Variational Quantum Sensing

• Variational quantum sensing is a hybrid approach that uses parameterized quantum circuits and classical estimators to design adaptable sensing protocols.
• It integrates tailored probe design, measurement strategies, and estimation techniques across platforms like qubit, photonic, and continuous-variable systems.
• By optimizing metrics such as Fisher Information and Bayesian risk, VQS enhances precision and adapts to practical hardware and noise limitations.

Variational quantum sensing (VQS) denotes hybrid quantum-classical sensing protocols in which parameterized quantum circuits, and in some settings trainable classical estimators, are optimized for a sensing task under explicit hardware and noise constraints. Across qubit, photonic, bosonic, and continuous-variable settings, the recurring structure is to prepare a trainable probe, imprint an unknown parameter through a quantum channel, apply a trainable measurement or decoding stage, and optimize against a metrological objective such as Classical Fisher Information (CFI), Bayes Mean Squared Error (BMSE), Wineland spin-squeezing, overlap-based loss, or long-term coverage risk (MacLellan et al., 2024, Nikoloska et al., 29 May 2025).

1. Formal structure of variational quantum sensing

A standard VQS formulation specifies an unknown parameter or parameter vector, a probe family, an encoding channel, a measurement model, and a classical objective. In one widely used notation, an $n$-qubit probe is prepared as

$$\sigma_{\theta}=U(\theta)\,\lvert 0\cdots 0\rangle\!\langle 0\cdots 0\rvert\,U(\theta)^\dagger,$$

the sensing channel produces

$$\rho_{\theta}(x_t)=\mathcal E(x_t)\bigl(\sigma_\theta\bigr),$$

and a fixed POVM $\{\Pi^s\}$ yields outcome probabilities

$$p_\theta(s\,|\,x_t)=\mathrm{Tr}\bigl[\Pi^s\,\rho_\theta(x_t)\bigr].$$

This formulation supports sequential estimation, set-valued prediction, and adaptive control (Nikoloska et al., 29 May 2025).

A closely related single-shot Bayesian formulation parameterizes both probe preparation and measurement. A pure probe

$$\lvert\psi(\boldsymbol\theta)\rangle = U(\boldsymbol\theta)\lvert 0\rangle^{\otimes n}$$

is exposed to a channel $\mathcal E(x)$, then measured by a parameterized POVM $\mathcal M=\{\Pi_s(\boldsymbol\mu)\}$, with

$$p\bigl(s\mid x,\boldsymbol\theta,\boldsymbol\mu\bigr)=\mathrm{Tr}\bigl[\Pi_s(\boldsymbol\mu)\,\rho(x,\boldsymbol\theta)\bigr].$$

The sensing action $a=(\boldsymbol\theta,\boldsymbol\mu)$ can then be chosen by maximizing active information gain, namely the mutual information between $$\sigma_{\theta}=U(\theta)\,\lvert 0\cdots 0\rangle\!\langle 0\cdots 0\rvert\,U(\theta)^\dagger,$$0 and the as-yet unobserved outcome $$\sigma_{\theta}=U(\theta)\,\lvert 0\cdots 0\rangle\!\langle 0\cdots 0\rvert\,U(\theta)^\dagger,$$1 (Nikoloska et al., 22 Jul 2025).

End-to-end formulations extend the variational scope beyond probe design to include the classical estimator. In that setting, a parameterized quantum circuit prepares the probe, a channel $$\sigma_{\theta}=U(\theta)\,\lvert 0\cdots 0\rangle\!\langle 0\cdots 0\rvert\,U(\theta)^\dagger,$$2 imprints the unknown scalar parameter, a parameterized POVM $$\sigma_{\theta}=U(\theta)\,\lvert 0\cdots 0\rangle\!\langle 0\cdots 0\rvert\,U(\theta)^\dagger,$$3 yields samples $$\sigma_{\theta}=U(\theta)\,\lvert 0\cdots 0\rangle\!\langle 0\cdots 0\rvert\,U(\theta)^\dagger,$$4, and a neural network $$\sigma_{\theta}=U(\theta)\,\lvert 0\cdots 0\rangle\!\langle 0\cdots 0\rvert\,U(\theta)^\dagger,$$5 maps outcomes to a discrete posterior over phase bins. This makes probe preparation, measurement, and estimation part of a single trainable sensing pipeline (MacLellan et al., 2024).

