Paper deep dive
Multivariate Time Series Forecasting with Adaptive Non-Local Observables
Yu-Ting Lee, Huan-Hsin Tseng, Samuel Yen-Chi Chen
Intelligence
Status: not_run | Model: - | Prompt: - | Confidence: 0%
Entities (0)
Relation Signals (0)
No relation signals yet.
Cypher Suggestions (0)
No Cypher suggestions yet.
Abstract
Abstract:Multivariate time series forecasting (MTSF) predicts future values of multiple variables from historical data. While quantum neural networks have been increasingly applied to this task, they typically rely on fixed local measurements, which restrict their expressivity. We propose MTSF-ANO, a simple hybrid model for MTSF that integrates variational quantum circuits with adaptive non-local observables (ANO). On the four ETT datasets, MTSF-ANO ranks first or second in MSE in 17 of 20 settings, improving over the strongest baseline by up to 20% on ETTh1, and outperforms or matches its fixed local observable counterpart across all settings. Our ablations show how the quantum circuit design and ANO non-locality affect performance. These results suggest that ANO is a promising direction for quantum time series forecasting.
Tags
Links
- Source: https://arxiv.org/abs/2607.24399v1
- Canonical: https://arxiv.org/abs/2607.24399v1
Trouble viewing inline? Open PDF directly →
Full Text
23,278 characters extracted from source content.
Expand or collapse full text
MULTIVARIATE TIME SERIES FORECASTING WITH ADAPTIVE NON-LOCAL OBSERVABLES Yu-Ting Lee 1 Huan-Hsin Tseng 2 Samuel Yen-Chi Chen 3 1 Graduate Institute of Communication Engineering, National Taiwan University, Taipei, Taiwan 2 Brookhaven National Laboratory, AI & ML Department, Upton, NY, USA 3 Wells Fargo, New York, NY, USA r14942088@ntu.edu.tw, htseng@bnl.gov, yen-chi.chen@wellsfargo.com ABSTRACT Multivariate time series forecasting (MTSF) predicts future values of multiple variables from historical data. While quan- tum neural networks have been increasingly applied to this task, they typically rely on fixed local measurements, which restrict their expressivity. We propose MTSF-ANO, a simple hybrid model for MTSF that integrates variational quantum cir- cuits with adaptive non-local observables (ANO). On the four ETT datasets, MTSF-ANO ranks first or second in MSE in 17 of 20 settings, improving over the strongest baseline by up to 20% on ETTh1, and outperforms or matches its fixed local observable counterpart across all settings. Our ablations show how the quantum circuit design and ANO non-locality affect performance. These results suggest that ANO is a promising direction for quantum time series forecasting. Index Terms— Quantum machine learning, Variational quantum circuits, Quantum neural networks, Non-local ob- servables, Time series forecasting 1. INTRODUCTION Multivariate time series forecasting (MTSF), which predicts multiple variables from historical data, is critical for strategic planning in domains such as energy management, weather modeling, and finance. Notably, the high-dimensional correla- tions in the data require models to learn complex distributions. Quantum machine learning (QML) enhances classical ma- chine learning by leveraging the representational expressiv- ity stemming from quantum phenomena such as superposi- tion, entanglement, and quantum interference [1, 2]. Among QML frameworks, quantum neural networks (QNNs) are in- creasingly applied to complex machine learning tasks, such as reinforcement learning [3–7], classification [8, 9], data com- The