QARIMA: A Quantum Approach To Classical Time Series Analysis

Nishikanta Mohanty; Bikash K. Behera; Badshah Mukherjee; Pravat Dash; Giuseppe Sergioli; Roberto Giuntini

quant-phApr 9, 2026

QARIMA: A Quantum Approach To Classical Time Series Analysis

Authors: Nishikanta Mohanty, Bikash K. Behera, Badshah Mukherjee, Pravat Dash, Giuseppe Sergioli, Roberto Giuntini

Organizations: Centre for Quantum Software and Information, University of Technology Sydney, Sydney, NSW 2007, Australia · Bikash’s Quantum (OPC) Pvt. Ltd., Mohanpur, West Bengal 741246, India · Universit`a degli Studi di Cagliari, Cagliari 09123, Italy · Independent Researcher, Dubai 9262, United Arab Emirates ·

Abstract

We present QARIMA, a quantum state-similarity-based reconstruction of the classical ARIMA modelling pipeline. Rather than using a quantum circuit as a standalone forecaster, QARIMA preserves ARIMA's interpretable forecasting structure while reformulating its core building blocks through analogous quantum-compatible modules. The framework integrates quantum differencing assessment, QACF/QPACF lag discovery, compact-swap-test state projection, swap-test/VQC-based AR and MA coefficient estimation, and weak-lag refinement within a single ARIMA forecasting workflow. QACF and QPACF serve the functional roles of ACF and PACF for MA and AR lag discovery, but construct lag relevance through quantum measurement geometry rather than direct classical correlation. Given screened candidate orders $(p,d,q)$ , AR and MA coefficients are estimated through state-alignment losses incorporating cosine alignment, entropy regularization, phase correction, and norm control. We evaluate QARIMA across environmental, climatic, industrial, and weather time-series datasets using rolling-origin out-of-sample testing against automated classical ARIMA baselines, with performance assessed through MSE, MAPE, and Diebold--Mariano tests. The results show that quantum state-similarity modules can produce competitive and, in several cases, improved forecasting behaviour while preserving ARIMA's transparency, modularity, and interpretability. QARIMA therefore establishes a distinct pathway for quantum-enhanced statistical forecasting: its modules remain functionally analogous to classical ARIMA subroutines, but are not algebraic replicas; they serve the same modelling roles through state overlap, projection, and measurement-driven parameter estimation.

Figures & tables

Figure 1: Classical ARIMA forecasting workflow. The process consists of stationarity transformation, autocorrelation and partial-autocorrelation analysis for lag discovery, estimation of autoregressive and moving-average coefficients, and forecast generation using the fitted ARIMA model.

Figure 2: QARIMA forecasting workflow. The major stages of classical ARIMA are reformulated using quantum-compatible primitives including quantum differencing, swap-test-driven QACF/QPACF lag discovery, quantum-assisted coefficient estimation, and VQC-guided weak-lag refinement while preserving the ARIMA forecasting structure.

Component	Classical ARIMA	QARIMA
Dependency Discovery	ACF/PACF	QACF/QPACF
AR Estimation	OLS/MLE	Swap-Test Estimation
Weak Lag Refinement	None	VQC-guided
MA Estimation	Residual-based	Swap-Test Estimation
Forecasting Engine	ARIMA Equation	ARIMA Equation

Table 1: Key distinctions between classical ARIMA and QARIMA.

Approach	Main idea	Difference from QARIMA
VQC/PQC forecasting	Quantum circuits are used as direct regression or sliding-window predictors [ 6 , 7 , 8 ] .	QARIMA preserves the ARIMA forecasting equation while reformulating its internal modelling stages through quantum state-similarity modules.
Quantum recurrent models	Quantum circuits augment RNN/LSTM-style sequence learning [ 9 , 10 , 11 ] .	QARIMA remains statistically interpretable and does not rely on hidden-state recurrent learning.
Quantum reservoir computing	Quantum dynamics provide temporal memory and nonlinear readout features [ 12 , 13 , 14 ] .	QARIMA uses explicit QACF/QPACF and AR/MA estimation instead of reservoir dynamics.
Long-horizon and multivariate quantum forecasting	Hybrid quantum models target longer-range or multivariate forecasting applications [ 15 ] .	QARIMA focuses on a quantum-compatible reconstruction of the ARIMA pipeline.
Quantum forecasting benchmarks	Quantum forecasting models are compared against classical baselines under controlled validation protocols [ 16 ] .	QARIMA follows this discipline through fixed VQC configurations, rolling-origin evaluation, MSE/MAPE reporting, and Diebold-Mariano testing.

