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arXiv:2510.08500

Learning and certification of local time-dependent quantum dynamics and noise

Daniel Stilck França, Cambyse Rouzé, Albert H. Werner, Tim Möbus

Published Oct 9, 2025• 0 citations
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Abstract

Hamiltonian learning protocols are becoming valuable tools for benchmarking and verifying quantum computers and simulators, yet few rigorous protocols address explicitly time-dependent dynamics. We give a scalable protocol for learning the time-dependent coefficients of a locally interacting $n$-qubit system on a graph $\mathsf{G}$ of effective dimension $D$. The unknown generator contains geometrically local Hamiltonian terms and weak onsite Pauli-dephasing terms with time-dependent, nonnegative rates. Its coefficient functions and their derivatives admit a certified simultaneous $C^1$ approximation in a known finite-dimensional function class that supports stable interpolation. The only experimental primitives are preparation of product Pauli eigenstates, evolution for chosen times, and product-Pauli measurements. Under an explicit weak-dissipation condition, the Lieb--Robinson estimates proved here, and a certified effective approximation/interpolation degree $m_{\rm eff}$, the protocol reconstructs every coefficient uniformly on a constant interval with error $ε$ and success probability $1-δ$ using $\widetilde{\mathcal{O}}\!\left(ε^{-2}\operatorname{poly}(m_{\rm eff})\log(nδ^{-1})\right)$ samples and polynomial classical processing. The method estimates derivatives of finite-time channel overlaps by stable interpolation, uses process shadows to learn the required local transfer matrices, and constructs sparse diagonally dominant systems through localized semidefinite programs.

DOI
10.48550/arxiv.2510.08500

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We present an efficient protocol for learning an unknown kkk-local Lindblad generator on nnn qubits using only product-state preparations, short-time evolution, and single-qubit Pauli measurements, without prior knowledge of the interaction structure. For fixed kkk and a supplied weighted interaction-strength bound αˉ\barααˉ, the protocol estimates all Hamiltonian and dissipative Pauli--GKSL coefficients with total coefficient error at each site at most ε\varepsilonε with probability at least 1−δ1-δ1−δ using O~k(αˉ2n2k−2ε−2log⁡(n/δ))\widetilde{\mathcal O}_k(\barα^2n^{2k-2}\varepsilon^{-2}\log(n/δ))Ok​(αˉ2n2k−2ε−2log(n/δ)) samples and polylogarithmically many distinct evolution times. This guarantee requires no degree, sparsity, or tail promise. A convex optimization converts these estimates into a valid kkk-local Lindblad generator with diamond-norm error at most ε\varepsilonε using O~k(αˉ2n2kε−2log⁡(n/δ))\widetilde{\mathcal O}_k(\barα^2n^{2k}\varepsilon^{-2}\log(n/δ))Ok​(αˉ2n2kε−2log(n/δ)) samples and polynomial-time classical postprocessing. With a supplied local effective-sparsity bound rrr and a sufficiently small sitewise subthreshold tail Ok(ε)\mathcal O_k(\varepsilon)Ok​(ε), coefficient recovery requires only O~k(αˉ2r2ε−2log⁡(n/δ))\widetilde{\mathcal O}_k(\barα^2r^2\varepsilon^{-2}\log(n/δ))Ok​(αˉ2r2ε−2log(n/δ)) samples, without a supplied threshold or coefficient locations and without a coefficient-gap assumption. In particular, exactly sparse bounded-degree models have logarithmic sample dependence on nnn. We also provide guard-band support recovery and complementary supplied-candidate guarantees, extend the guarantees to model misspecification, and prove complementary sample-complexity lower bounds.
Quantum devices are open systems whose dynamics interleave coherent evolution with dissipation, and benchmarking, error mitigation, and error correction all rest on a faithful model of both. Existing characterization protocols either assume prior knowledge of the interaction and noise structure, or demand ancillas, entangled probes, or mid-circuit control, or capture only the Pauli-diagonal part of the noise. Here, we present a protocol that reconstructs an arbitrary sparse Markovian generator, including every Hamiltonian together with the jump operator coefficients, using only product Pauli state preparation, single uninterrupted forward evolutions, and product Pauli measurements. Given a sparsity budget M0M_0M0​ and a strength bound ΓΓΓ of the Lindbladian, every coefficient is learned to precision εεε from O~(Γ2M02/ε4)\widetilde{O}(Γ^2M_0^2/ε^4)O(Γ2M02​/ε4) experiments and O~(ΓM02/ε2)\widetilde{O}(ΓM_0^2/ε^2)O(ΓM02​/ε2) total evolution time, with both supports identified from data without locality assumptions. The protocol runs at a logarithmic number of positive evolution times on a hardware clock lattice and is provably robust to calibrated state-preparation and measurement errors.
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We study Hamiltonian learning for kkk-local Hamiltonians at large step size, using only forward time evolution at integer multiples of a fixed time interval. For this task, our new technique, which we call eigenphase engineering, achieves Heisenberg-limited learning with a step size that is optimal up to logarithmic factors. For an unknown sss-sparse Hamiltonian with fixed locality kkk and ∥H∥op≤Λ\|H\|_{\mathrm{op}}\leqΛ∥H∥op​≤Λ, our algorithm recovers its Pauli coefficients to ℓ2\ell_2ℓ2​ error ε\varepsilonε in total evolution time O~(s/ε)\widetilde{O}(s/\varepsilon)O(s/ε), matching the best known scaling while allowing a near-maximal step size Θ~(1/Λ)\widetildeΘ(1/Λ)Θ(1/Λ). As an independent contribution, we strengthen the evolution-time lower bound to Ωk(s1−1/(2k)/ε)Ω_k(s^{1-1/(2k)}/\varepsilon)Ωk​(s1−1/(2k)/ε), placing our algorithm within a factor O~(s1/(2k))\widetilde{O}(s^{1/(2k)})O(s1/(2k)) of being optimal. Eigenphase engineering departs from approaches based on dynamical approximation or iteratively canceling the unknown Hamiltonian. It encodes energy expectation differences in the perturbative response of eigenphases of the actual controlled evolution, then extracts them through phase estimation and high-order extrapolation. This avoids the need for polynomially shorter evolution steps as the target precision improves. A version using only single-qubit operations retains the same step size with a n\sqrt nn​ overhead in ℓ2\ell_2ℓ2​ learning time for nnn qubits. The method also enables tolerant Hamiltonian certification and estimation of a specified Pauli coefficient, the latter without locality or sparsity assumptions. These results establish eigenphase engineering as a versatile approach to extracting Hamiltonian information with infrequent control.