views expressed in this article are those of the authors and do not represent the views of Wells Fargo. This article is for informational purposes only. Nothing contained in this article should be construed as investment advice. Wells Fargo makes no express or implied warranties and expressly disclaims all legal, tax, and accounting implications related to this article. pression [10], anomaly detection [11], and time-series predic- tion [12–15]. However, QNNs are typically constructed from variational quantum circuits (VQCs) and rely on local measurements, such as Pauli observables. This reliance on local measure- ments restricts the network’s expressivity and its ability to learn complex data distributions. To overcome this bottleneck, recent research suggests jointly optimizing circuit parame- ters alongside trainable observables [16, 17]. Specifically, the adaptive non-local observables (ANO) framework [18] em- ploys trainable multi-qubit Hermitian observables, showing strong potential across super-resolution [19], reinforcement learning [20], and classification tasks [18]. In this work, we propose MTSF-ANO, a simple hybrid model for MTSF that integrates ANO into a data re-uploading VQC [9, 21]. 1 We benchmark MTSF-ANO against strong classical and quantum baselines, and introduce a channel- independent variant that excels at longer lookback windows. We conduct ablations on circuit design and ANO non-locality. Our findings establish ANO-based hybrid models as novel and effective solutions for time-series. Our contributions are: •We introduce MTSF-ANO, a simple hybrid model for MTSF that utilizes trainable non-local observables to enhance forecasting performance. • On the four ETT datasets, MTSF-ANO ranks first or sec- ond in 17 out of 20 settings, outperforming or matching the fixed local observable counterpart in all of them. •Ablations show that non-local measurement is the key driver of gains, with entanglement, shallow depth, and moderate non-locality beneficial. 2. METHODOLOGY 2.1. Problem Formulation Consider a multivariate time series dataset withCvariates (channels). LetLdenote the size of the lookback window and 1 Code at https://github.com/Yu-TingLee/MTSF-ANO. arXiv:2607.24399v1 [quant-ph] 27 Jul 2026 Hthe forecasting horizon. Given historical dataX t ∈R C×L , the goal of MTSF is to predict future values b Y t ∈R C×H . The corresponding ground truth is denoted Y t ∈R C×H . 2.2. Variational Quantum Circuits Variational quantum circuits (VQCs), or parameterized quan- tum circuits (PQCs), are trainable quantum models that process classical data in three stages. First, a data encoding unitary circuitU(x)maps a classical inputxinto ann-qubit sys- tem, yielding the encoded statesU(x)|0⟩ ⊗n , where|0⟩ ⊗n is the ground state. Next, a parameterized unitary circuitV (θ) evolves the encoded states intoV (θ)U(x)|0⟩ ⊗n . This vari- ational circuitV (θ)typically consists of alternating layers of trainable single-qubit rotations and multi-qubit entangling gates. Finally, a measurement layer is applied to extract classi- cal information by evaluating the expectation values of a fixed Hermitian observableH. The computation of a VQC can be summarized as a quantum function f VQC (x;θ): f VQC (x;θ) =⟨0| ⊗n U † (x)V † (θ)HV (θ)U(x)|0⟩ ⊗n .