Table 2: Positioning of QARIMA relative to related quantum forecasting approaches.

Figure 3: Compact swap test estimating state similarity between encoded states $\ket{\phi}$ and $\ket{\psi}$ . The control qubit is prepared in $|0\rangle$ , followed by Hadamard, controlled-swap, Hadamard, and measurement. The measured probability $p_{0}^{\mathrm{meas}}$ is used to compute the overlap-magnitude score $\widehat{s}_{\mathrm{swap}}=\sqrt{2p_{0}^{\mathrm{meas}}-1}$ . Entropy regularization in the AR and MA losses uses a separate overlap-derived probability $\pi_{t}$ , not the raw measurement probability $p_{0}^{\mathrm{meas}}$ .

Symbol	Name	Description
$d_{\max}$	Max differencing depth	Maximum number of differencing levels tested.
$p$	Lag order	Number of past lags used to construct the delay matrix.
$\epsilon$	Convergence threshold	Threshold for detecting stabilization of the fitted $\gamma_{d}$ values across successive differencing levels.
$T$	Patience	Number of recent differencing levels used to confirm stabilization of $\gamma_{d}$ .
$I_{\max}$	Max optimizer iterations	Maximum optimizer iterations used to estimate $(\alpha_{d},\gamma_{d})$ at each fixed differencing level.
$\mathcal{L}$	Loss metric	Primary evaluation metric, such as MSE or MAE.

Table 3: Hyperparameters for Estimating Differencing Order

d

Symbol	Name	Description
$K$	max_lag	Maximum residual lag evaluated by QACF.
$S$	shots	Number of compact swap-test measurements per lag.
$\tau_{\mathrm{QACF}}$	acf_threshold	Empirical cutoff for retaining candidate MA orders; default value $0.15$ .
$\mathcal{Q}_{0}$	fallback q range	Bounded MA-order set used when no QACF lag passes the cutoff.
$\widehat{\rho}^{\,\mathrm{Q}}_{k}$	QACF score	State-similarity score between aligned residual vectors at lag $k$ .
$\mathcal{Q}_{\mathrm{cand}}$	candidate orders	MA-order set passed to the MA loss evaluation.

Table 4: Hyperparameters for Quantum-Inspired ACF Estimation

Symbol	Name	Description
$K$	max lag	Maximum lag evaluated by the QPACF screening routine.
$S$	shots	Number of compact swap-test measurements used for each aligned lag pair.
$T_{m}$	threshold mode	Thresholding strategy used to select candidate AR lags: static, percentile, or std.
$Z$	static numerator	Reference numerator for the static screening band, typically $1.96$ for a PACF-inspired $95\%$ reference band.
$\xi$	percentile level	Percentile level used for dynamic thresholding of QPACF magnitudes.
$c_{\sigma}$	std multiplier	Multiplier applied to the standard deviation of QPACF magnitudes in std mode.

Table 5: Hyperparameters for QPACF Partial-Lag Screening

Symbol	Name	Description
$S$	shots	Number of swap test measurement shots used for cosine similarity estimation.
$\lambda_{\text{cos}}$	lambda_cosine_align	Penalty weight applied to alignment difference between classical dot and swap-based cosine.
$\lambda_{\text{ent}}$	lambda_entropy	Penalty weight for quantum entropy derived from swap test probability.
$\omega$	phase_weight	Scaling factor to correct the phase discrepancy between classical and quantum cosine.
$\sigma$	noise_std	Standard deviation of added Gaussian noise to simulate quantum fluctuations.
$P$	padded_dim	Size of padded vector, adjusted to nearest power-of-two for swap test register.

