Learning a Local Hamiltonian from Local Measurements

Recovering an unknown Hamiltonian from measurements is an increasingly important task for certification of noisy quantum devices and simulators. Recent works have succeeded in recovering the Hamiltonian of an isolated quantum system with local interactions from long-ranged correlators of a single eigenstate. Here, we show that such Hamiltonians can be recovered from local observables alone, using computational and measurement resources scaling linearly with the system size. In fact, to recover the Hamiltonian acting on each finite spatial domain, only observables within that domain are required. The observables can be measured in a Gibbs state as well as a single eigenstate; furthermore, they can be measured in a state evolved by the Hamiltonian for a long time, allowing us to recover a large family of time-dependent Hamiltonians. We derive an estimate for the statistical recovery error due to approximation of expectation values using a finite number of samples, which agrees well with numerical simulations.

Figure 1

Recovery from local measurements. (Left) Our method recovers $H_L$ (light blue), using only measurements of observables residing in $L$ (solid blue line), a subregion of the whole system $\mathrm{\Lambda }$. The interior $L_0\subseteq L$ (dashed blue line) consists of sites interacting only within $L$. (Right) Our simulations are performed on $|\mathrm{\Lambda }|=12$ chains, recovering $H_L$ on the eight middle spins.

Figure 2

Quality of Hamiltonian reconstruction as a function of the number of measured constraints $N$. We generated ground states of random spin chains [Eq. (14)] and measured local observables $i[A_n,S_m]$ on the eight middle spins $L$ to recover $H_L$. When the number $N$ of constraints $A_n$ exceeded the number $M$ of possible Hamiltonian terms $S_m$ (dashed vertical line), $\mathcal{M}$ opened a gap (light blue; ${\lambda }_0=0$ here). The reconstruction error [red, see Eq. (12)] was solely due to the addition of a small Gaussian noise with standard deviation $\epsilon ={10}^{-12}$ to each measurement and followed closely an estimate obtained from the spectrum of $K$ [dashed purple, see Eq. (13)]. We used all $k$-local constraints $A_n$ up to $k=4$ in an increasing order of support size $k$. The solid vertical lines denote the transition to $k=2$, 3, 4, respectively, and within each $k$ we chose the constraints in random order. Results were averaged over 200 random Hamiltonians; the means and standard deviations were calculated after taking the log.

Figure 3

Quality of Hamiltonian reconstruction from Gibbs states $\rho =(1/Z)e^{-\beta H}$. We reconstructed $H_L$ on the $L=8$ middle spins of random spin chains (14) of length $|\mathrm{\Lambda }|=12$ (see Fig. 1). (Left) As a function of temperature $T={\beta }^{-1}$, using four-local constraints $A_n$. The gap of the correlation matrix $\mathcal{M}$ decreased with temperature (light blue). Correspondingly, the reconstruction error (red) due to a small measurement uncertainty ($\epsilon ={10}^{-12}$) increased according to the estimate (13) obtained from the spectrum of $K$ (dashed purple). (Right) Reconstruction with two-local measurements only, using single-site constraints $A_n$ and multiple Gibbs states of different temperatures. We generated Gibbs states at temperatures in the range ${\beta }^{-1}=[{10}^0,{10}^2]$, chosen with uniform spacings (in log space), which decreased with the number of states. A few different states sufficed; additional states improved the reconstruction quality. Results were averaged over 200 randomizations.

Figure 4

Reconstruction from dynamics, as a function of time, of the final Hamiltonian following a quench at $t=0$. (Left) Reconstruction of a time-independent Hamiltonian from ${\rho }_{\mathrm{av}}(t)=(1/t){\int }_0^t\rho (t^{\prime })d{t}^{\prime }$. While the first excited eigenvalue ${\lambda }_1$ of the correlation matrix $\mathcal{M}$ saturated (light blue), its lowest eigenvalue ${\lambda }_0$ decayed with time (green), leading to a decrease in the reconstruction error $\mathrm{\Delta }$ (red). (Right) Reconstruction of a time-oscillating Hamiltonian $\widehat{H}(t)={\widehat{H}}^{(0)}+J\widehat{V}\cos \omega t$. Here ${\lambda }_1$ decreased with time due to heating, leading to a larger reconstruction error $\mathrm{\Delta }$ compared to the time-independent case (red for ${\widehat{H}}^{(0)}$, purple for $\widehat{V}$). Results were averaged over 50 randomizations.