2. Trainable probe and measurement architectures

VQS is not tied to a single ansatz class. The literature uses hardware-aware, physics-informed, and measurement-centric parameterizations, with the trainable degrees of freedom matched to the platform and sensing objective.

Setting	Trainable structure	Objective
Structured linear function estimation (Srivastava et al., 29 Jul 2025)	Dipolar-interacting layers + global rotations	CFI
Spin squeezing on programmable quantum sensors (Kaubruegger et al., 2019)	$$\sigma_{\theta}=U(\theta)\,\lvert 0\cdots 0\rangle\!\langle 0\cdots 0\rvert\,U(\theta)^\dagger,$$6, $$\sigma_{\theta}=U(\theta)\,\lvert 0\cdots 0\rangle\!\langle 0\cdots 0\rvert\,U(\theta)^\dagger,$$7, $$\sigma_{\theta}=U(\theta)\,\lvert 0\cdots 0\rangle\!\langle 0\cdots 0\rvert\,U(\theta)^\dagger,$$8 layers	$$\sigma_{\theta}=U(\theta)\,\lvert 0\cdots 0\rangle\!\langle 0\cdots 0\rvert\,U(\theta)^\dagger,$$9
Photonic multiparameter estimation (Cimini et al., 2023)	Variational phase block $$\rho_{\theta}(x_t)=\mathcal E(x_t)\bigl(\sigma_\theta\bigr),$$0 in a four-mode unitary	$$\rho_{\theta}(x_t)=\mathcal E(x_t)\bigl(\sigma_\theta\bigr),$$1
Continuous-variable optical phase sensing (Nielsen et al., 2023)	Displacement phase $$\rho_{\theta}(x_t)=\mathcal E(x_t)\bigl(\sigma_\theta\bigr),$$2, homodyne angle $$\rho_{\theta}(x_t)=\mathcal E(x_t)\bigl(\sigma_\theta\bigr),$$3	$$\rho_{\theta}(x_t)=\mathcal E(x_t)\bigl(\sigma_\theta\bigr),$$4
VISTA (Novak et al., 5 May 2026)	Physics-informed twin parameters $$\rho_{\theta}(x_t)=\mathcal E(x_t)\bigl(\sigma_\theta\bigr),$$5	Swap-test loss

On spin platforms, ansatzes are usually built from native collective rotations and native entanglers. For structured linear function estimation, the probe starts in $$\rho_{\theta}(x_t)=\mathcal E(x_t)\bigl(\sigma_\theta\bigr),$$6 and applies $$\rho_{\theta}(x_t)=\mathcal E(x_t)\bigl(\sigma_\theta\bigr),$$7 layers combining global single-qubit rotations and evolution under a dipolar Hamiltonian

$$\rho_{\theta}(x_t)=\mathcal E(x_t)\bigl(\sigma_\theta\bigr),$$8

so that each layer carries three real parameters $$\rho_{\theta}(x_t)=\mathcal E(x_t)\bigl(\sigma_\theta\bigr),$$9 and the full probe has $\{\Pi^s\}$0 trainable parameters (Srivastava et al., 29 Jul 2025). In programmable optical-tweezer sensors, a variational squeezer

$\{\Pi^s\}$1

uses Rydberg-dressing-generated interactions to engineer metrologically useful squeezing with only $\{\Pi^s\}$2 real parameters, independent of atom number $\{\Pi^s\}$3 (Kaubruegger et al., 2019).

Other qubit architectures emphasize graph-based or hardware-efficient layouts. End-to-end VQS studies use a Hardware-Efficient Ansatz with nearest-neighbor controlled-phase gates, a Trapped-Ion Ansatz with Mølmer–Sørensen entanglers, and a bundled-graph ansatz (MacLellan et al., 2024). Multiparameter magnetometry under dephasing employs star-graph, ring-graph, and squeezing ansätze for both state preparation and variational POVMs (Le et al., 2023). Bayesian one-axis-twist metrology introduces Arbitrary-Axis Twist ansatzes built from

$\{\Pi^s\}$4

interleaved with global rotations, reducing the number of twists needed to reach a target estimation error (Thurtell et al., 2022).