Table 6: Hyperparameters for Quantum-Inspired AR Loss Estimation

\mathbf{b}^{(0)}\leftarrow\arg\min_{\mathbf{b}}\left\|\mathbf{y}^{(p)}-\mathbf{X}^{(p)}\mathbf{b}\right\|_{2}^{2}

Algorithm 6 VQC-Based AR Order Estimation

Symbol	Name	Description
$\mathcal{P}$	candidate orders	QPACF-screened candidate AR orders evaluated by the VQC refinement stage.
$d^{\star}$	selected differencing order	Differencing order selected by quantum state-similarity differencing assessment.
$\mathbf{X}^{(p)}$	AR design matrix	Lagged design matrix constructed for candidate order $p$ .
$\mathbf{y}^{(p)}$	aligned target vector	Target vector aligned with $\mathbf{X}^{(p)}$ after differencing and lag construction.
$\mathbf{b}^{(0)}$	OLS initialization	Initial AR coefficient vector obtained by ordinary least squares for each candidate order $p$ .
$\boldsymbol{\beta}$	VQC parameters	Trainable VQC rotation-parameter vector used to refine AR coefficients.

Table 7: Hyperparameters and Coefficient Map for VQC-Based AR Order Estimation

Figure 4: VQC for AR order estimation with $p$ qubits (one per lag). The first rotation layer uses OLS initialization $R_{y}(b_{i}^{(0)})$ ; then $r=2$ variational layers (shown) alternate trainable $R_{y}(\beta_{i}^{(\ell)})$ rotations with a linear entangling pattern (CNOT ladder). The optimized VQC parameters are mapped to the AR coefficient vector through $\mathbf{b}(\boldsymbol{\beta})=\mathrm{Proj}_{\mathcal{B}}(\boldsymbol{\beta})$ . These coefficients are subsequently used in the norm-scaled phase-corrected projection to compute the AR prediction, while training minimizes $\mathcal{L}_{\mathrm{AR}}$ consisting of prediction error, cosine-alignment, and entropy regularization.

Symbol	Name	Description
$p^{\star}$	best_p	AR order chosen by PACF + VQC refinement.
$\mathbf{b}^{\star}$	anchor_coeffs	Fixed anchor coefficients for AR( $p^{\star}$ ).
$k$	weak_count	Number of weak lags added to form $p^{\prime}=p^{\star}+k$ .
$\mathbf{w}$	weak_inits	Weak-lag initialization magnitudes (from summary, excluding $\|\mathbf{b}^{\star}\|$ ).
$r$	vqc_depth	Number of entangling layers in the weak-lag VQC block.
$T_{\max}$	max_iter	Max COBYLA iterations for minimizing $\mathcal{L}_{\mathrm{weak}}$ .

Table 8: Hyperparameters for VQC-Based Weak Lag Refinement

Figure 5: VQC for weak-lag refinement with $p^{\star}$ frozen anchors in the top block and $k$ trainable weak-lag qubits in the bottom block. Anchors are initialized as $R_{y}\big(b^{(p^{\star})}_{i}\big)$ and kept fixed, shown as a grouped frozen block. Only weak-lag qubits carry trainable $R_{y}(\beta^{(\ell)}_{\mathrm{w},j})$ rotations and are entangled through a CNOT ladder restricted to the weak block. Training minimizes $\mathcal{L}_{\text{weak}}=\mathcal{L}_{\mathrm{AR}}+\lambda_{\text{dev}}\lVert\mathbf{b}_{\text{weak}}-\mathbf{w}\rVert_{2}^{2}+\lambda_{\text{mag}}\lVert\mathbf{b}_{\text{weak}}\rVert_{1}$ .

Symbol	Name	Description
$\mathbf{w}$	weak_lags	Ordered list of weak-lag initialization values.
$r$	vqc_depth	Number of entangling layers in weak-lag VQC.
$T_{\max}$	max_iter	Maximum COBYLA iterations per refinement step.
$\lambda_{\mathrm{dev}},\lambda_{\mathrm{mag}}$	penalty_weights	Adaptive regularization weights based on $\mathcal{L}_{\mathrm{base}}$ .

