Quantum computational sensing using quantum signal processing, quantum neural networks, and Hamiltonian engineering

npj Quantum Inf. 12, 119 (2026) · DOI: 10.1038/s41534-026-01235-w · arXiv: 2507.15845

License: CC BY 4.0.

Abstract

Combining quantum sensing with quantum computing can lead to quantum computational sensing (QCS) protocols that are able to more efficiently extract task-specific information from physical signals than is possible otherwise. In this paper, we present, in theory and numerical simulations, the application of two quantum algorithms—quantum signal processing and quantum neural networks—to various binary and multiclass machine-learning classification tasks in sensing. Here, sensing operations are interleaved with computing operations, giving rise to nonlinear functions of the sensed signals. Our approach to optimizing QCS protocols takes into account quantum sampling noise and allows us to engineer protocols that can yield accurate results with as few as just a single measurement shot. In all cases, we have been able to show a regime of operation where a quantum computational sensor can achieve higher accuracy than a conventional quantum sensor for a given budget of sensing time, with a simulated accuracy advantage of >20 percentage points for some tasks. We also present protocols for performing nonlinear tasks using Hamiltonian-engineered bosonic systems and quantum signal processing with hybrid qubit-bosonic systems, and empirically show an advantage when the received signal has a limited mean photon number.

Figures

69 panels with data across 5 figures. Each panel page shows the plot, its columns and its files; each data.csv begins with a header naming the paper, the panel, the source, the license and the provenance route.

Fig. 1

Illustrative figure, no extractable data. Shown in the paper PDF. Page 2 of the PDF.

Fig. 2

• panel (a): Target function $\mathcal{F}^{\star}(\theta)$ of the single-variable binary classification task against the phase $\theta$ from 0 to $\pi/4$: 0 on the Class 1 intervals and 1 on the Class 2 intervals, which the coloured line at 0.5 marks (blue Class 1, orange Class 2). data.csv
• panel (c): Expected measurement outcome $x_{\mathrm{QS}}$ of the conventional quantum sensor (Ramsey interferometer) against $\theta$, with the target $\mathcal{F}^{\star}(\theta)$ in grey. The print colours the curve by its value on the qubit excitation probability scale; here it is one colour. data.csv
• panel (e-1): Expected measurement outcome $x_{\mathrm{QCS}}$ of the trained quantum computational sensor with $L = 2^4$ layers ($S = 1$) against $\theta$, with the target $\mathcal{F}^{\star}(\theta)$ in grey. The print colours the curve by its value; here it is one colour. data.csv
• panel (e-2): Expected measurement outcome $x_{\mathrm{QCS}}$ of the trained quantum computational sensor with $L = 2^5$ layers ($S = 1$) against $\theta$, with the target $\mathcal{F}^{\star}(\theta)$ in grey. The print colours the curve by its value; here it is one colour. data.csv
• panel (e-3): Expected measurement outcome $x_{\mathrm{QCS}}$ of the trained quantum computational sensor with $L = 2^6$ layers ($S = 1$) against $\theta$, with the target $\mathcal{F}^{\star}(\theta)$ in grey. The print colours the curve by its value; here it is one colour. data.csv
• panel (f): Classification error (%) against the number of sensing periods $N$ (log-log) for conventional sensing with a Ramsey interferometer (QS (R)) and with quantum phase estimation (QS (QPE), dashed, open circles) and for quantum computational sensing with $L = 2^3$ to $2^6$ layers. Error bars are the spread over trained protocols; the grey line marks $N = 2^6$. data.csv
• panel (f-inset): Inset of panel (f): classification error (%) at fixed $N = 2^6$ for QS (R), QS (QPE) and quantum computational sensing with $L = 2^3$ to $2^6$, with error bars over trained protocols. The print hatches the QS (QPE) bar; here it is drawn lighter. data.csv