Continuous-variable and bosonic settings use different trainable primitives. Small-angle optical phase sensing varies the displacement phase $\{\Pi^s\}$5 and homodyne angle $\{\Pi^s\}$6 of a displaced squeezed vacuum (Nielsen et al., 2023). Bosonic sensor networks use qubit rotations and echoed conditional-displacement gates,

$\{\Pi^s\}$7

to realize Gaussian plus non-Gaussian control for physical-layer classification tasks (Liao et al., 2024). VISTA takes a different route: the probe is a GHZ state left to evolve passively under a Lindbladian, while a shallow “quantum twin” circuit is trained to mimic the same pure-state or noisy dynamics (Novak et al., 5 May 2026).

3. Objectives, losses, and optimization loops

The dominant VQS objectives are Fisher-information based, but the field also uses Bayesian, decision-theoretic, and task-specific losses. For structured single-parameter sensing, a common choice is the Classical Fisher Information

$\{\Pi^s\}$8

with $\{\Pi^s\}$9 assembled from local parameter shifts when the phase is encoded across multiple qubits with known weights (Srivastava et al., 29 Jul 2025). End-to-end sensing likewise optimizes CFI or QFI at the circuit level, then trains a classical estimator by cross-entropy on sampled outcomes (MacLellan et al., 2024). Photonic multiparameter estimation minimizes

$$p_\theta(s\,|\,x_t)=\mathrm{Tr}\bigl[\Pi^s\,\rho_\theta(x_t)\bigr].$$0

while variational multiparameter metrology under dephasing uses the relative-difference cost

$$p_\theta(s\,|\,x_t)=\mathrm{Tr}\bigl[\Pi^s\,\rho_\theta(x_t)\bigr].$$1

which vanishes when the scalar classical and quantum Cramér–Rao bounds coincide (Cimini et al., 2023, Le et al., 2023).

Several papers depart from Fisher-information objectives. Spin-squeezing protocols minimize the Wineland parameter

$$p_\theta(s\,|\,x_t)=\mathrm{Tr}\bigl[\Pi^s\,\rho_\theta(x_t)\bigr].$$2

which directly controls phase sensitivity via $$p_\theta(s\,|\,x_t)=\mathrm{Tr}\bigl[\Pi^s\,\rho_\theta(x_t)\bigr].$$3 (Kaubruegger et al., 2019). Bayesian vector-field sensing minimizes BMSE, where the optimal estimator is the posterior mean and the optimal cost is the average posterior variance (Kaubruegger et al., 2023). Adaptive Bayesian single-shot sensing maximizes active information gain at each time step (Nikoloska et al., 22 Jul 2025). Online conformal VQS defines a coverage loss $$p_\theta(s\,|\,x_t)=\mathrm{Tr}\bigl[\Pi^s\,\rho_\theta(x_t)\bigr].$$4 for sequential estimation sets and updates the threshold by

$$p_\theta(s\,|\,x_t)=\mathrm{Tr}\bigl[\Pi^s\,\rho_\theta(x_t)\bigr].$$5

yielding $$p_\theta(s\,|\,x_t)=\mathrm{Tr}\bigl[\Pi^s\,\rho_\theta(x_t)\bigr].$$6 in the worst case (Nikoloska et al., 29 May 2025).

Optimization strategies are correspondingly heterogeneous. Gradient-based methods rely heavily on the parameter-shift rule, including generalized shift rules for photonic probes with generator spectrum $$p_\theta(s\,|\,x_t)=\mathrm{Tr}\bigl[\Pi^s\,\rho_\theta(x_t)\bigr].$$7, Gaussian parameter-shift rules in continuous-variable optics, and stochastic parameter shifts in VISTA (Cimini et al., 2023, Nielsen et al., 2023, Novak et al., 5 May 2026). Gradient-free methods remain prominent when the objective is noisy or expensive: CMA-ES is used for structured linear function estimation, DIRECT for spin squeezing, Nelder–Mead for photonic FIM minimization, COBYLA and Powell for hardware learning on IBM devices, and Gaussian-process Bayesian optimization for optical phase sensing (Srivastava et al., 29 Jul 2025, Kaubruegger et al., 2019, Cimini et al., 2023, Ma et al., 2020, Nielsen et al., 2023). Taken together, these results suggest that VQS is best viewed as a family of optimization-driven sensing workflows rather than a single algorithm.