Table 9: Hyperparameters for Progressive VQC-Based Weak Lag Refinement

Symbol	Name	Description
$\lambda_{\mathrm{cos}}$	cosine alignment weight	Penalty for misalignment between classical dot-product projection and phase-corrected swap-test projection.
$\lambda_{\mathrm{ent}}$	entropy weight	Weight for binary entropy regularization based on the overlap-derived MA uncertainty probability $\pi_{t}^{\mathrm{MA}}$ .
$\omega$	phase weight	Phase-correction factor for angular adjustment.
$S$	shots	Number of measurements used in the swap-test simulation.
$\lambda_{\mathrm{L2}}$	L2 weight	$\ell_{2}$ regularization weight on the magnitude of $\boldsymbol{\theta}$ .

Table 10: Hyperparameters for Quantum MA Loss

q^{\star}\leftarrow\arg\min_{q\in\mathcal{Q}_{\mathrm{cand}}}\mathcal{L}_{\mathrm{MA}}^{(q)\star}

Algorithm 11 Quantum MA Order Estimation

Symbol	Name	Description
$q_{\min},q_{\max}$	q range	Bounds of the fallback MA-order search.
$\mathcal{Q}_{\mathrm{QACF}}$	screened orders	Orders satisfying $\widehat{\rho}^{\,\mathrm{Q}}_{k}\geq\tau_{\mathrm{QACF}}$ .
$\mathcal{Q}_{0}$	fallback set	Bounded order set used when QACF selects no lag.
$\mathcal{Q}_{\mathrm{cand}}$	candidate orders	Final MA-order set passed to loss evaluation.
$\boldsymbol{\theta}^{(q,0)}$	theta init	Standard Gaussian initialization $\mathcal{N}(\mathbf{0},\mathbf{I}_{q})$ used during candidate-order screening.
$\mathcal{L}_{\mathrm{MA}}^{(q)}$	MA loss	Quantum MA loss for candidate order $q$ .

Table 11: Hyperparameters for MA Order Estimation

\boldsymbol{\eta}^{\star}\leftarrow\arg\min_{\boldsymbol{\eta}}\mathcal{L}_{\mathrm{MA}}\left(\boldsymbol{\theta}(\boldsymbol{\eta})\right)

Algorithm 12 VQC-Refined MA Training for

q^{\star}

Symbol	Name	Description
$\boldsymbol{\theta}^{(0)}$	MA coefficient initialization	Initial MA coefficient vector obtained from conditional least squares, clipped to $[-1,1]$ , with fallback to $\mathcal{U}(-1,1)$ .
$\boldsymbol{\eta}$	VQC parameters	Trainable VQC rotation-parameter vector used to refine the MA coefficients.
$\boldsymbol{\eta}^{(0)}$	VQC initialization	Initial VQC parameter vector, set from the MA coefficient initialization as $\boldsymbol{\eta}^{(0)}=\boldsymbol{\theta}^{(0)}$ .
$\boldsymbol{\theta}(\boldsymbol{\eta})$	MA coefficient map	MA coefficient vector obtained from VQC parameters through $\boldsymbol{\theta}(\boldsymbol{\eta})=\mathrm{Proj}_{\Theta}(\boldsymbol{\eta})$ .
$\mathrm{Proj}_{\Theta}(\cdot)$	coefficient projection	Elementwise projection or clipping operation enforcing the coefficient bounds $[-1,1]$ .
$[-1,1]$	coefficient bounds	Elementwise bounds imposed on each MA coefficient $\theta_{j}$ for numerical stability.

Table 12: Hyperparameters and Coefficient Map for VQC-Refined MA Training

Figure 6: Schematic VQC ansatz for MA coefficient refinement using build_vqc_ma with arguments $q^{\star}$ and $r$ . Each qubit $q_{i}$ receives a trainable single-qubit $R_{y}(\eta_{i})$ rotation, followed by an entangling CNOT chain. The block is repeated $r$ times, indicated by $\cdots$ . The trainable VQC parameter vector $\boldsymbol{\eta}$ is mapped to the MA coefficient vector through $\boldsymbol{\theta}(\boldsymbol{\eta})=\mathrm{Proj}_{\Theta}(\boldsymbol{\eta})$ . Compact swap-test projections are then used inside the MA loss evaluation for $\mathcal{L}_{\mathrm{MA}}$ .