(1) 2.3. Adaptive Non-Local Observables Adaptive non-local observables (ANO) [18] replace the fixed observable of a generic VQC with a trainable HermitianH(φ) parameterized by φ. A k-local observable takes the form: H(φ) = c 11 a 12 + ib 12 a 13 + ib 13 · a 1K + ib 1K ∗c 22 a 23 + ib 23 · a 2K + ib 2K ∗ ∗c 33 · a 3K + ib 3K . . . . . . . . . . . . . . . ∗ ∗ · c K (2) wherek ≤ n,K = 2 k , andφ = (a ij ,b ij ,c i ) K i,j=1 is a set of K 2 real parameters. Making the observable trainable strictly enlarges the func- tion class: a standard VQC with fixed local observables is provably a special case of ANO [18]. Moreover, ak-local observableH(φ)acts jointly onkqubits, coupling features across distant qubits and promoting an information mixture that single-qubit Pauli measurements cannot express. This expressivity makes ANO well-suited to the complex, high- dimensional correlations in MTSF. 2.4. MTSF-ANO MTSF-ANO has three parts: instance normalization, a data re- uploading VQC (DRVQC) with ANO, and a prediction head. 2.4.1. Instance Normalization We employ instance normalization [22], which is commonly used in prior forecasting work, to address the potential distri- |0⟩ 0 H R z (θ 0,1 )R y (θ 0,2 )R y (w 0,1 h 0 )R z (w 0,2 h 0 ) |0⟩ 1 H R z (θ 1,1 )R y (θ 1,2 )R y (w 1,1 h 1 )R z (w 1,2 h 1 ) |0⟩ 2 H R z (θ 2,1 )R y (θ 2,2 )R y (w 2,1 h 2 )R z (w 2,2 h 2 ) |0⟩ 3 H R z (θ 3,1 )R y (θ 3,2 )R y (w 3,1 h 3 )R z (w 3,2 h 3 ) Variational V (θ)Encoding U(h, w)H(φ) Fig. 1: VQC architecture of MTSF-ANO. Each layer con- sists a variational circuit of parameterizedR z ,R y rotations, cir- cular CNOT gates, and a data re-uploading encodingU(h, w) with trainable input scalingw. For measurement, trainable k-local observables H(φ) are employed. bution shift. Given X t ∈R C×L , we normalize each channel: X ′ t = (X t −μ)⊙ (σ 2 +ε) −1/2 .(3) Here,μ,σ 2 ∈R C are the per-channel mean and variance respectively,εensures numerical stability, and⊙denotes element-wise multiplication. 2.4.2. Data Re-Uploading VQC with ANO The normalized input is flattened and projected by a linear layer to a latent representationh t ∈R n , wherenis the num- ber of qubits. Starting from a layer of Hadamard gates, a DRVQC then transformsh t . Each VQC layer consists of three parts: parameterizedR z andR y rotation gates, a circular topology of CNOT gates for entanglement, along withR y and R z encoding gates that re-upload the latent input scaled by trainable parametersw(Fig. 1). For measurement, we use combinatorial measurement withk-local observables. Specifi- cally, we measure all n k combinations ofkqubits out of the navailable, producing an output value per combination and thereby accounting for multi-qubit correlations. 2.4.3. Prediction Head To transform the quantum representations into forecasts, we apply a linear layer followed by de-normalization: b Y t = (W out f VQC (h t ;θ,φ) + b out )⊙ p σ 2 +ε +μ,(4) wheref VQC (h t ;θ,φ)∈R ( n k ) is the vector of expectation val- ues from the quantum circuit. 3. EXPERIMENTAL SETTINGS 3.1. Dataset We conduct experiments on four widely-used real-world datasets from the Electricity Transformer Temperature (ETT) benchmark [23]. Statistics of the datasets are reported in Table 1. Following standard practice, we split each dataset into training, validation, and test sets with a 6:2:2 ratio. Table 1: Statistics of the datasets. DatasetsETTh1 & ETTh2ETTm1 & ETTm2 Variates77 Timesteps17,42069,680 Sample rate1 hour5 min 3.2. Evaluation Metrics Following prior work [12, 23], we report mean squared error (MSE) and mean absolute error (MAE), calculated asMSE = 1 C·H ∥Y t − b Y t ∥ 2 F andMAE = 1 C·H ∥Y t − b Y t ∥ 1 . Here,∥·∥ F and∥·∥ 1 denote the Frobenius and theL 1 -norm, respectively. 