4. Single-parameter metrology and structured estimands

A clear illustration of VQS beyond uniform phase sensing is structured linear function estimation. In a spin-$$p_\theta(s\,|\,x_t)=\mathrm{Tr}\bigl[\Pi^s\,\rho_\theta(x_t)\bigr].$$8 array, each qubit acquires a local phase $$p_\theta(s\,|\,x_t)=\mathrm{Tr}\bigl[\Pi^s\,\rho_\theta(x_t)\bigr].$$9, reducing the task to estimation of a single scalar through the collective generator

$$\lvert\psi(\boldsymbol\theta)\rangle = U(\boldsymbol\theta)\lvert 0\rangle^{\otimes n}$$0

For uniform encoding $$\lvert\psi(\boldsymbol\theta)\rangle = U(\boldsymbol\theta)\lvert 0\rangle^{\otimes n}$$1, the entanglement-enhanced limit is $$\lvert\psi(\boldsymbol\theta)\rangle = U(\boldsymbol\theta)\lvert 0\rangle^{\otimes n}$$2; for weighted-central encoding with $$\lvert\psi(\boldsymbol\theta)\rangle = U(\boldsymbol\theta)\lvert 0\rangle^{\otimes n}$$3 and $$\lvert\psi(\boldsymbol\theta)\rangle = U(\boldsymbol\theta)\lvert 0\rangle^{\otimes n}$$4, the limit is $$\lvert\psi(\boldsymbol\theta)\rangle = U(\boldsymbol\theta)\lvert 0\rangle^{\otimes n}$$5. Optimized circuits of depth $$\lvert\psi(\boldsymbol\theta)\rangle = U(\boldsymbol\theta)\lvert 0\rangle^{\otimes n}$$6 already achieve $$\lvert\psi(\boldsymbol\theta)\rangle = U(\boldsymbol\theta)\lvert 0\rangle^{\otimes n}$$7 up to $$\lvert\psi(\boldsymbol\theta)\rangle = U(\boldsymbol\theta)\lvert 0\rangle^{\otimes n}$$8 in the uniform case, and at depth 3 the weighted-central case comes within $$\lvert\psi(\boldsymbol\theta)\rangle = U(\boldsymbol\theta)\lvert 0\rangle^{\otimes n}$$9 of $\mathcal E(x)$0 for $\mathcal E(x)$1. The optimized states have fidelity $\mathcal E(x)$2–$\mathcal E(x)$3 with the corresponding GHZ-like probes (Srivastava et al., 29 Jul 2025).

End-to-end VQS studies show that full-pipeline optimization can retain metrological performance under realistic device restrictions. For $\mathcal E(x)$4 and depth $\mathcal E(x)$5, three hardware-relevant ansätze all achieve CFI $\mathcal E(x)$6 in the noise-free case, i.e. the Heisenberg limit for that system size. Under iid dephasing noise per two-qubit gate, VQS converges to states that preserve higher CFI than GHZ, remaining above the SQL for dephasing up to $\mathcal E(x)$7. The trained neural estimator has near-zero bias and variance approaching $\mathcal E(x)$8 for $\mathcal E(x)$9 (MacLellan et al., 2024).

Real-hardware learning on superconducting qubits gives a different perspective. On IBM “Paris,” adaptive circuit learning optimized the encoder, decoder, and interrogation time using a CFI-based objective

$\mathcal M=\{\Pi_s(\boldsymbol\mu)\}$0

The reported gains reached up to $\mathcal M=\{\Pi_s(\boldsymbol\mu)\}$1 CFI improvement over the classical limit and up to $\mathcal M=\{\Pi_s(\boldsymbol\mu)\}$2 SNR improvement over the best GHZ protocol on 15 qubits. The learned circuits never entrain more than 5 qubits into a single GHZ-like chain, instead forming local clusters adapted to connectivity and noise (Ma et al., 2020).

