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arXiv:1809.02965 · IEEE Transactions on Automatic Control

Quantum Hamiltonian Identifiability via a Similarity Transformation Approach and Beyond

Yuanlong Wang, Daoyi Dong, Akira Sone, Ian R. Petersen, Hidehiro Yonezawa, Paola Cappellaro

Published Mar 6, 2020• 41 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1109/tac.2020.2973582

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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.
Identification of the Hamiltonian is vital for characterizing the dynamical evolution of a quantum system. The dimension of a multi-qubit system increases exponentially with the qubit number, which usually leads to daunting computational complexity for general Hamiltonian identification algorithms. In this paper, we design an efficient quantum Hamiltonian identification method based on periodical sampling. The computational complexity is O(M2 + MN2), where M is the number of unknown parameters to be identified in the Hamiltonian and N is the length of the sampling data. Numerical results with different data lengths demonstrate the effectiveness of the proposed identification algorithm.
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.
Quantum Hamiltonian identification (QHI) is important for characterizing the dynamics of quantum systems, calibrating quantum devices, and achieving precise quantum control. In this paper, an effective two-step optimization (TSO) QHI algorithm is developed within the framework of quantum process tomography. In the identification method, different probe states are input into quantum systems and the output states are estimated using the quantum state tomography protocol via linear regression estimation. The time-independent system Hamiltonian is reconstructed based on the experimental data for the output states. The Hamiltonian identification method has computational complexity O(d6), where d is the dimension of the system Hamiltonian. An error upper bound O( d3/√N ) is also established, where N is the resource number for the tomography of each output state, and several numerical examples demonstrate the effectiveness of the proposed TSO Hamiltonian identification method.
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.
Identifying unknown Hamiltonians from their quantum dynamics is a pivotal challenge in quantum technologies. In this paper, we introduce Hamiltonian recognition, a framework that bridges quantum hypothesis testing and quantum metrology, aiming to identify the Hamiltonian governing quantum dynamics from a known set of Hamiltonians. To identify H for an unknown qubit quantum evolution exp⁡(−iHθ)\exp (-iH\theta )exp(−iHθ) with unknown θ, from two or three orthogonal Hamiltonians, we develop a quantum algorithm for coherent function simulation, built on two quantum signal processing (QSP) structures. It can simultaneously realize a target polynomial based on measurement results regardless of the chosen signal unitary for the QSP. Utilizing semidefinite optimization and group representation theory, we prove that our methods achieve the optimal average success probability, taken over possible Hamiltonians H and parameters θ, decays as O(1/k) with k queries of the unknown unitary transformation. Furthermore, we demonstrate the validity of our protocol on a superconducting quantum processor. We also investigate a physically motivated recognition task for Heisenberg Hamiltonians, providing numerical evidence for effective multi-qubit quantum system recognition. This work presents an efficient method to recognize Hamiltonians from limited queries of the dynamics, opening new avenues in composite channel discrimination and quantum metrology.
In this brief, we present a Hamiltonian-identification method for a closed quantum system whose time-trace observables are measured with colored measurement noise. The dynamics of the quantum system are described by a Liouville equation that can be converted into a coherence vector representation. Since the measurement process is disturbed by classical colored noise, we introduce an augmented system model to describe the total dynamics, where the classical colored noise is parameterized. Based on the augmented system model as well as the measurement data, we can find a realization of the quantum system with unknown parameters by employing an eigenstate-realization algorithm. The unknown parameters can be identified using a transfer-function-based technique. An example of a two-qubit system with colored measurement noise is shown to verify the effectiveness of our method.