Column	Type	Description
$p$	Integer	AR order selected or retained after QPACF-guided VQC refinement.
$d^{\star}$	Integer	Differencing order selected by the quantum state-similarity differencing assessment.
$q_{p}^{\star}$	Integer	MA order selected for the residual sequence associated with AR order $p$ .
$\boldsymbol{b}^{(p)}$	Vector	VQC-refined AR coefficient vector for order $p$ .
$\boldsymbol{\theta}^{(q_{p}^{\star})\star}$	Vector	VQC-refined MA coefficient vector for the selected residual MA order.
$\sigma_{\mathrm{AR}}^{(p)}$	Float	Standard deviation of the AR residuals before MA correction.

Table 13: Output Structure for Final ARMA Summary

Sunspots Classical Vs Quantum OOS MSE MAPE
Model	N	$MSE$	MAPE
Classical (pmdarima) (2, 0, 0) seasonal=None	128	2181.589	1.661
Quantum (pdq=(3, 1, 1))	128	2146.926	1.790
Quantum (pdq=(10, 1, 1))	128	2160.998	1.534
Quantum (pdq=(7, 1, 3))	128	2174.622	1.629
Quantum (pdq=(4, 1, 1))	128	2179.405	1.705

Table 14: Sunspots Classical Vs Quantum MSE MAPE

Panel of Figure 7 — Figure 7: Sunspots OOS MSE, MAPE. QARIMA vs. classical.

Panel of Figure 8 — Figure 8: Sunspots DM (MSE) p-values with $\alpha$ reference line; significant QARIMA bars appear in green.

Panel of Figure 9 — Figure 9: Sunspots DM (MAE) $-\log_{10}(p)$ with threshold at $-\log_{10}(\alpha)$ .

CO2 Classical Vs Quantum OOS
Model	N	MSE	MAPE
Classical (pmdarima non-seasonal) (5, 1, 0)	120	78.37204	0.022805
Quantum (pdq=(10, 1, 1))	120	10.02575	0.007472
Quantum (pdq=(9, 1, 1))	120	10.17655	0.007496
Quantum (pdq=(8, 1, 1))	120	10.45898	0.007573
Quantum (pdq=(3, 1, 1))	120	79.3785	0.022943

Table 15: CO2 Classical Vs Quantum OOS

Panel of Figure 10 — Figure 10: CO2 OOS MSE, MAPE. QARIMA vs. classical.

Panel of Figure 11 — Figure 11: CO2 DM (MSE) p-values with $\alpha$ reference line; significant QARIMA bars appear in green.

Panel of Figure 12 — Figure 12: CO2 DM (MAE) $-\log_{10}(p)$ with threshold at $-\log_{10}(\alpha)$ .

Model	N	MSE	MAPE
Classical (pmdarima non-seasonal) (0, 1, 1)	8	1491.762063	0.080529
Quantum (pdq=(7, 1, 1))	8	59.792337	0.016178
Quantum (pdq=(10, 1, 1))	8	76.190098	0.017387
Quantum (pdq=(9, 1, 1))	8	84.130823	0.017418
Quantum (pdq=(6, 1, 1))	8	95.932282	0.018874
Quantum (pdq=(3, 1, 3))	8	96.029641	0.017407

Table 16: AusBeer classical vs. quantum OOS (last 8 quarters).

Panel of Figure 13 — Figure 13: AusBeer OOS MSE and MAPE (last 8 quarters). QARIMA vs. classical ARIMA(0,1,1).

Panel of Figure 14 — Figure 14: AusBeer DM (MSE) p-values with $\alpha=0.05$ reference line; significant QARIMA models appear below the line.

Panel of Figure 15 — Figure 15: AusBeer DM (MAE) $-\log_{10}(p)$ with threshold at $-\log_{10}(\alpha)$ .