Continuous-variable optical phase sensing shows that VQS is not restricted to qubit circuits. In a squeezed-light experiment, gradient descent reduced the cost from $\mathcal M=\{\Pi_s(\boldsymbol\mu)\}$3 to $\mathcal M=\{\Pi_s(\boldsymbol\mu)\}$4 over 24 epochs, corresponding to a $\mathcal M=\{\Pi_s(\boldsymbol\mu)\}$5 dB improvement in the phase-estimation variance, while a gradient-free Bayesian optimizer found a comparable minimum $\mathcal M=\{\Pi_s(\boldsymbol\mu)\}$6 in 50 epochs. The learned operating points included minima “not predicted by any model,” indicating direct adaptation to the true noisy apparatus (Nielsen et al., 2023).

Noisy open-system single-parameter sensing has motivated physics-informed protocols. VISTA evolves a GHZ probe under a Lindbladian master equation and compares it, via the Swap test, to a shallow “quantum twin” with trainable $\mathcal M=\{\Pi_s(\boldsymbol\mu)\}$7 and $\mathcal M=\{\Pi_s(\boldsymbol\mu)\}$8. With $\mathcal M=\{\Pi_s(\boldsymbol\mu)\}$9 shots, simulations report empirical scaling $$p\bigl(s\mid x,\boldsymbol\theta,\boldsymbol\mu\bigr)=\mathrm{Tr}\bigl[\Pi_s(\boldsymbol\mu)\,\rho(x,\boldsymbol\theta)\bigr].$$0, and the framework can transiently scale as $$p\bigl(s\mid x,\boldsymbol\theta,\boldsymbol\mu\bigr)=\mathrm{Tr}\bigl[\Pi_s(\boldsymbol\mu)\,\rho(x,\boldsymbol\theta)\bigr].$$1 for $$p\bigl(s\mid x,\boldsymbol\theta,\boldsymbol\mu\bigr)=\mathrm{Tr}\bigl[\Pi_s(\boldsymbol\mu)\,\rho(x,\boldsymbol\theta)\bigr].$$2 up to $$p\bigl(s\mid x,\boldsymbol\theta,\boldsymbol\mu\bigr)=\mathrm{Tr}\bigl[\Pi_s(\boldsymbol\mu)\,\rho(x,\boldsymbol\theta)\bigr].$$3–$$p\bigl(s\mid x,\boldsymbol\theta,\boldsymbol\mu\bigr)=\mathrm{Tr}\bigl[\Pi_s(\boldsymbol\mu)\,\rho(x,\boldsymbol\theta)\bigr].$$4 before noise drives the scaling back toward SQL. Quasi-Normalization sharpens the gradients when learning the noise rate $$p\bigl(s\mid x,\boldsymbol\theta,\boldsymbol\mu\bigr)=\mathrm{Tr}\bigl[\Pi_s(\boldsymbol\mu)\,\rho(x,\boldsymbol\theta)\bigr].$$5, and the same framework extends to simultaneous estimation of $$p\bigl(s\mid x,\boldsymbol\theta,\boldsymbol\mu\bigr)=\mathrm{Tr}\bigl[\Pi_s(\boldsymbol\mu)\,\rho(x,\boldsymbol\theta)\bigr].$$6 in transverse-field vector metrology (Novak et al., 5 May 2026).

Adaptive Bayesian single-shot sensing targets a non-asymptotic regime in which one probe is deployed at each time step. In a magnetometry case study with a 6-qubit $$p\bigl(s\mid x,\boldsymbol\theta,\boldsymbol\mu\bigr)=\mathrm{Tr}\bigl[\Pi_s(\boldsymbol\mu)\,\rho(x,\boldsymbol\theta)\bigr].$$7-equivariant ansatz, the adaptive policy tracked a sawtooth-varying phase over 100 steps almost exactly in the noise-free regime. Under Gaussian perturbations $$p\bigl(s\mid x,\boldsymbol\theta,\boldsymbol\mu\bigr)=\mathrm{Tr}\bigl[\Pi_s(\boldsymbol\mu)\,\rho(x,\boldsymbol\theta)\bigr].$$8 on the parameters at each step, the adaptive policy kept estimation error within a few degrees, and multi-agent fusion with $$p\bigl(s\mid x,\boldsymbol\theta,\boldsymbol\mu\bigr)=\mathrm{Tr}\bigl[\Pi_s(\boldsymbol\mu)\,\rho(x,\boldsymbol\theta)\bigr].$$9 achieved an RMSE reduction of $a=(\boldsymbol\theta,\boldsymbol\mu)$0–$a=(\boldsymbol\theta,\boldsymbol\mu)$1 compared to a single sensor with three repeated probes (Nikoloska et al., 22 Jul 2025).