Fig. 3

• panel (a): Training set of the binary Logspirals task: phase pairs $(\theta_1, \theta_2)$ sampled for Class 1 (blue) and Class 2 (orange). data.csv
• panel (d-1): Expected probability $x_{10}$ of measuring bitstring 10 over the phases $(\theta_1, \theta_2)$ for the conventional QS scheme with $N = S = 2^5$. data.csv
• panel (d-2): Expected probability $x_{01}$ of measuring bitstring 01 over the phases $(\theta_1, \theta_2)$ for the conventional QS scheme with $N = S = 2^5$. data.csv
• panel (d-3): Expected probability $x_{11}$ of measuring bitstring 11 over the phases $(\theta_1, \theta_2)$ for the conventional QS scheme with $N = S = 2^5$. data.csv
• panel (d-4): Expected probability $x_{00}$ of measuring bitstring 00 over the phases $(\theta_1, \theta_2)$ for the conventional QS scheme with $N = S = 2^5$. data.csv
• panel (e-1): Expected probability $x_{10}$ of measuring bitstring 10 over the phases $(\theta_1, \theta_2)$ for the QCS scheme with $N = L \times S = 2^5$, $L = 2^5$, $S = 1$. data.csv
• panel (e-2): Expected probability $x_{01}$ of measuring bitstring 01 over the phases $(\theta_1, \theta_2)$ for the QCS scheme with $N = L \times S = 2^5$, $L = 2^5$, $S = 1$. data.csv
• panel (e-3): Expected probability $x_{11}$ of measuring bitstring 11 over the phases $(\theta_1, \theta_2)$ for the QCS scheme with $N = L \times S = 2^5$, $L = 2^5$, $S = 1$. data.csv
• panel (e-4): Expected probability $x_{00}$ of measuring bitstring 00 over the phases $(\theta_1, \theta_2)$ for the QCS scheme with $N = L \times S = 2^5$, $L = 2^5$, $S = 1$. data.csv
• panel (f): Class predicted by the conventional QS protocol, $N = S = 2^5$ with classical postprocessing over $(\theta_1, \theta_2)$ (shaded), with test points in their class colour where classified correctly and black where not. One sampled realization, not the printed one. data.csv
• panel (f-inset): Inset of panel (f): zoom of the predicted-class map and test points near the origin, misclassified points black. One sampled realization, not the printed one. data.csv
• panel (g): Class predicted by the QCS protocol, $L = 2^5$, $S = 1$ with classical postprocessing over $(\theta_1, \theta_2)$ (shaded), with test points in their class colour where classified correctly and black where not. One sampled realization, not the printed one. data.csv
• panel (g-inset): Inset of panel (g): zoom of the predicted-class map and test points near the origin, misclassified points black. One sampled realization, not the printed one. data.csv
• panel (h): Classification error (%) against the number of sensing periods $N$ (log-log) for the binary Logspirals task: QS with a two-qubit Ramsey-variational interferometer (QS (R-V)), QS with quantum phase estimation (QS (QPE), dashed, open circles) and QCS with $L = 2^2$ to $2^5$; error bars over trained protocols. The grey line marks $N = 2^5$. data.csv
• panel (h-inset): Inset of panel (h): classification error (%) at fixed $N = 2^5$ for QS (R-V), QS (QPE) and QCS with $L = 2^2$ to $2^5$, with error bars. The print hatches the QS (QPE) bar; here it is drawn lighter. data.csv
• panel (i): Training set of the four-class Logspirals task: phase pairs $(\theta_1, \theta_2)$ sampled for Classes 1 to 4 (blue, orange, teal, red). data.csv
• panel (l-1): Expected probability $x_{10}$ of measuring bitstring 10 over the phases $(\theta_1, \theta_2)$ for the conventional QS scheme with $N = S = 2^6$, four-class task. data.csv
• panel (l-2): Expected probability $x_{01}$ of measuring bitstring 01 over the phases $(\theta_1, \theta_2)$ for the conventional QS scheme with $N = S = 2^6$, four-class task. data.csv
• panel (l-3): Expected probability $x_{11}$ of measuring bitstring 11 over the phases $(\theta_1, \theta_2)$ for the conventional QS scheme with $N = S = 2^6$, four-class task. data.csv
• panel (l-4): Expected probability $x_{00}$ of measuring bitstring 00 over the phases $(\theta_1, \theta_2)$ for the conventional QS scheme with $N = S = 2^6$, four-class task. data.csv
• panel (m-1): Expected probability $x_{10}$ of measuring bitstring 10 over the phases $(\theta_1, \theta_2)$ for the QCS scheme with $N = L \times S = 2^6$, $L = 2^6$, $S = 1$, four-class task. data.csv
• panel (m-2): Expected probability $x_{01}$ of measuring bitstring 01 over the phases $(\theta_1, \theta_2)$ for the QCS scheme with $N = L \times S = 2^6$, $L = 2^6$, $S = 1$, four-class task. data.csv
• panel (m-3): Expected probability $x_{11}$ of measuring bitstring 11 over the phases $(\theta_1, \theta_2)$ for the QCS scheme with $N = L \times S = 2^6$, $L = 2^6$, $S = 1$, four-class task. data.csv
• panel (m-4): Expected probability $x_{00}$ of measuring bitstring 00 over the phases $(\theta_1, \theta_2)$ for the QCS scheme with $N = L \times S = 2^6$, $L = 2^6$, $S = 1$, four-class task. data.csv
• panel (n): Class predicted by the conventional QS protocol, $N = S = 2^6$ with classical postprocessing over $(\theta_1, \theta_2)$ (shaded), with test points in their class colour where classified correctly and black where not. One sampled realization, not the printed one. data.csv
• panel (n-inset): Inset of panel (n): zoom of the predicted-class map and test points near the origin, misclassified points black. One sampled realization, not the printed one. data.csv
• panel (o): Class predicted by the QCS protocol, $L = 2^6$, $S = 1$ with classical postprocessing over $(\theta_1, \theta_2)$ (shaded), with test points in their class colour where classified correctly and black where not. One sampled realization, not the printed one. data.csv
• panel (o-inset): Inset of panel (o): zoom of the predicted-class map and test points near the origin, misclassified points black. One sampled realization, not the printed one. data.csv
• panel (p): Classification error (%) against the number of sensing periods $N$ (log-log) for the four-class Logspirals task: QS (R-V), QS (QPE) (dashed, open circles) and QCS with $L = 2^3$ to $2^6$; error bars over trained protocols. The grey line marks $N = 2^6$. data.csv
• panel (p-inset): Inset of panel (p): classification error (%) at fixed $N = 2^6$ for QS (R-V), QS (QPE) and QCS with $L = 2^3$ to $2^6$, with error bars. The print hatches the QS (QPE) bar; here it is drawn lighter. data.csv

