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Scalable Bayesian Hamiltonian learning
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abstract
As the size of quantum devices continues to grow, the development of scalable methods to characterise and diagnose noise is becoming an increasingly important problem. Recent methods have shown how to efficiently estimate Hamiltonians in principle, but they are poorly conditioned and can only characterize the system up to a scalar factor, making them difficult to use in practice. In this work we present a Bayesian methodology, called Bayesian Hamiltonian Learning (BHL), that addresses both of these issues by making use of any or all, of the following: well-characterised experimental control of Hamiltonian couplings, the preparation of multiple states, and the availability of any prior information for the Hamiltonian. Importantly, BHL can be used online as an adaptive measurement protocol, updating estimates and their corresponding uncertainties as experimental data become available. In addition, we show that multiple input states and control fields enable BHL to reconstruct Hamiltonians that are neither generic nor spatially local. We demonstrate the scalability and accuracy of our method with numerical simulations on up to 100 qubits. These practical results are complemented by several theoretical contributions. We prove that a $k$-body Hamiltonian $H$ whose correlation matrix has a spectral gap $\Delta$ can be estimated to precision $\varepsilon$ with only $\tilde{O}\bigl(n^{3k}/(\varepsilon \Delta)^{3/2}\bigr)$ measurements. We use two subroutines that may be of independent interest: First, an algorithm to approximate a steady state of $H$ starting from an arbitrary input that converges factorially in the number of samples; and second, an algorithm to estimate the expectation values of $m$ Pauli operators with weight $\le k$ to precision $\epsilon$ using only $O(\epsilon^{-2} 3^k \log m)$ measurements, which quadratically improves a recent result by Cotler and Wilczek.
Forward citations
Cited by 7 Pith papers
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Characterizing Arbitrary Lindbladian Dynamics with a Few Pauli Measurements
A control-free protocol using only product-Pauli preparations and measurements reconstructs arbitrary sparse Lindbladian generators, identifying supports from data with O~(Γ²M0²/ε⁴) samples and O~(ΓM0²/ε²) total evolu...
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Near-Optimal Learning of Local Lindbladians
Near-optimal algorithm learns local Lindbladians via finite-time probes and classical shadows with Õ(Λ²/ε²) channel uses and matching lower bounds showing dissipative terms block Heisenberg-limited scaling.
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Heisenberg-limited Hamiltonian learning continuous variable systems via engineered dissipation
An engineered-dissipation protocol learns general low-intersection bosonic Hamiltonians with O(epsilon^{-1} log(m/delta)) total evolution time, achieving Heisenberg-limited scaling.
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Learning Arbitrary Lindbladians from Time Evolution
Arbitrary Lindbladians of strength ≤Λ are learned entrywise to error ε with Õ(Λ²/ε²) ancilla-free, control-free experiments and Õ(Λ/ε²) total evolution time.
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Optimal Ansatz-free Hamiltonian Learning In Situ
Ansatz-free Hamiltonian learning with product Pauli states and no control achieves optimal total evolution time Θ(Λ/ε² log(Λ/ε)), with a matching new lower bound over all control-free protocols.
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Pauli Measurements Are Near-Optimal for Single-Qubit Tomography
Single-qubit measurements need Ω(10^N/(√N ε²)) copies for N-qubit tomography, matching the Pauli-measurement upper bound up to a √N factor.
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Heisenberg-Limited Quantum Hamiltonian Learning via Randomly Spread Product-States
Randomly spread product states and random Pauli measurements activate all spectral gaps of a Hamiltonian, giving a finite-time quadratic Fisher-information window and enabling simultaneous, beyond-Standard-Quantum-Lim...
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