5. Multiparameter, distributed, and task-generalized variants

VQS has become especially important for multiparameter sensing, where exhaustive analytic design is difficult. In Bayesian $a=(\boldsymbol\theta,\boldsymbol\mu)$2 interferometry for 2D and 3D vector fields, the optimal quantum sensor minimizes the average posterior variance, while low-depth variational entanglers and decoders approximate that optimum on realistic hardware. For $a=(\boldsymbol\theta,\boldsymbol\mu)$3 in 2D sensing, a variational decoder with $a=(\boldsymbol\theta,\boldsymbol\mu)$4 quantitatively approaches the 2-partition quantum sensor and nearly saturates the van Trees bound. The same framework yields 2D and 3D “quantum compass” sensors whose estimators lie on concentric phase-space rings (Kaubruegger et al., 2023).

Under dephasing noise, multiparameter VQS can optimize both probe state and measurement basis. In 3D magnetic-field sensing with $a=(\boldsymbol\theta,\boldsymbol\mu)$5, the star-graph ansatz saturates $a=(\boldsymbol\theta,\boldsymbol\mu)$6 for all $a=(\boldsymbol\theta,\boldsymbol\mu)$7, while ring and squeezing ansätze do so only up to $a=(\boldsymbol\theta,\boldsymbol\mu)$8. The optimized $a=(\boldsymbol\theta,\boldsymbol\mu)$9 remains below the standard-quantum-limit benchmark $$\sigma_{\theta}=U(\theta)\,\lvert 0\cdots 0\rangle\!\langle 0\cdots 0\rvert\,U(\theta)^\dagger,$$00, and in the Ornstein–Uhlenbeck case the non-Markovian bound is systematically lower than the purely Markovian one (Le et al., 2023).

Integrated photonics provides an experimental realization of variational multiparameter metrology. A universal four-mode interferometer $$\sigma_{\theta}=U(\theta)\,\lvert 0\cdots 0\rangle\!\langle 0\cdots 0\rvert\,U(\theta)^\dagger,$$01 with unknown sensing phases $$\sigma_{\theta}=U(\theta)\,\lvert 0\cdots 0\rangle\!\langle 0\cdots 0\rvert\,U(\theta)^\dagger,$$02 and trainable offsets $$\sigma_{\theta}=U(\theta)\,\lvert 0\cdots 0\rangle\!\langle 0\cdots 0\rvert\,U(\theta)^\dagger,$$03 is optimized by reconstructing the full classical FIM from parameter-shift data. With two-photon probes and $$\sigma_{\theta}=U(\theta)\,\lvert 0\cdots 0\rangle\!\langle 0\cdots 0\rvert\,U(\theta)^\dagger,$$04 pairs, the variationally optimized measurement achieves quadratic loss within $$\sigma_{\theta}=U(\theta)\,\lvert 0\cdots 0\rangle\!\langle 0\cdots 0\rvert\,U(\theta)^\dagger,$$05 of the noisy QCRB for all 10 tested phase configurations. Under added phase noise up to $$\sigma_{\theta}=U(\theta)\,\lvert 0\cdots 0\rangle\!\langle 0\cdots 0\rvert\,U(\theta)^\dagger,$$06 rad or full photon distinguishability, the optimized strategy still yields a $$\sigma_{\theta}=U(\theta)\,\lvert 0\cdots 0\rangle\!\langle 0\cdots 0\rvert\,U(\theta)^\dagger,$$07–$$\sigma_{\theta}=U(\theta)\,\lvert 0\cdots 0\rangle\!\langle 0\cdots 0\rvert\,U(\theta)^\dagger,$$08 error reduction over a fixed-basis strategy (Cimini et al., 2023).