Fig. 4

• panel (a): Example MEG signals $s_j(t)$ from $M = 7$ sensors against time (s), for class 1 (blue, top) and class 2 (orange, bottom). The print has no amplitude scale: amplitudes are in arbitrary units, one unit per stacking step, and each trace keeps its printed offset (class 2 at 0 to 6, class 1 at 8 to 14, $s_1$ on top). The two printed axes are drawn as one. data.csv
• panel (d): Classification error (%) on a log scale against signal strength (average imparted phase $\theta$) for $M = 7$ qubits: Ramsey interferometry and the temporally-, spatially- and spatiotemporally-coherent schemes, with error bars. The grey line marks the strength used in panel (e). data.csv
• panel (e): Classification error (%) at the signal strength marked in panel (d), for $M = 3$, 5 and 7 sensing qubits, for Ramsey interferometry and the temporally-, spatially- and spatiotemporally-coherent schemes, with error bars. The print draws horizontal bars and shades each scheme's gain over Ramsey; here the bars are vertical and unshaded. data.csv

Fig. 5

• panel (d-1): Single-shot estimates $\mathcal{F}_{\mathrm{QS}}$ (dots) of the cubic $\alpha_x^3$ against $\alpha_x$, with the band $\mathbb{E} \pm \sqrt{\mathrm{Var}}$ and the target $\mathcal{F}^{\star}$ (red dashed). The dots are one sampled realization, not the printed one. data.csv
• panel (d-2): Single-shot estimates $\mathcal{F}_{\mathrm{QS}}$ (dots) of the quintic $\alpha_x^5$ against $\alpha_x$, with the band $\mathbb{E} \pm \sqrt{\mathrm{Var}}$ and the target $\mathcal{F}^{\star}$ (red dashed). The dots are one sampled realization, not the printed one. data.csv
• panel (d-3): Single-shot estimates $\mathcal{F}_{\mathrm{QS}}$ (dots) of the degree-7 polynomial $\alpha_x^7 + \alpha_x^6 - 2\alpha_x^5 + \alpha_x^3 - 2\alpha_x^2$ against $\alpha_x$, with the band $\mathbb{E} \pm \sqrt{\mathrm{Var}}$ and the target $\mathcal{F}^{\star}$ (red dashed). The dots are one sampled realization, not the printed one. data.csv
• panel (d-4): Single-shot estimates $\mathcal{F}_{\mathrm{QCS}}$ (dots) of the cubic $\alpha_x^3$ against $\alpha_x$, with the band $\mathbb{E} \pm \sqrt{\mathrm{Var}}$ and the target $\mathcal{F}^{\star}$ (red dashed). The dots are one sampled realization, not the printed one. data.csv
• panel (d-5): Single-shot estimates $\mathcal{F}_{\mathrm{QCS}}$ (dots) of the quintic $\alpha_x^5$ against $\alpha_x$, with the band $\mathbb{E} \pm \sqrt{\mathrm{Var}}$ and the target $\mathcal{F}^{\star}$ (red dashed). The dots are one sampled realization, not the printed one. data.csv
• panel (d-6): Single-shot estimates $\mathcal{F}_{\mathrm{QCS}}$ (dots) of the degree-7 polynomial $\alpha_x^7 + \alpha_x^6 - 2\alpha_x^5 + \alpha_x^3 - 2\alpha_x^2$ against $\alpha_x$, with the band $\mathbb{E} \pm \sqrt{\mathrm{Var}}$ and the target $\mathcal{F}^{\star}$ (red dashed). The dots are one sampled realization, not the printed one. data.csv
• panel (d-7): Expected mean-squared error $\mathbb{E}[\mathrm{MSE}]$ (log scale) of the QS (yellow) and QCS (blue) estimators for $\mathcal{F}^{\star} = \alpha_x^2$ to $\alpha_x^6$ and the degree-7 polynomial. data.csv
• panel (e-1): Training set of the XOR task in the plane of the sensed displacement $\alpha = \alpha_x + i\alpha_y$: Class 1 (blue) and Class 2 (orange). data.csv
