We establish the ultimate precision limits for estimating a function of multiple Hamiltonian parameters in the presence of Markovian noise. By reducing multiparameter function estimation to an optimization over effective single-parameter embeddings, we derive tight bounds on the quantum Fisher information. We identify a necessary and sufficient functional Hamiltonian-not-in-Lindblad-span condition for Heisenberg-limited scaling in time. When this condition holds, we construct a code that simultaneously removes the noise and nuisance Hamiltonian parameters while preserving the target signal. When it fails, we derive the optimal standard quantum-limit coefficient and show that it is asymptotically attainable using approximate quantum error correction. Finally, we demonstrate that direct function estimation can substantially outperform estimating all parameters individually and subsequently evaluating the function. This advantage can scale with both the evolution time and the number of parameters.
We establish the ultimate precision limits for estimating a function of multiple Hamiltonian parameters in the presence of Markovian noise. By reducing multiparameter function estimation to an optimization over effective single-parameter embeddings, we derive tight bounds on the quantum Fisher information. We identify a necessary and sufficient functional Hamiltonian-not-in-Lindblad-span condition for Heisenberg-limited scaling in time. When this condition holds, we construct a code that simultaneously removes the noise and nuisance Hamiltonian parameters while preserving the target signal. When it fails, we derive the optimal standard quantum-limit coefficient and show that it is asymptotically attainable using approximate quantum error correction. Finally, we demonstrate that direct function estimation can substantially outperform estimating all parameters individually and subsequently evaluating the function. This advantage can scale with both the evolution time and the number of parameters.
Estimation of physical parameters encoded in a Hamiltonian is a central task in quantum sensing and learning. While the ultimate precision limit for estimating a single parameter coupled to a single generator is well established, the corresponding bound for estimating a function of multiple parameters-each coupled to distinct and possibly non-commuting generators-remains unknown in general. Here, we derive the ultimate quantum limit and present an estimation protocol for any function of parameters in a general Hamiltonian that attains this bound. We show that, although the task is fundamentally a multiparameter problem, our tight bound reduces to an optimized single-parameter quantum Cramér-Rao bound, even for arbitrary generator sets. Our result unifies and extends previous works, providing a general framework for optimal function estimation in quantum systems.
We derive ultimate precision bounds for estimating parameters encoded in \emph{time-dependent} Hamiltonians in the presence of general Markovian noise, allowing for arbitrary adaptive protocols with fast controls and noiseless ancillas. Extending the minimization-over-purifications framework to time-varying continuous channels, we obtain a differential upper bound on the achievable quantum Fisher information (QFI) that can be evaluated at all times via semidefinite programming. For parameter-independent noise, we prove a universal long-time scaling law: if the coherent (noiseless) dynamics yields Qcoh(T)∼T2k, then under Markovian noise the QFI scales at most as Q(T)∼T2k in the DHNLS regime, whereas in the DHLS regime it is fundamentally limited to Q(T)∼T2k−1. We illustrate these behaviors on paradigmatic driven-qubit sensors, exhibiting T4 and T3 scalings under dephasing and spontaneous emission, respectively. Finally, we provide explicit continuous exact and approximate quantum error correction constructions -- supplemented by spin-squeezed probes -- that asymptotically saturate the bounds, establishing their tightness.
Measuring stochastic signals ("noise metrology") constitutes a central task in quantum sensing and the characterization of open quantum systems. Here we establish ultimate precision bounds for multiparameter estimation of stochastic signals encoded through Markovian Lindblad dynamics, allowing for arbitrary quantum control and noiseless ancillae. Although Markovianity enforces standard-quantum-limit scaling with sensing time T, our bounds reveal Heisenberg-type scaling in the number of dissipative channels, R: when the stochastic signal exhibits high-rank correlations across the R channels and the probe is entangled, the average variance (per parameter) scales no better than Ω(1/(TR2)). For collective k-body dissipation, R=Θ(Nk), signifying super-Heisenberg scaling with the system size N. We further show that, when the unknown parameters enter through the dissipative eigenrates, a Rapid Prepare-and-Measure (RPM) protocol that tracks many distinct quantum jumps in parallel attains these limits. In this regime, the estimation problem reduces to a multi-Poisson counting model, providing a conceptually clean route to optimal quantum noise metrology. We illustrate the breadth of the framework with applications to networked noise metrology, collective many-body dissipation, learning Pauli noise, and subdiffraction quantum imaging.
We derive a necessary and sufficient condition for the possibility of achieving the Heisenberg scaling in general adaptive multi-parameter estimation schemes in presence of Markovian noise. In situations where the Heisenberg scaling is achievable, we provide a semidefinite program to identify the optimal quantum error correcting (QEC) protocol that yields the best estimation precision. We overcome the technical challenges associated with potential incompatibility of the measurement optimally extracting information on different parameters by utilizing the Holevo Cramér-Rao (HCR) bound for pure states. We provide examples of significant advantages offered by our joint-QEC protocols, that sense all the parameters utilizing a single error-corrected subspace, over separate-QEC protocols where each parameter is effectively sensed in a separate subspace.
For single-parameter sensing, Greenberger-Horne-Zeilinger (GHZ) probes achieve optimal quantum-enhanced precision across the unknown parameter range, solely relying on parameter-independent separable measurement strategies for all values of the unknown parameter. However, in the multiparameter setting, a single GHZ probe not only fails to achieve quantum advantage but also the corresponding optimal measurement becomes complex and dependent on the unknown parameters. Here, we provide a recipe for multiparameter sensing with GHZ probes using quantum error correction techniques by treating all but one unknown parameters as noise, whose effects can be corrected. This strategy restores the core advantage of single parameter GHZ-based quantum sensing, namely reaching optimally quantum-enhanced precision for all unknown parameter values while keeping the measurements separable and fixed. Specifically, given one shielded ancilla qubit per GHZ probe, our protocol extracts optimal possible precision for any probe size. While this optimal precision is shot-noise limited for a single GHZ probe, we recover the Heisenberg scaling through use of multiple complementary GHZ probes. We demonstrate the effectiveness of the protocol with Bayesian estimation.
Unitary encoding of a single parameter provides quadratic enhancement in precision, with the quantum Fisher information scaling quadratically with the encoding time. However, when estimating multiple parameters simultaneously, this fundamental scaling is not guaranteed. Here, we establish a universal geometric obstruction that dictates when multiparameter quantum metrology fails to achieve simultaneous t−2 scaling. By decomposing the Hamiltonian derivatives into components that commute and do not commute with the system Hamiltonian, we prove that linear dependence among the commuting components inevitably generates a slow parameter direction whose Fisher information remains bounded as O(t0), limiting the overall estimation precision. We demonstrate this mechanism in both discrete- and continuous-variable setups, including collective spin magnetometry and a generalized quantum harmonic oscillator, and contrast it with the Lipkin--Meshkov--Glick model where t−2 decay is preserved. Remarkably, while the slow direction fundamentally limits the achievable precision, the measurement incompatibility between fast and slow directions decays as 1/t, rendering the symmetric logarithmic derivative bound asymptotically saturable. Our framework provides a readily computable diagnostic, given by the Gram matrix of the diagonal generators, for identifying such obstructions in arbitrary multiparameter estimation problems. We further show that the bottleneck can be circumvented by relegating slow directions to nuisance parameters or by employing adaptive quantum control.