Several recent works broaden VQS beyond conventional parameter estimation. Quantum integrated sensing and communication (QISAC) uses a variational two-qudit post-processing unitary $$\sigma_{\theta}=U(\theta)\,\lvert 0\cdots 0\rangle\!\langle 0\cdots 0\rvert\,U(\theta)^\dagger,$$09 to trade off communication throughput against sensing accuracy; for intermediate rate back-off, the learned measurement outperforms the fixed Bell measurement (Nikoloska et al., 20 Nov 2025). Quantum neural compressive sensing for ghost imaging embeds the forward model directly into the loss and reports, for the “butterfly” object at $$\sigma_{\theta}=U(\theta)\,\lvert 0\cdots 0\rangle\!\langle 0\cdots 0\rvert\,U(\theta)^\dagger,$$10, PSNR $$\sigma_{\theta}=U(\theta)\,\lvert 0\cdots 0\rangle\!\langle 0\cdots 0\rvert\,U(\theta)^\dagger,$$11 and SSIM $$\sigma_{\theta}=U(\theta)\,\lvert 0\cdots 0\rangle\!\langle 0\cdots 0\rvert\,U(\theta)^\dagger,$$12, versus $$\sigma_{\theta}=U(\theta)\,\lvert 0\cdots 0\rangle\!\langle 0\cdots 0\rvert\,U(\theta)^\dagger,$$13 and $$\sigma_{\theta}=U(\theta)\,\lvert 0\cdots 0\rangle\!\langle 0\cdots 0\rvert\,U(\theta)^\dagger,$$14 for a classical CNN and $$\sigma_{\theta}=U(\theta)\,\lvert 0\cdots 0\rangle\!\langle 0\cdots 0\rvert\,U(\theta)^\dagger,$$15 and $$\sigma_{\theta}=U(\theta)\,\lvert 0\cdots 0\rangle\!\langle 0\cdots 0\rvert\,U(\theta)^\dagger,$$16 for differential GI (Zhai et al., 25 Feb 2025). Variational quantum RF sensing trains a 10-qubit probe on ray-tracer data and then classifies locations without classical CSI at deployment, reaching $$\sigma_{\theta}=U(\theta)\,\lvert 0\cdots 0\rangle\!\langle 0\cdots 0\rvert\,U(\theta)^\dagger,$$17 accuracy in the LOS task and $$\sigma_{\theta}=U(\theta)\,\lvert 0\cdots 0\rangle\!\langle 0\cdots 0\rvert\,U(\theta)^\dagger,$$18 in the NLOS task, against a classical LSTM baseline of $$\sigma_{\theta}=U(\theta)\,\lvert 0\cdots 0\rangle\!\langle 0\cdots 0\rvert\,U(\theta)^\dagger,$$19 and $$\sigma_{\theta}=U(\theta)\,\lvert 0\cdots 0\rangle\!\langle 0\cdots 0\rvert\,U(\theta)^\dagger,$$20 (Nikoloska, 10 Mar 2026).

Bosonic sensor networks make the generalization still more explicit. In cavity-QED-based supervised learning assisted by an entangled sensor network, the cost is the average classification error after a data-encoding displacement $$\sigma_{\theta}=U(\theta)\,\lvert 0\cdots 0\rangle\!\langle 0\cdots 0\rvert\,U(\theta)^\dagger,$$21, a probe VQC, and a measurement VQC. For binary displacement discrimination, the optimal error shows a threshold phenomenon: above a critical signal strength $$\sigma_{\theta}=U(\theta)\,\lvert 0\cdots 0\rangle\!\langle 0\cdots 0\rvert\,U(\theta)^\dagger,$$22, an appropriate non-Gaussian probe can achieve literally zero error, whereas below threshold the error decays continuously. The paper gives analytical expressions for $$\sigma_{\theta}=U(\theta)\,\lvert 0\cdots 0\rangle\!\langle 0\cdots 0\rvert\,U(\theta)^\dagger,$$23 and for residual error under weak Gaussian noise (Liao et al., 2024). This suggests that VQS is increasingly being used as a general design framework for physical-layer inference, not only for asymptotically unbiased estimation.

