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arXiv:1609.09446 · Physical Review A

Hamiltonian identifiability assisted by a single-probe measurement

Akira Sone, Paola Cappellaro

Published Feb 24, 2017• 64 citations
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Abstract

We study the Hamiltonian identifiability of a many-body spin-$1/2$ system assisted by the measurement on a single quantum probe based on the eigensystem realization algorithm approach employed in Zhang and Sarovar, Phys. Rev. Lett. 113, 080401 (2014). We demonstrate a potential application of Gr\"obner basis to the identifiability test of the Hamiltonian, and provide the necessary experimental resources, such as the lower bound in the number of the required sampling points, the upper bound in total required evolution time, and thus the total measurement time. Focusing on the examples of the identifiability in the spin-chain model with nearest-neighbor interaction, we classify the spin-chain Hamiltonian based on its identifiability, and provide the control protocols to engineer the nonidentifiable Hamiltonian to be an identifiable Hamiltonian.

DOI
10.1103/physreva.95.022335

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We study the Hamiltonian identifiability of a many-body spin-1/21/21/2 system assisted by the measurement on a single quantum probe based on the eigensystem realization algorithm approach employed in Zhang and Sarovar, Phys. Rev. Lett. 113, 080401 (2014). We demonstrate a potential application of Gr"obner basis to the identifiability test of the Hamiltonian, and provide the necessary experimental resources, such as the lower bound in the number of the required sampling points, the upper bound in total required evolution time, and thus the total measurement time. Focusing on the examples of the identifiability in the spin-chain model with nearest-neighbor interaction, we classify the spin-chain Hamiltonian based on its identifiability, and provide the control protocols to engineer the nonidentifiable Hamiltonian to be an identifiable Hamiltonian.
Hamiltonian learning provides a systematic framework for reconstructing unknown quantum dynamics. However, existing protocols typically assume direct measurement access to the entire system. With fast single-qubit control and a connected reference backbone, we show that a single measurable qubit suffices to learn all O(N)O(N)O(N) independent parameters of a bounded-degree two-body Hamiltonian on NNN qubits at the Heisenberg limit. Crucially, our protocol uses robust SWAP gates synthesized by quantum signal processing, enabling coherent transfer of states evolving under distant Hamiltonian parameters to the measurable qubit. This transfer requires no prior calibration of the Hamiltonian parameters of the intermediate links. A parallel learning architecture achieves total query time O~(N)\widetilde{O}(N)O(N) for an NNN-qubit chain, while retaining Heisenberg-limited precision scaling. On the chain, these scalings match the fundamental precision and information-propagation lower bounds up to logarithmic factors. The framework further extends to arbitrary bounded-degree interaction graphs. Our results establish a scalable route to learning an extensive number of many-body Hamiltonian parameters through only a local measurement interface.
The identifiability of a system is concerned with whether the unknown parameters in the system can be uniquely determined with all the possible data generated by a certain experimental setting. A test of quantum Hamiltonian identifiability is an important tool to save time and cost when exploring the identification capability of quantum probes and experimentally implementing quantum identification schemes. In this article, we generalize the identifiability test based on the similarity transformation approach (STA) in classical control theory and extend it to the domain of quantum Hamiltonian identification. We employ the STA to prove the identifiability of spin-1/2 chain systems with arbitrary dimension assisted by single-qubit probes. We further extend the traditional STA method by proposing a structure preserving transformation (SPT) method for nonminimal systems. We use the SPT method to introduce an indicator for the existence of economic quantum Hamiltonian identification algorithms, whose computational complexity directly depends on the number of unknown parameters (which could be much smaller than the system dimension). Finally, we give an example of such an economic Hamiltonian identification algorithm and perform simulations to demonstrate its effectiveness.
Characterizing the Hamiltonian that a quantum processor actually implements is central to calibrating and validating current quantum hardware. Many devices, however, operate with generators that are time dependent by design. Here we develop a rigorous and experimentally friendly protocol for learning time-dependent many-body Hamiltonians from continuous weak measurement records. The key observation is that interaction sparsity reduces the global reconstruction to a set of local inverse problems, whose number is controlled by the interaction connectivity rather than by the system size. Pure separable probe states suffice to drive these inversions, and a graph-coloring construction embeds them into a small number of global product-state preparations. We derive explicit reconstruction-error bounds and a sample-complexity theorem that cleanly separates the finite-sampling statistical noise from the deterministic bias of the iterative state update, and we validate the protocol on time-dependent spin chains with up to n=8n=8n=8 qubits. Beyond these results, our analysis provides a rigorous foundation for time-dependent Hamiltonian learning from continuous monitoring in many-body systems, establishing a framework that extends naturally to many platforms and probe ensembles.