3.3. Baselines We benchmark MTSF-ANO against several classical and quantum baselines. Following prior works, we evaluate classical and quantum recurrent networks: LSTM, quantum LSTM (QLSTM) [13], and quantum fast weight programmers (QFWP) [14]. We also include state-of-the-art linear methods DLinear, NLinear, and Linear [24], which are strong baselines known to outperform many Transformer-based methods in long-term MTSF. We further compare against QuLTSF [12], a state-of-the-art quantum method for long-term MTSF. Finally, to isolate the effect of ANO measurement, we include an MTSF-PZ baseline that replaces thek-local measurement in MTSF-ANO with fixed Pauli-Z observables. The LSTM baseline uses 3 layers with 7 hidden units. QLSTM, QFWP, QuLTSF, and the three linear baselines are rerun using their official code. For a fair comparison, all recur- rent models use a linear prediction head for direct multi-step forecasting, and all baselines utilize instance normalization. 3.4. Hyperparameters For all quantum methods, the number of qubits equals to the channel countC, and the circuit depth is 3 layers. We train for 100 epochs with Adam (learning rate 0.001) using the MSE loss. Early stopping is applied with a patience of 10 epochs. 4. EXPERIMENTS 4.1. Main Results Table 2 summarizes the forecasting performance at a fixed lookbackL = 16, reporting the best result overk ∈1,..., 7 for each setting. Fig. 2 reports MSE as the lookback window varies atH = 48on ETTh1, likewise reporting the best result overk ∈3, 5. MTSF-ANO ranks first or second in terms of MSE in 17 of the 20 settings. Its advantage is largest on ETTh1, where it improves over the strongest baseline by 9–20%, and it remains competitive on ETTm1 and ETTm2. This advantage narrows as the lookbackLgrows, with the linear baselines starting to surpass MTSF-ANO. This is likely because project- ing a longer input down tonqubits becomes a bottleneck. To address it, we further introduce a channel-independent vari- ant, MTSF-ANO-CI, whose input and output projections use per-channel shared-weight linear layers. As Fig. 2 shows, MTSF-ANO and MTSF-ANO-CI are the strongest models for L∈8, 16andL = 24toL = 96, respectively. Finally, the comparisons against the MTSF-PZ baseline confirm that ANO effectively enhances MTSF capabilities (Table 3). Fig. 2: Impact of lookback window size (ETTh1). We report MSE withH = 48and varying lookback fromL = 8to L = 96. MTSF-ANO and its channel-independent variant, MTSF-ANO-CI, outperform all other models, with MTSF- ANO best atL∈8, 16and MTSF-ANO-CI best forL > 16. 4.2. Impact of ANO Non-Locality Table 4 reports the improvement over MTSF-PZ, averaged across the ETT datasets, as the non-localitykvaries. Perfor- mance increases steadily fromk = 1and reaches its peak betweenk = 3andk = 5, where MTSF-ANO demonstrates a 6.4% to 8.5% average improvement over MTSF-PZ. Inter- estingly, the gains collapse sharply atk = 7. We attribute this to our combinatorial ANO scheme: atk = 7, only a single expected value is evaluated and transformed into forecasts. These results also indicate that a moderate non-localitykis the sweet spot for our propose model. 