6. Noise, guarantees, misconceptions, and open questions

A central theme of VQS is adaptation to realistic noise and imperfect modeling. Spin-squeezing optimization on programmable quantum sensors reports graceful degradation under Gaussian control noise, average squeezing within one $$\sigma_{\theta}=U(\theta)\,\lvert 0\cdots 0\rangle\!\langle 0\cdots 0\rvert\,U(\theta)^\dagger,$$24 of the full-filling case for a random half-filled array, and spontaneous-emission error $$\sigma_{\theta}=U(\theta)\,\lvert 0\cdots 0\rangle\!\langle 0\cdots 0\rvert\,U(\theta)^\dagger,$$25 for realistic Rydberg-dressing parameters (Kaubruegger et al., 2019). In experimental photonic sensing, imperfections are not modeled away but are “learned into” the cost function, making the optimization self-correcting against static fabrication errors and slow drifts (Cimini et al., 2023). Quantum neural compressive sensing reports nearly identical imaging performance under depolarizing rates $$\sigma_{\theta}=U(\theta)\,\lvert 0\cdots 0\rangle\!\langle 0\cdots 0\rvert\,U(\theta)^\dagger,$$26 beyond modest sampling (Zhai et al., 25 Feb 2025). Continuous-variable optical sensing found minima not captured by the analytical or simple numerical noise models, again emphasizing adaptation to the real device (Nielsen et al., 2023).

A recurring misconception is that VQS is synonymous with unconstrained black-box circuit search. Several works explicitly move in the opposite direction: VISTA restricts the search space to the physical parameters of interest and states that this circumvents barren plateaus (Novak et al., 5 May 2026); structured linear function estimation uses dipolar-interaction layers matched to the sensor geometry (Srivastava et al., 29 Jul 2025); arbitrary-axis twist metrology uses a low-parameter family built from one-axis twists and collective rotations (Thurtell et al., 2022). Another misconception is that GHZ is universally optimal. GHZ probes are central in noiseless or specially structured settings, and VISTA even argues “GHZ is All You Need” for its twin-ansatz protocol, but hardware experiments on superconducting qubits show GHZ protocols can fall below the classical limit for $$\sigma_{\theta}=U(\theta)\,\lvert 0\cdots 0\rangle\!\langle 0\cdots 0\rvert\,U(\theta)^\dagger,$$27, while learned local-cluster circuits continue to improve with system size (Novak et al., 5 May 2026, Ma et al., 2020). A plausible implication is that the optimal probe family is generally task-, architecture-, and noise-model dependent.

Rigorous guarantees remain relatively rare. Online conformal VQS provides a deterministic long-term coverage guarantee,

$$\sigma_{\theta}=U(\theta)\,\lvert 0\cdots 0\rangle\!\langle 0\cdots 0\rvert\,U(\theta)^\dagger,$$28

with no distributional assumptions, and dynamic tuning produces smaller, more informative sets than static baselines in the reported magnetometry experiments (Nikoloska et al., 29 May 2025). VQFIE addresses a different guarantee problem by estimating lower and upper fidelity-based bounds on the QFI, producing a range in which the true QFI lies without requiring knowledge of the explicit sensor dynamics; the lower TQFI bound tightens as the state purity increases and is reported to be tighter than comparison bounds in the tested magnetometry setup (Beckey et al., 2020).

Open problems are explicit across the literature. Full-wavefunction evaluation of $$\sigma_{\theta}=U(\theta)\,\lvert 0\cdots 0\rangle\!\langle 0\cdots 0\rvert\,U(\theta)^\dagger,$$29 can scale as $$\sigma_{\theta}=U(\theta)\,\lvert 0\cdots 0\rangle\!\langle 0\cdots 0\rvert\,U(\theta)^\dagger,$$30, and on hardware many probabilities must be estimated from shots (Srivastava et al., 29 Jul 2025). Online conformal VQS currently assumes a discrete grid of $$\sigma_{\theta}=U(\theta)\,\lvert 0\cdots 0\rangle\!\langle 0\cdots 0\rvert\,U(\theta)^\dagger,$$31, full feedback after each step, and incurs online-training overhead for the learned posterior model (Nikoloska et al., 29 May 2025). Continuous-parameter extensions, multi-parameter conformal sets, adaptive measurement bases, decoherence-aware training objectives, error-mitigation inside the variational loop, and real-device demonstrations on NV arrays or Rydberg platforms are all identified as potential extensions (Nikoloska et al., 29 May 2025, Srivastava et al., 29 Jul 2025). Taken together, the existing results define VQS less as a finalized sensing doctrine than as a rapidly expanding variational methodology for co-designing probe states, measurements, and estimators under realistic experimental constraints.