Figure 7: The impact of $N_{\ell}$ for TNQE-unitary. To facilitate a comparison with other methods, $N_{\ell}$ converts into the corresponding circuit depth on the x-axis and $1-\text{MSE}$ on the y-axis. The detailed numerical results are provided in Table 7 .

QPUs	Methods	MSE	BCE	PSNR	SSIM
ibm_torino (Heron r1)	Amp.	0.112202	0.488010	9.6688	0.008421
	Auto.	0.050782	0.268235	13.2173	0.554541
	Ours	0.034157	0.193016	15.0322	0.735706
ibm_fez (Heron r2)	Amp.	0.107725	0.666679	9.9193	0.000325
	Auto.	0.053146	0.270768	13.2744	0.567423
	Ours	0.029948	0.183303	15.6425	0.773744

Table 5: Quantitative comparisons on real quantum hardware.

Figure 8: Visual comparisons on ibm_boston with Heron r3.

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 9: Confusion matrix of (a) Automatic encoding, (b) TNQE-full, (c) TNQE-core, (d) TNQE-unitary on MNIST classification task in Figure 5 d using pre-trained MLP (Top) and CNN (Bottom) classifiers.

Methods	Rank	1	2	3	4	5	8	9	16
Full	BCE	1.137407	0.504207	0.248129	0.125140	0.100464	0.067908	0.065382	0.093536
	PSNR	10.9041	12.9945	15.5608	19.0217	21.5318	29.9344	33.4708	34.0668
	SSIM	0.287041	0.583822	0.780099	0.906428	0.946004	0.989724	0.993186	0.969076
	Structure	(10, 7)	(11, 195)	(12, 945)	(12, 945)	(13, 4185)	(13, 4185)	(14, 17625)	(14, 17625)
Core	BCE	1.150839	0.496339	0.249040	0.124900	0.100473	0.068111	0.065349	0.092729
	PSNR	10.8891	12.9989	15.5508	19.0257	21.5332	29.8655	33.2292	33.9262

Appendix

Table 6: Ablation study on different ranks using TNQE on MNIST. The circuit structure is shown in parentheses as (qubits, depth).

Layers	1	2	3	4	5	6	8	10	12	14	16	18	20
MSE	0.045022	0.021625	0.015167	0.011249	0.008361	0.006421	0.003857	0.002621	0.001746	0.001147	0.000838	0.000652	0.000504
BCE	0.202573	0.123985	0.103954	0.092333	0.083855	0.077791	0.069254	0.064738	0.061277	0.058689	0.057237	0.056265	0.055501
PSNR	13.6188	17.2647	19.1014	20.5837	21.8803	23.1861	25.7583	27.8214	30.1595	32.8591	34.9306	37.0738	38.8589
SSIM	0.694205	0.840451	0.885914	0.915903	0.938019	0.952568	0.971858	0.981073	0.987587	0.992111	0.994204	0.995560	0.996577
Depth	21	41	61	81	101	121	161	201	241	281	321	361	401

Appendix

Table 7: The impact of number of layers in TNQE-unitary on image approximation quality.

Figure 10: Comparison of simulated and real quantum results on ibm_kawasaki with Heron r2.

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