• panel (e-2): Single instance of the estimator $\mathcal{F}_{\mathrm{QS}}$ for the XOR task passed through the thresholding function (shaded class) over $(\alpha_x, \alpha_y)$, with the training points. One sampled realization, not the printed one. data.csv
• panel (e-3): Squared error $(\mathcal{F}_{\mathrm{QS}} - \mathcal{F}^{\star})^2$ of that thresholded instance, normalized to the QS maximum, over $(\alpha_x, \alpha_y)$ (white 0, black 1). One sampled realization, not the printed one. data.csv
• panel (e-4): Target $\mathcal{F}^{\star}$ of the XOR task passed through the thresholding function, over $(\alpha_x, \alpha_y)$: Class 1 blue, Class 2 orange. data.csv
• panel (e-5): Single instance of the estimator $\mathcal{F}_{\mathrm{QCS}}$ for the XOR task passed through the thresholding function (shaded class) over $(\alpha_x, \alpha_y)$, with the training points. One sampled realization, not the printed one. data.csv
• panel (e-6): Squared error $(\mathcal{F}_{\mathrm{QCS}} - \mathcal{F}^{\star})^2$ of that thresholded instance, normalized to the QS maximum, over $(\alpha_x, \alpha_y)$ (white 0, black 1). One sampled realization, not the printed one. data.csv
• panel (e-7): Classification error (%) of QS (yellow) and QCS (blue) for the XOR task. Computed from sampled measurements, so it can differ slightly from the printed bars. data.csv
• panel (e-8): Expected mean-squared error $\mathbb{E}[\mathrm{MSE}]$ over the input dataset of QS (yellow) and QCS (blue) for the XOR task. data.csv
• panel (f-1): Training set of the Spirals task in the plane of the sensed displacement $\alpha = \alpha_x + i\alpha_y$: Class 1 (blue) and Class 2 (orange). data.csv
• panel (f-2): Single instance of the estimator $\mathcal{F}_{\mathrm{QS}}$ for the Spirals task passed through the thresholding function (shaded class) over $(\alpha_x, \alpha_y)$, with test points in their class colour, misclassified ones black. One sampled realization, not the printed one. data.csv
• panel (f-3): Squared error $(\mathcal{F}_{\mathrm{QS}} - \mathcal{F}^{\star})^2$ of that thresholded instance, normalized to the QS maximum, over $(\alpha_x, \alpha_y)$ (white 0, black 1). One sampled realization, not the printed one. data.csv
• panel (f-4): Target $\mathcal{F}^{\star}$ of the Spirals task passed through the thresholding function, over $(\alpha_x, \alpha_y)$: Class 1 blue, Class 2 orange. data.csv
• panel (f-5): Single instance of the estimator $\mathcal{F}_{\mathrm{QCS}}$ for the Spirals task passed through the thresholding function (shaded class) over $(\alpha_x, \alpha_y)$, with test points in their class colour, misclassified ones black. One sampled realization, not the printed one. data.csv
• panel (f-6): Squared error $(\mathcal{F}_{\mathrm{QCS}} - \mathcal{F}^{\star})^2$ of that thresholded instance, normalized to the QS maximum, over $(\alpha_x, \alpha_y)$ (white 0, black 1). One sampled realization, not the printed one. data.csv
• panel (f-7): Classification error (%) of QS (yellow) and QCS (blue) for the Spirals task. Computed from sampled measurements, so it can differ slightly from the printed bars. data.csv
• panel (f-8): Expected mean-squared error $\mathbb{E}[\mathrm{MSE}]$ over the input dataset of QS (yellow) and QCS (blue) for the Spirals task. data.csv

