Exact dimension estimation of interacting qubit systems assisted by a single quantum probe

Abstract

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.

Learning a Local Hamiltonian from Local Measurements

“…Given a region L whose Hamiltonian we wish to learn, our method is therefore as follows: The lowest right-singular vector of K is the numerical solution to Eq. (7), the vector that minimizes Kc . Namely, it is the ground-state of the correlation matrix,…”

“…Various methods have been suggested for recovering a Hamiltonian based on its dynamics [2][3][4][5][6][7][8] or Gibbs state [9][10][11]. The system-size scaling of the recovery efficiency can be improved using a trusted quantum simulator [12][13][14][15][16], manipulations of the investigated system [17], or accurate measurements of short-time dynamics [18,19].…”

Pulsed Quantum-State Reconstruction of Dark Systems

“…However, even with the rapid progress in the coherent manipulation and quantum-state tomography of several quantum systems, such as photons [6,7], electron spins [8][9][10], atomic qubits [11], superconducting circuits [12,13], and mechanical resonators [14,15], many quantum systems still remain difficult to access for a direct observation of their state, systems we will refer to as dark. In order to circumvent the requirement of such a direct access, a promising technique is to employ an auxiliary quantum system as a measurement probe, on which measurements as well as coherent manipulations can be performed [16][17][18][19][20][21][22][23]. Interferometry [24] based on such a measurement probe allows us to extract information on a target system [25][26][27][28][29][30].…”

Quantum Hamiltonian Identifiability via a Similarity Transformation Approach and Beyond

“…We have the following lemma (the proof is given in Appendix D) to show that the system is minimal. Lemma 4: With (27), (28), (35) and (36), both the controllability matrix CM = [B AB · · · A n B] and the observability matrix OM = [C T A T C T · · · A nT C T ] T have full rank for almost any value of θ .…”

“…We assume the dimension [28] and structure (e.g., the coupling types) [29] of the Hamiltonian is already determined, and the task is to identify unknown parameters in the Hamiltonian. It is natural to resort to identifiability test methods in classical (non-quantum) control field to tackle the quantum Hamiltonian identifiability problem.…”