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

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.

Figure 1

Dimension estimation conceptual scheme. We first derive the function $f$ such that $N=f\left(n\right)$. Through the measurement, we can determine the model order $n$, yielding the system dimension as $\text{dim}\left(\mathcal{H}\right)=2^N=2^{f\left(n\right)}$.

Figure 2

$n=4$: Actual model order is $n=4$. We plot the median of the ratios between the singular values ${\{{\lambda }_k/{\lambda }_{k+1}\}}_{k=1}^{15}$ over 500 random Hamiltonian realizations as a function of the possible model order $k$. We choose the sampling time $dt$ by assuming that the number of qubits $N$ could be at most 10, and all the coupling strengths take the possible maximum number 100 (this assumptions set ${\omega }_{\text{max}}$). For each Hamiltonian, we construct $100\times 100$ Hankel matrix ${\mathbf{H}}_{100}$ and estimate $\{{\lambda }_k/{\lambda }_{k+1}\}$ 100 times to evaluate the average. Solid line with circles: ${\sigma }^2={10}^{-7}$; dashed line with circles: variance ${\sigma }^2={10}^{-6}$; dotted line: variance ${\sigma }^2={10}^{-5}$; dashed-dotted line: variance ${\sigma }^2={10}^{-4}$. The error bars are the median of the standard deviation of the singular value ratio over 500 random Hamiltonian realizations. The final sharp peak occurs at $k=4$, and from $k=5$ the ratios are almost the same. The final peak becomes sharper as the variance of the noise becomes smaller.

Figure 3

$n=5$: Actual model order is $n=5$. Simulation details are the same as in Fig. 2. The final sharp peak occurs at $k=5$, and from $k=6$ the ratios are almost same. The final peak becomes sharper as the variance of the noise becomes smaller.

Figure 4

$n=6$: Actual model order is $n=6$. Simulation details are the same as in Fig. 2. We can see that when SNR $\ge 25\text{dB}$ the final peak occurs at $k=6$, and we can see the flatting part from $k=7$. The final peak becomes much sharper as the variance of the noise becomes smaller.