Characterizing the features of a Hamiltonian that governs a quantum system serves as a fundamental subroutine of quantum device calibration, signal sensing, and error correction. Recent works have proposed protocols achieving the optimal Heisenberg-limited scaling learning ansatz-free Hamiltonians from their real-time evolutions without fully specifying interaction structures. However, these protocols rely on both deep circuits with interleaving probes and control, and extremely short time resolution, making them difficult to implement on near- and intermediate-term in situ quantum experiments. In this work, we propose a computationally efficient, control-free, and ancilla-free algorithm that uses only Pauli product state preparation and measurement, and learns an ansatz-free Hamiltonian HHH with ∣∣H∣∣≤Λ||H||\leqΛ∣∣H∣∣≤Λ in total evolution time of Θ(Λε2log⁡(Λε))Θ(\fracΛ{ε^2}\log(\fracΛε))Θ(ε2Λ​log(εΛ​)). The evolution time cost of our algorithm is optimal for any control-free protocols as we further prove a lower bound of Ω(Λε2log⁡(Λε))Ω(\fracΛ{ε^2}\log(\fracΛε))Ω(ε2Λ​log(εΛ​)). Technically, our method introduces a randomized-sampling framework that combines band-limited kernel-based time sampling with a displacement sieve for Hamiltonian structure learning. The characteristic probe time resolution depends only on ΛΛΛ instead of ε\varepsilonε, which makes our protocol especially appealing in the high-precision regime for sensing and calibration applications. We also show that the algorithm maintains the same asymptotic total evolution time in the presence of state-preparation-and-measurement (SPAM) noise when the Hamiltonian is local after calibration. Our results demonstrate the fundamental cost of experimentally friendly Hamiltonian learning and provide a practical route to rigorous in situ characterization of near-term quantum platforms.
Characterizing quantum many-body systems is a fundamental problem across physics, chemistry, and materials science. While significant progress has been made, many existing Hamiltonian learning protocols demand digital quantum control over the entire system, creating a disconnect from many real-world settings that provide access only through small, local probes. Motivated by this, we introduce and formalize the problem of quantum probe tomography, where one seeks to learn the parameters of a many-body Hamiltonian using a single local probe access to a small subsystem of a many-body thermal state undergoing time evolution. We address the identifiability problem of determining which Hamiltonians can be distinguished from probe data through a new combination of tools from algebraic geometry and smoothed analysis. Using this approach, we prove that generic Hamiltonians in various physically natural families are identifiable up to simple, unavoidable structural symmetries. Building on these insights, we design the first efficient end-to-end algorithm for probe tomography that learns Hamiltonian parameters to accuracy ε\varepsilonε, with query complexity scaling polynomially in 1/ε1/\varepsilon1/ε and classical post-processing time scaling polylogarithmically in 1/ε1/\varepsilon1/ε. In particular, we demonstrate that translation- and rotation-invariant nearest-neighbor Hamiltonians on square lattices in one, two, and three dimensions can be efficiently reconstructed from single-site probes of the Gibbs state, up to inversion symmetry about the probed site. Our results demonstrate that robust Hamiltonian learning remains achievable even under severely constrained experimental access.
Estimating the dimension of an Hilbert space is an important component of quantum system identification. In quantum technologies, the dimension of a quantum system (or its corresponding accessible Hilbert space) is an important resource, as larger dimensions determine, e.g., the performance of quantum computation protocols or the sensitivity of quantum sensors. Despite being a critical task in quantum system identification, estimating the Hilbert space dimension is experimentally challenging. While there have been proposals for various dimension witnesses capable of putting a lower bound on the dimension from measuring collective observables that encode correlations, in many practical scenarios, especially for multiqubit systems, the experimental control might not be able to engineer the required initialization, dynamics, and observables. Here we propose a more practical strategy that relies not on directly measuring an unknown multiqubit target system, but on the indirect interaction with a local quantum probe under the experimenter's control. Assuming only that the interaction model is given and the evolution correlates all the qubits with the probe, we combine a graph-theoretical approach and realization theory to demonstrate that the system dimension can be exactly estimated from the model order of the system. We further analyze the robustness in the presence of background noise of the proposed estimation method based on realization theory, finding that despite stringent constrains on the allowed noise level, exact dimension estimation can still be achieved.