4.3. Impact of VQC Architectures Table 5 ablates three architectural choices and reports MSE at H ∈1, 48 on ETTh1. First, entanglement helps: removing Table 2: Multivariate forecasting results. Results are averaged over 10 runs with different random seeds. Lookback window sizeL = 16and prediction horizonH ∈1, 5, 16, 32, 48. Lower MSE and MAE indicate better performance. The best result is highlighted in bold and the second best is highlighted withunderline. IMP. is the improvement between MTSF-ANO and the best baseline, where a larger value indicates a better improvement. MethodsIMP.MTSF-ANOMTSF-PZQuLTSFQLSTMQFWPLSTMDLinearNLinearLinear MetricMSEMSEMAEMSEMAEMSEMAEMSEMAEMSEMAEMSEMAEMSEMAEMSEMAEMSEMAE ETTh1 115.4%0.137 0.2430.1670.2720.1620.2540.3950.4100.3970.4060.2380.3320.1640.2580.1620.2560.1990.284 520.1%0.338 0.3710.4230.4180.4310.4090.6860.5200.7000.5220.4780.4490.4780.4290.4890.4320.5100.443 1611.2%0.388 0.4050.4530.4390.4370.4220.6070.4930.6620.5140.4690.4470.4590.4280.4870.4410.4740.435 3211.4%0.426 0.4270.4970.4600.4810.4470.6220.5010.6700.5230.4990.4630.4940.4470.5180.4580.5070.453 489.9%0.427 0.4250.4850.4530.4740.4410.6040.4980.6450.5140.4930.4590.4810.4390.5020.4500.4900.443 ETTh2 1-6.8%0.0790.1730.0900.1910.0740.1670.1120.2220.1170.2250.0970.2000.074 0.1660.074 0.1660.0760.170 50.8%0.121 0.2180.1310.2330.1220.2220.1500.2550.1550.2610.1360.2380.1260.2250.1260.2260.1270.228 16-1.7%0.180 0.2680.1810.2690.1770.2710.1920.2830.1930.2840.1830.2720.1770.2690.1790.2710.1780.270 32-1.4%0.2250.2950.2250.2950.2220.2970.2370.3070.2430.3140.2270.2970.222 0.2940.2230.2960.2220.295 48-1.9%0.2640.3180.2640.3180.2590.3180.2730.3270.2770.3310.2650.3190.259 0.3150.2610.3170.2590.316 ETTm1 14.0%0.048 0.1350.0500.1370.0510.1360.0730.1760.1050.2010.0540.1490.0520.1380.0520.1370.0520.138 56.8%0.110 0.2040.1180.2100.1280.2100.1820.2660.1980.2710.1190.2160.1330.2140.1320.2140.1330.214 164.7%0.328 0.3420.3700.3600.4320.3690.4560.4000.5200.4230.3440.3540.4510.3780.4510.3780.4520.378 322.0%0.594 0.4620.6690.4920.7880.5170.7710.5310.8590.5520.6060.4760.8350.5340.8360.5340.8360.534 48-1.8%0.7200.5220.8080.5530.9560.5880.9130.5900.9860.6080.7070.5251.0180.6101.0190.6101.0190.610 ETTm2 1-6.2%0.0340.1060.0350.1060.032 0.0950.0440.1270.0510.1360.0370.1130.032 0.0950.0320.0960.0320.096 5-1.7%0.0590.1410.0600.1440.058 0.1380.0710.1640.0730.1660.0620.1480.0600.1420.0600.1420.0600.142 162.9%0.100 0.1910.1030.1960.1040.1960.1080.2020.1180.2160.1040.1970.1080.2010.1080.2010.1080.201 324.6%0.146 0.2380.1530.2450.1580.2500.1610.2530.1680.2630.1540.2460.1620.2550.1620.2550.1620.255 483.8%0.179 0.2670.1860.2740.1950.2840.1960.2830.2050.2950.1860.2740.1980.2880.1980.2880.1980.288 Table 3: Improvement over MTSF-PZ. We report the IMP. of MTSF-ANO over MTSF-PZ from Table 2. Positive values indicate MTSF-ANO is better. Settings H = 1 H = 5 H = 16 H = 32 H = 48 ETTh118.0%20.1%14.3%14.3%12.0% ETTh212.2%7.6%0.6%0.0%0.0% ETTm14.0%6.8%11.4%11.2%10.9% ETTm22.9%1.7%2.9%4.6%3.8% Table 4: Impact of ANO non-locality. We report the IMP. over MTSF-PZ, averaged across the four ETT datasets, as the non-locality k varies. Bold indicates the best k for each H . Settings H = 1 H = 5 H = 16 H = 32 H = 48 k = 10.3%0.2%-0.3%-0.1%-0.7% k = 25.4%4.8%3.1%3.7%3.3% k = 37.4%8.5%5.4%5.7%5.2% k = 47.2%7.5%6.5%6.8%6.4% k = 58.0%7.8%7.3%7.4%6.3% k = 67.2%7.1%5.4%5.1%3.6% k = 7-97.5%-39.7%-24.9%-19.4%-18.0% the CNOT gates (w/o CNOT) degrades performance in nearly every setting, and the gap widens at higher non-locality and greater depth. Second, the trainable input scalingwyields marginally better results in most settings, but the difference is negligible. Third, compared to the default 3 layers, decreasing to 1 layer generally decreases MSE, while increasing layers generally increases it. Overall, entanglement