Hamiltonian identifiability assisted by a single-probe measurement

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 Gröbner 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.

Figure 1

Hamiltonian identification model. A quantum probe is coupled to one end of the spin chain. Apart from the quantum sensor (red spin), the rest of the spins (blue spins) are initially in the maximally mixed state. We further assume that we only have selective control on the quantum probe and global control on the spin chain.

Figure 2

Identifiability with external control. By applying a periodic control pulse sequence $n$ times in the limit of a very small $J_k\delta t\ll 1$, we can transfer $H_{\text{is}}={\sum }_{k=1}^{N-1}\frac{J_k}{2}{S}_k^{\alpha }{S}_{k+1}^{\alpha }$ to $H_{\text{ex}}={\sum }_{k=1}^{N-1}\frac{J_k}{2}(S_k^{\alpha }{S}_{k+1}^{\alpha }+S_k^{\beta }{S}_{k+1}^{\beta })$ so that we can use $2N$ sampling points to identify the parameters $J_k$.

Figure 3

Estimation error with fixed time step $\mathrm{\Delta }t$. Median of the estimation error $\left\{\langle \epsilon \left(J_i\right)\rangle \right\}$ over 500 random Hamiltonian realizations as a function of the total number of measurements. For each Hamiltonian, we repeated the ERA estimation 100 times, to evaluate the average error $\langle \epsilon \left(J_i\right)\rangle$. Solid lines with circles: $4\times 4$ Hankel matrix; dashed lines with solid square: $8\times 8$; dotted lines with squares: $40\times 40$. The error bars are the absolute median deviation. We plot results for the three $J$ couplings, $J_1$ (black), $J_2$ (blue, gray), and $J_3$ (orange, light gray).

Figure 4

Estimation error with fixed total time $T$. Median of the estimation error $\left\{\langle \epsilon \left(J_i\right)\rangle \right\}$ over 500 random realizations of the Hamiltonian as a function of the total number of measurements. For each Hamiltonian, we repeated the ERA estimation 100 times, to evaluate the averaged error. Solid lines with circles: $4\times 4$ Hankel matrix; dashed lines with solid square: $8\times 8$. The error bars are the absolute median deviation.