Fig. 6

• panel (a): Training set of the Circles task in the plane of the sensed displacement $\alpha = \alpha_x + i\alpha_y$: Class 1 (blue) and Class 2 (orange). data.csv
• panel (b-1): Left map of panel (b): mean output quadrature $\langle \hat{X}_a \rangle$ of the conventional QS sensor over the sensed displacement $(\alpha_x, \alpha_y)$, divided by $\max|\alpha_{x,y}|$ so the colour scale runs from $-1$ to $1$. The print labels the scale only as $-\max|\alpha_{x,y}|$ to $\max|\alpha_{x,y}|$ and prints no axis numbers. data.csv
• panel (b-2): Right map of panel (b): mean output quadrature $\langle \hat{P}_b \rangle$ of the conventional QS sensor over the sensed displacement $(\alpha_x, \alpha_y)$, divided by $\max|\alpha_{x,y}|$ so the colour scale runs from $-1$ to $1$. The print labels the scale only as $-\max|\alpha_{x,y}|$ to $\max|\alpha_{x,y}|$ and prints no axis numbers. data.csv
• panel (c): Qubit excitation probability $x_{\mathrm{QCS}}$ of the trained hybrid quantum computational sensor ($D = 16$) over the sensed displacement $(\alpha_x, \alpha_y)$. data.csv
• panel (d-1): Class predicted by the conventional bosonic sensor (two-mode squeezing, beam splitter, quadrature readout) with a classical neural network over $(\alpha_x, \alpha_y)$ (shaded), with test points in their class colour where classified correctly and black where not. One sampled realization, not the printed one. data.csv
• panel (d-2): Class predicted by the hybrid quantum computational sensor ($D = 16$) with a classical neural network over $(\alpha_x, \alpha_y)$ (shaded), with test points in their class colour where classified correctly and black where not. One sampled realization, not the printed one. data.csv

Cite

Saeed A. Khan, Sridhar Prabhu, Logan G. Wright, Peter L. McMahon. Quantum computational sensing using quantum signal processing, quantum neural networks, and Hamiltonian engineering. npj Quantum Inf. 12, 119 (2026). https://doi.org/10.1038/s41534-026-01235-w

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