and shallow depth stand out as the most beneficial architectural factors. Table 5: Impact of VQC architectures (ETTh1). We report MSE with and without trainable input scaling (left|right) and CNOT gates, and varying non-locality k and circuit depth. (a)H = 1 Architecturek = 3k = 5 Layerw/ CNOTw/o CNOTw/ CNOTw/o CNOT 10.137| 0.1380.138| 0.1390.139| 0.139 0.138| 0.140 30.141| 0.1420.142| 0.142 0.137| 0.1380.143| 0.142 50.142| 0.1450.153| 0.153 0.139| 0.1400.172| 0.175 (b)H = 48 Architecturek = 3k = 5 Layerw/ CNOTw/o CNOTw/ CNOTw/o CNOT 10.430| 0.4310.432| 0.4330.431| 0.4300.433| 0.433 30.437| 0.4360.440| 0.4390.431| 0.4300.441| 0.441 50.436| 0.4400.453| 0.454 0.431| 0.4310.462| 0.459 5. CONCLUSION This work introduced MTSF-ANO, a hybrid model that inte- grates adaptive non-local observables into a VQC for multi- variate time series forecasting. MTSF-ANO utilizes trainable non-local measurements and yields the best or second-best MSE in 17 of 20 settings across the four ETT datasets, sur- passing or matching its fixed local observable counterpart in all settings. A channel-independent variant further extends these gains to longer lookback windows. Our ablations iden- tify non-local measurement as the main driver of these gains, with entanglement and shallow depth also beneficial and non- locality showing a sweet spot. Our results establish ANO as a promising direction for quantum time series forecasting. 6. REFERENCES [1]M. Cerezo, Andrew Arrasmith, Ryan Babbush, Simon C. Ben- jamin, Suguru Endo, Keisuke Fujii, Jarrod R. McClean, Kosuke Mitarai, Xiao Yuan, Lukasz Cincio, and Patrick J. Coles, “Vari- ational quantum algorithms,” Nature Reviews Physics, vol. 3, no. 9, p. 625–644, Sep 2021. [2]Jarrod R McClean, Jonathan Romero, Ryan Babbush, and Al ́ an Aspuru-Guzik, “The theory of variational hybrid quantum- classical algorithms,” New Journal of Physics, vol. 18, no. 2, p. 023023, feb 2016. [3]Sofiene Jerbi, Casper Gyurik, Simon Marshall, Hans Briegel, and Vedran Dunjko, “Parametrized quantum policies for re- inforcement learning,” in Advances in Neural Information Processing Systems, M. Ranzato, A. Beygelzimer, Y. Dauphin, P.S. Liang, and J. Wortman Vaughan, Eds. 2021, vol. 34, p. 28362–28375, Curran Associates, Inc. [4]Gyu Seon Kim, Samuel Yen-Chi Chen, Soohyun Park, and Joongheon Kim, “Quantum reinforcement learning for coordi- nated satellite systems,” in ICASSP 2025 - 2025 IEEE Interna- tional Conference on Acoustics, Speech and Signal Processing (ICASSP), 2025, p. 1–5. [5]Yu-Ting Lee, Samuel Yen-Chi Chen, and Fu-Chieh Chang, “Quantum hierarchical reinforcement learning via variational quantum circuits,” 2026. [6]Samuel Yen-Chi Chen, Chao-Han Huck Yang, Jun Qi, Pin-Yu Chen, Xiaoli Ma, and Hsi-Sheng Goan, “Variational Quantum Circuits for Deep Reinforcement Learning,” IEEE Access, vol. 8, p. 141007–141024, 2020. [7]Owen Lockwood and Mei Si, “Reinforcement learning with quantum variational circuit,” Proceedings of the AAAI Confer- ence on Artificial Intelligence and Interactive Digital Entertain- ment, vol. 16, no. 1, p. 245–251, Oct. 2020. [8]Edward Farhi and Hartmut Neven, “Classification with quantum neural networks on near term processors,” 2018. [9]Adri ́ an P ́ erez-Salinas, Alba Cervera-Lierta, Elies Gil-Fuster, and Jos ́ e I. Latorre, “Data re-uploading for a universal quantum classifier,” Quantum, vol. 4, p. 226, Feb. 2020. [10]Jonathan Romero, Jonathan P Olson, and Alan Aspuru-Guzik, “Quantum autoencoders for efficient compression