Quantum superresolution and noise spectroscopy with quantum computing

Abstract: Quantum metrology of an incoherent signal is a canonical sensing problem related to superresolution and noise spectroscopy. We show that quantum computing can accelerate searches for a weak incoherent signal when the signal and noise are not precisely known. In particular, we consider weak Schur sampling, density matrix exponentiation, and quantum signal processing for testing the rank, purity, and spectral gap of the unknown quantum state to detect the incoherent signal. We show that these algorithms are faster than full-state tomography, which scales with the dimension of the Hilbert space. We apply our results to detecting exoplanets, stochastic gravitational waves, ultralight dark matter, geontropic quantum gravity, and Pauli noise.

• The paper introduces robust quantum metrology protocols using weak Schur sampling and DME-QSP to overcome the Rayleigh curse in incoherent sensing.
• It demonstrates that leveraging unitarily invariant spectral properties reduces sample complexity in detecting weak signals amid unknown noise.
• The results have practical implications for exoplanet detection, noise spectroscopy, and quantum diagnostics in advanced physics experiments.

Quantum Superresolution and Noise Spectroscopy with Quantum Computing

Overview and Problem Formulation

The paper "Quantum superresolution and noise spectroscopy with quantum computing" (2602.17862) addresses the canonical problem of sensing weak incoherent signals where neither the signal nor the noise are precisely known. The authors formalize the problem in terms of distinguishing between the null hypothesis ($H_0$: no signal, $\theta = 0$) and the alternative ($H_1$: weak signal present, $\theta = \theta_0$ with $0 < \theta_0 \ll 1$) using $M$ independent copies of the unknown state $\rho(\theta) = \rho_n + \theta \Delta$, where $\rho_n$ and $\rho_s$ correspond to noise and signal substates, respectively, and $\Delta = \rho_s - \rho_n$.

A central motivation is that classical metrological solutions require precise characterization of noise, which is often unavailable in practical physics applications such as exoplanet detection, stochastic gravitational wave searches, ultralight dark matter sensing, geontropic quantum gravity, and Pauli noise spectroscopy. The major technical advance is to exploit unitarily invariant properties (rank, purity, spectral gap) of the noise, crafting quantum protocols that yield robustness to nuisance transformations (e.g., spatial or temporal misalignment).

Figure 1: Illustration of bosonic sources of incoherent signals and quantum computer circuit for sensing, highlighting robustness to misalignment via unitarily invariant signal properties.

Quantum Hypothesis Testing and Rayleigh Curse Analysis

The theoretical analysis leverages quantum hypothesis testing bounds, specifically the quantum Chernoff-Stein lemma and its composite extensions, to establish that error exponents and sample complexity depend on the quantum Kullback-Leibler divergence $D(\rho_n \| \rho(\theta_0))$. The manuscript rigorously distinguishes support-extending perturbations (where the signal state $\rho_s$ is orthogonal to noise $\rho_n$) from support-preserving cases, illustrating that only the former allows linear scaling in $\theta_0$ and thus avoidance of the Rayleigh curse. The latter scenario, where signal does not extend the support, necessarily leads to quadratic scaling in error exponent (Rayleigh curse), irrespective of quantum improvement.

Algorithms: Weak Schur Sampling, Density Matrix Exponentiation, Quantum Signal Processing

The paper evaluates several quantum algorithms tailored for incoherent sensing under minimal prior knowledge about noise:

• Full-State Tomography: Requires $\mathcal{O}(rd \log(1/\beta) / \theta_0^2)$ samples; scaling is prohibitive in Hilbert space dimension $d$.
• Weak Schur Sampling (WSS): Projects onto the Schur basis, estimating spectral properties (rank, purity) without eigenvector information. Sample complexity for rank testing is $\Theta(r_n^2 \log(1/\beta)/\theta_0)$ and for purity is $\Theta(\log(1/\beta)/\theta_0)$, both independent of $d$ and superior to tomography for low rank.
• Spectral Gap Testing via WSS: When a gap $\Lambda$ separates signal and noise eigenvalues, sample complexity is $$\mathcal{O}(r^2 \log(1/\beta)/\min(\theta_0^2, \Lambda^2))$$. Upon exploitation of a large gap ($\Lambda^2 \gg \theta_0$) and noise rank, the scaling reverts to the favorable $\Theta(r_n^2 \log(1/\beta)/\theta_0)$.
• Density Matrix Exponentiation (DME) & Quantum Signal Processing (QSP): DME-QSP provides conditional block-diagonalization with constant quantum memory, enabling spectral gap testing with a sample complexity independent of $d$, specifically $$\mathcal{O}(\log(1/\delta)^2 \log(1/\beta)/(\Lambda^2 \varepsilon \theta_0))$$.

The optimality of WSS is proved in a minimax sense for unitarily invariant composite hypotheses, exploiting permutation and unitary averaging. Furthermore, DME-QSP achieves practical advantage for larger systems due to minimal quantum memory requirements despite slightly worse scaling.

Performance Summary

The results demonstrate that WSS and DME-QSP protocols decisively outperform full-state tomography for incoherent sensing tasks with unknown noise structure. For rank, purity, and spectral gap priors, the sample complexity scales only polynomially with rank and linearly with inverse signal amplitude, provided perturbations are support-extending. When noise becomes dominant or spectral information is absent, the Rayleigh curse is unavoidable even for quantum algorithms.

Application Domains

The authors discuss several high-impact domains:

• Exoplanet Detection: WSS protocols generalize from pure single-star systems (qubit model) to mixed sources and multi-star configurations, exploiting spectral gap or rank priors.
• Noise Spectroscopy: Robust quantum metrology protocols apply to stochastic gravitational wave detection and ultralight dark matter search, accommodating unknown state preparation errors by focusing on unitarily invariant spectral features.
• Lindblad and Kraus Operator Detection: Rank testing enables identification of weak jump or Kraus operators amidst unknown noise backgrounds in quantum technologies and NMR.

Implications and Future Directions

The theoretical framework presented unifies quantum superresolution and noise spectroscopy within the paradigm of quantum hypothesis testing and unitarily invariant signal extraction. The results imply that quantum computing platforms—by applying WSS or DME-QSP—can accelerate searches for weak incoherent signals, even in the presence of substantial noise uncertainty. Practically, this enables scalable quantum metrology applicable to telescopic astronomy, gravitational wave physics, quantum error correction diagnostics, and dark matter experiments.

Future work should rigorously optimize scaling constants and explore the precise quantum composite Chernoff-Stein lemma for composite hypotheses, especially for low-rank and constrained-memory regimes. Technological development of efficient quantum memory architectures and further acceleration of DME-QSP (e.g., via virtual DME) are forecasted to deepen these advantages.

Conclusion

This work delineates the boundaries of quantum-enhanced incoherent sensing, highlighting that the Rayleigh curse can be surmounted for support-extending signals with quantum computing metrological protocols, provided minimal spectral priors. The investigation substantially narrows the gap between theoretical optimality and practical implementability by scrutinizing sample complexity, memory requirements, and robustness to nuisance processes. The insights offer a foundation for future advancements in quantum sensing and hypothesis testing for physics beyond the standard model, quantum gravity, and advanced quantum technologies.