Model	N	MSE	MAPE
Classical (pmdarima non-seasonal) (6, 1, 0)	55	528229.5064	0.105291
Quantum (pdq=(9, 1, 1))	55	533331.3927	0.104924
Quantum (pdq=(6, 1, 1))	55	530506.1527	0.105351
Quantum (pdq=(8, 1, 1))	55	538097.7794	0.105399
Quantum (pdq=(5, 1, 1))	55	544440.1589	0.107432
Quantum (pdq=(10, 1, 1))	55	555406.3154	0.107201

Table 17: Woolyarn Classical vs Quantum OOS MSE & MAPE

Panel of Figure 16 — Figure 16: Woolyarn OOS MSE (left) and MAPE (right): QARIMA vs. classical ARIMA(6,1,0). The classical model is best; several quantum models are close but not better.

Panel of Figure 17 — Figure 17: Woolyarn DM (MSE): almost all quantum models are significantly different from classical; the difference is mostly in the direction of higher loss.

Panel of Figure 18 — Figure 18: Woolyarn DM (MAE): most quantum models remain significant; Q(6,1,1) and especially Q(5,1,1) are the only ones close to “no significant difference”.

Model	N	MSE	MAPE
Classical (pmdarima non-seasonal) (2, 0, 1)	336	11.435905	0.127427
Quantum (pdq=(3, 1, 1))	336	11.356242	0.126633
Quantum (pdq=(4, 1, 1))	336	11.393498	0.127034
Quantum (pdq=(5, 1, 1))	336	11.412022	0.127216
Quantum (pdq=(6, 1, 1))	336	11.423452	0.127318
Quantum (pdq=(9, 1, 1))	336	11.702693	0.129946

Table 18: Sydney 2024 Summer Temp: Classical vs. Quantum OOS (MSE & MAPE)

Panel of Figure 19 — Figure 19: Sydney Summer 2024 temperature: OOS MSE and MAPE, QARIMA vs. classical.

Panel of Figure 20 — Figure 20: Sydney Summer 2024 temperature: DM (MSE) p-values and $-\log_{10}(p)$ with $\alpha=0.05$ . Q(4,1,1) and Q(3,1,1) are significant.

Panel of Figure 21 — Figure 21: Sydney Summer 2024 temperature: DM (MAE) p-values and $-\log_{10}(p)$ . MA-heavy Q(1,1,6) and Q(2,1,6) beat the classical baseline.

Appendix figures & tables11 assets

Supplementary material from the paper’s appendix.

Appendix

\widehat{s}_{\mathrm{swap}}\leftarrow\sqrt{\max\left(2p_{0}^{\mathrm{meas}}-1,0\right)}

Appendix

Algorithm 15 Compact Swap-Test State-Similarity Projection

Sunspots Classical VS Quantum DM Stats MSE
Quantum (pdq)	blocks_used	loss	DM_stat	p_value	classical_mean_loss	quantum_mean_loss	delta_mean_loss
(10, 1, 1)	10	MSE	4.941708	0.0000007744	2411.531786	1563.023210	848.508576
(9, 1, 3)	10	MSE	4.831355	0.0000013561	2411.531786	1608.609316	802.922470
(8, 1, 3)	10	MSE	4.812065	0.0000014938	2411.531786	1613.363812	798.167975
(6, 1, 3)	10	MSE	4.680626	0.0000028600	2411.531786	1603.130658	808.401129
(7, 1, 3)	10	MSE	4.608440	0.0000040570	2411.531786	1629.559531	781.972255

Appendix

Table 19: Sunspots Classical VS Quantum - DM Stats MSE

Sunspots Classical VS Quantum DM Stats MAE
Quantum pdq	blocks_used	loss	DM_stat	p_value	classical_mean_loss	quantum_mean_loss	delta_mean_loss
(10, 1, 1)	10	MAE	4.908669	0.0000009170	35.734117	28.387620	7.346497
(8, 1, 3)	10	MAE	4.609235	0.0000040415	35.734117	29.106819	6.627298
(9, 1, 3)	10	MAE	4.602818	0.0000041681	35.734117	29.066595	6.667522
(6, 1, 3)	10	MAE	4.539380	0.0000056420	35.734117	28.866935	6.867182
(7, 1, 3)	10	MAE	4.324537	0.0000152853	35.734117	29.306929	6.427188