of quantum data,” Quantum Science and Technology, vol. 2, no. 4, p. 045001, aug 2017. [11] Marco Casalbore, Leonardo Lavagna, Antonello Rosato, and Massimo Panella, “Time series anomaly detection with quantum variational methods and set covering,” in ICASSP 2026 - 2026 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2026, p. 1846–1850. [12] Hari Hara Suthan Chittoor, Paul Robert Griffin, Ariel Neufeld, Jayne Thompson, and Mile Gu, “Qultsf: Long-term time series forecasting with quantum machine learning,” in Proceedings of the 17th International Conference on Agents and Artificial Intelligence - Volume 1: QAIO. INSTICC, 2025, p. 824–829, SciTePress. [13]Samuel Yen-Chi Chen, Shinjae Yoo, and Yao-Lung L. Fang, “Quantum long short-term memory,” in ICASSP 2022 - 2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), 2022, p. 8622–8626. [14]Samuel Yen-Chi Chen, “Learning to program variational quan- tum circuits with fast weights,” in 2024 International Joint Conference on Neural Networks (IJCNN), 2024, p. 1–9. [15]Andrea Ceschini, Antonello Rosato, Massimo Panella, and Samuel Yen-Chi Chen, “Quantum fast weight programming for time series prediction,” in ICASSP 2026 - 2026 IEEE Interna- tional Conference on Acoustics, Speech and Signal Processing (ICASSP), 2026, p. 22032–22036. [16] Samuel Yen-Chi Chen, Huan-Hsin Tseng, Hsin-Yi Lin, and Shinjae Yoo, “Learning to program quantum measurements for machine learning,” in 2025 IEEE International Conference on Quantum Computing and Engineering (QCE), 2025, vol. 01, p. 1826–1836. [17]Samuel Yen-Chi Chen, Huan-Hsin Tseng, Hsin-Yi Lin, and Shinjae Yoo, “Learning to measure quantum neural networks,” in 2025 IEEE International Conference on Acoustics, Speech, and Signal Processing Workshops (ICASSPW), 2025, p. 1–5. [18]Hsin-Yi Lin, Huan-Hsin Tseng, Samuel Yen-Chi Chen, and Shinjae Yoo, “Adaptive non-local observable on quantum neural networks,” in 2025 IEEE International Conference on Quantum Computing and Engineering (QCE), 2025, vol. 01, p. 1884– 1893. [19]Hsin-Yi Lin, Huan-Hsin Tseng, Samuel Yen-Chi Chen, and Shinjae Yoo, “Quantum super-resolution by adaptive non-local observables,” in ICASSP 2026 - 2026 IEEE International Con- ference on Acoustics, Speech and Signal Processing (ICASSP), 2026, p. 22027–22031. [20]Hsin-Yi Lin, Samuel Yen-Chi Chen, Huan-Hsin Tseng, and Shinjae Yoo, “Quantum reinforcement learning by adaptive non-local observables,” in 2025 IEEE International Conference on Quantum Computing and Engineering (QCE), 2025, vol. 02, p. 241–246. [21] Maria Schuld, Ryan Sweke, and Johannes Jakob Meyer, “Ef- fect of data encoding on the expressive power of variational quantum-machine-learning models,” Phys. Rev. A, vol. 103, p. 032430, Mar 2021. [22]Taesung Kim, Jinhee Kim, Yunwon Tae, Cheonbok Park, Jang- Ho Choi, and Jaegul Choo, “Reversible instance normalization for accurate time-series forecasting against distribution shift,” in International Conference on Learning Representations, 2022. [23] Haoyi Zhou, Shanghang Zhang, Jieqi Peng, Shuai Zhang, Jianxin Li, Hui Xiong, and Wancai Zhang, “Informer: Beyond efficient transformer for long sequence time-series forecasting,” Proceedings of the AAAI Conference on Artificial Intelligence, vol. 35, no. 12, p. 11106–11115, May 2021. [24]Ailing Zeng, Muxi Chen, Lei Zhang, and Qiang Xu, “Are transformers effective for time series forecasting?,” Proceedings of the AAAI Conference on Artificial Intelligence, vol. 37, no. 9, p. 11121–11128, Jun. 2023.