Appendix

Table 20: Sunspots Classical VS Quantum - DM Stats MAE

Woolyarn Classical VS Quantum DM Stats MSE
Quantum pdq	blocks_used	loss	DM_stat	p_value	classical_mean_loss	quantum_mean_loss	delta_mean_loss
(2, 1, 1)	1	MSE	-7.968394	0.0000000000	934889.823	1136324	-201434.0761
(1, 1, 1)	1	MSE	-7.952058	0.0000000000	934889.823	1185477	-250586.7362
(3, 1, 1)	1	MSE	-7.245462	0.0000000000	934889.823	1131699	-196809.3958
(4, 1, 1)	1	MSE	-6.427643	0.0000000001	934889.823	1042081	-107190.8507
(7, 1, 1)	1	MSE	-4.500098	0.0000067922	934889.823	1343679	-408789.1725

Appendix

Table 21: Woolyarn Classical VS Quantum - DM Stats MSE

Woolyarn Classical VS Quantum DM Stats MAE
Quantum pdq	blocks_used	loss	DM_stat	p_value	classical_mean_loss	quantum_mean_loss	delta_mean_loss
(3, 1, 1)	1	MAE	-7.36933	0.0000000000	826.718879	904.906818	-78.187938
(4, 1, 1)	1	MAE	-7.148529	0.0000000000	826.718879	865.442915	-38.724036
(2, 1, 1)	1	MAE	-6.434141	0.0000000001	826.718879	902.821988	-76.103109
(1, 1, 1)	1	MAE	-5.220583	0.0000001784	826.718879	926.307162	-99.588282
(7, 1, 1)	1	MAE	-4.272585	0.0000193220	826.718879	994.530279	-167.811399

Appendix

Table 22: Woolyarn Classical VS Quantum - DM Stats MAE

CO 2 Classical vs Quantum - DM Stats (MSE)
Quantum (pdq)	blocks_used	loss	DM_stat	p_value	classical_mean_loss	quantum_mean_loss	delta_mean_loss
(8, 1, 1)	174	MSE	8.681193	0.00E+00	9.080382	3.599858	5.480524
(9, 1, 1)	172	MSE	8.991140	0.00E+00	9.099370	3.282792	5.816578
(10, 1, 1)	174	MSE	9.194297	0.00E+00	9.080382	3.192490	5.887893
(5, 1, 1)	174	MSE	-6.729980	1.70E-11	9.080382	9.236257	-0.155875
(6, 1, 1)	174	MSE	-6.673577	2.50E-11	9.080382	9.247513	-0.167131

Appendix

Table 23: CO 2 Classical vs Quantum - DM Stats (MSE)

CO 2 Classical VS Quantum DM Stats MAE
Quantum pdq	blocks_used	loss	DM_stat	p_value	classical_mean_loss	quantum_mean_loss	delta_mean_loss
(8, 1, 1)	174	MAE	9.252065	0.00E+00	2.315727	1.406047	0.909681
(9, 1, 1)	172	MAE	9.561353	0.00E+00	2.314338	1.348458	0.965881
(10, 1, 1)	174	MAE	9.857719	0.00E+00	2.315727	1.324859	0.990868
(3, 1, 1)	174	MAE	-5.639472	1.71E-08	2.315727	2.338754	-0.023027
(7, 1, 1)	174	MAE	5.486284	4.10E-08	2.315727	1.887741	0.427986

Appendix

Table 24: CO 2 Classical VS Quantum - DM Stats MAE

AusBeer Classical VS Quantum DM Stats MSE
Quantum pdq	blocks_used	loss	DM_stat	p_value	classical_mean_loss	quantum_mean_loss	delta_mean_loss
(1, 1, 6)	84	MSE	11.878028	0.00E+00	2559.328583	1296.194617	1263.133966
(2, 1, 6)	84	MSE	15.353045	0.00E+00	2559.328583	564.606641	1994.721942
(7, 1, 1)	84	MSE	15.626542	0.00E+00	2559.328583	388.120011	2171.208572
(6, 1, 1)	84	MSE	15.626973	0.00E+00	2559.328583	412.991120	2146.337463
(5, 1, 1)	84	MSE	15.708202	0.00E+00	2559.328583	408.258903	2151.069680

Appendix

Table 25: AusBeer Classical VS Quantum - DM Stats (MSE)

AusBeer Classical VS Quantum DM Stats MAE
Quantum pdq	blocks_used	loss	DM_stat	p_value	classical_mean_loss	quantum_mean_loss	delta_mean_loss
(1, 1, 6)	84	MAE	11.772301	0.00E+00	41.666725	29.306057	12.360668
(2, 1, 6)	84	MAE	18.632163	0.00E+00	41.666725	18.651992	23.014733
(3, 1, 3)	84	MAE	19.375578	0.00E+00	41.666725	16.011428	25.655297
(6, 1, 1)	84	MAE	19.859260	0.00E+00	41.666725	16.104594	25.562131
(7, 1, 1)	84	MAE	20.302789	0.00E+00	41.666725	15.588372	26.078353

Appendix

Table 26: AusBeer Classical VS Quantum - DM Stats (MAE)

Sydney 2024 Summer Temp Classical VS Quantum DM Stats MSE
Quantum pdq	blocks_used	loss	DM_stat	p_value	classical_mean_loss	quantum_mean_loss	delta_mean_loss
(4, 1, 1)	160	MSE	-3.41981	6.27E-04	8.1677	9.023377	-0.855678
(3, 1, 1)	161	MSE	-2.943498	3.25E-03	8.177383	19.809067	-11.631684
(5, 1, 1)	161	MSE	-2.456801	1.40E-02	8.177383	8.713587	-0.536204
(6, 1, 1)	161	MSE	-1.544826	1.22E-01	8.177383	8.668534	-0.491151
(10, 1, 1)	155	MSE	0.610649	5.41E-01	8.230546	7.906464	0.324082

Appendix

Table 27: Sydney 2024 Summer Temp - Classical vs. Quantum DM Stats (MSE)

Sydney 2024 Summer Classical VS Quantum DM Stats MAE
Quantum pdq	blocks_used	loss	DM_stat	p_value	classical_mean_loss	quantum_mean_loss	delta_mean_loss
(3, 1, 1)	161	MAE	-4.038151	5.40E-05	2.183719	2.875055	-0.691336
(4, 1, 1)	160	MAE	-2.975823	2.92E-03	2.180798	2.280019	-0.099221
(5, 1, 1)	161	MAE	-1.775833	7.58E-02	2.183719	2.245571	-0.061852
(10, 1, 1)	155	MAE	1.205682	2.28E-01	2.193426	2.106318	0.087108
(6, 1, 1)	161	MAE	-1.158140	2.47E-01	2.183719	2.237213	-0.053494

Appendix

Table 28: Sydney 2024 Summer Classical VS Quantum DM Stats (MAE)

Time Series Forecasting

Time series forecasting predicts future values from ordered observations using statistical, neural, and probabilistic models to support decisions under uncertainty.

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Quantum-classical hybrid models based on error correction for time series forecasting

Jonathan H. A. de Carvalho, Filipe C. de L. Duarte, Fernando M. de Paula Neto, Paulo S. G. de Mattos Neto

Centro de Inform´atica, Universidade Federal de Pernambuco, Recife, Brazil · Departamento de Finan¸cas e Contabilidade, Universidade Federal da Para´ıba, Jo˜ao Pessoa, Brazil

Time Series Forecasting Quantum Machine Learning Hybrid Quantum-Classical ML

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Time series forecasting largely benefits from combining the strengths of different models, especially using a scheme where a model corrects another model by capturing supplementary patterns from forecasting errors. Concurrently, quantum models are providing a means to augment the classical capacity, including in time series forecasting, by acting alongside classical models in hybrid architectures. In this work, we propose the first forecasting system based on error correction that jointly uses quantum and classical models. Here, quantum models first extract patterns by exploring quantum phenomena, and classical models capture the remaining patterns from the quantum errors. Compared to classical single models and classical-classical hybrid models based on error correction, the complementary capacity that emerges from this quantum-classical system provided the best results in most of the addressed problems. Therefore, this work paves the way to introduce quantum models in established hybridization schemes for time series forecasting.

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Time Series Forecasting

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