Learning Conservation Laws in Unknown Quantum Dynamics

We present a learning algorithm for discovering conservation laws given as sums of geometrically local observables in quantum dynamics. This includes conserved quantities that arise from local and global symmetries in closed and open quantum many-body systems. The algorithm combines the classical shadow formalism for estimating expectation values of observable and data analysis techniques based on singular value decompositions and robust polynomial interpolation to discover all such conservation laws in unknown quantum dynamics with rigorous performance guarantees. Our method can be directly realized in quantum experiments, which we illustrate with numerical simulations, using closed and open quantum system dynamics in a ${\mathbb{Z}}_2$ gauge theory and in many-body localized spin chains.

Popular Summary

From the times of Kepler and Newton, physicists have been finding conservation laws in nature from experimental data. The rapid development of machine learning provides us with new tools to discover conservation laws in a systematic and efficient manner, as was demonstrated in recent work on learning conservation laws from classical dynamical systems. As we are living in a quantum mechanical world, it is natural to ask, Is it possible to discover conservation laws in quantum systems from measurement data? In this work, we answer this question in the affirmative. We propose a learning algorithm for discovering all local conservation laws in unknown quantum dynamical systems, which can be open or closed, with rigorous guarantees.

Our algorithm uses random Pauli measurements to estimate the expectation values of all local observables and then to extract a subspace of operators with a small amount of fluctuation, which corresponds to the conservation laws. Our algorithm is guaranteed to find approximate representations of all conservations laws with sample and computational complexities polynomially dependent on the system size and precision. Moreover, our method is practical for experimental implementation, requiring only random single-qubit measurements.

We anticipate our method to be useful for capturing conservation laws from quantum experiments as well as for certifying quantum simulation results.

Figure 1

Learning conservation laws in a ${\mathbb{Z}}_2$ gauge theory. In panel (a), we display the 30 smallest eigenvalues of the exact data matrix for various evolution times $T$ and numbers of initial states $N_I$. At sufficiently long times $T$, $N+2$ singular values are gapped out (light and dark blue). The use of multiple initial states (green triangles) allows us to shorten the evolution time considerably. Inset illustrates this effect, showing the gap as a function of the number of initial states for fixed time $T=20$. In panel (b), we display the 20 smallest eigenvalues of the noisy data matrix, constructed from $M={10}^4$ and $M={10}^5$ per initial state ($N_I=15$) and final time $T=20,N_T=41$. For $M={10}^4$ ($M={10}^5$), increasing $M$, two (six) singular values are gapped out. Testing these against independently obtained data yields small variations over time (inset). In panel (c), we display the Pauli basis expansion of the corresponding lowest six singular vectors. We identify the magnetization, Hamiltonian, and four Gauss laws. In all panels, $N=8$ and $N_T=2N_P/N_I=624$. Points with error bars are the average and standard deviation over 25 experiments with identical parameters.

Figure 2

Learning conservation laws in open quantum systems. In panel (a), we display the 20 smallest singular values of $W$ for both the Hamiltonian (blue) and Lindbladian dynamics (orange) with local dephasing with rate $\gamma =0.1$. While magnetization and Gauss laws are conserved in both cases, the Hamiltonian (energy) is only conserved for Hamiltonian dynamics (${\lambda }_2$ is absent for $\gamma =0.1$). We choose $M={10}^6$ measurements and use a Gaussian noise approximation to simulate the resulting shot noise. In all panels, $N=8$, $N_T=41$, $N_I=15$, and points with error bars are the average and standard deviation over 25 experiments with identical parameters. Panel (b) displays the singular values obtained from the data matrix $W_{[j-1,j,j-1]}$ restricted to three adjacent subsystem sites as a function of $j$. This allows us to learn the Gauss laws, corresponding to the gapped singular values at even $j$, with considerably fewer measurements $M={10}^4$ [cf. Fig. 1].

Figure 3

Learning conservation laws in disordered spin chains. (a) We display the ten lowest squared singular values of the data matrix $\hat{W}$ constructed using $N_I=17$ initial states evolved according to $H_{XXZ}$ with different disorder strengths $w=2,4,6$ (blue, orange, green) to a final time $T=40$ and evaluated at $N_T=41$ equally spaced time points. We add Gaussian noise, simulating shot noise arising from $M=5\times {10}^5$ measurements. Each point is averaged over 11 random disorder patterns; the error bars indicate the standard error of the mean. The two smallest singular values ${\lambda }_1,{\lambda }_2$ correspond to the magnetization and Hamiltonian, respectively. As the disorder strength $w$ becomes larger, the singular values ${\lambda }_{i\ge 3}$ decrease significantly, indicating an increasing number of approximate conservation laws. (b) We show the number of singular values below a threshold $ϵ=0.02$ as a function of $w$ for different system sizes. As in panel (a), we choose $M=5\times {10}^5,T=40,N_T=41$. The numbers of initial states for system sizes $N=6,10,14$ are $N_I=9,17,24$, respectively, to ensure that $N_I{N}_T\approx 2N_P$. Points correspond to the median of 11 random Hamiltonian configurations. We observe a sharp increase with increasing disorder strength.

Figure 4

The uniform approximation error $\underset{0\le t\le T}{max}|f(t)-p(t)|$ as a function of the number of samples of $f(t)$ using the Gauss-Chebyshev approach. Both axes are on the logarithmic scale. The gray dashed line has slope $-0.5$, representing the $1/\sqrt{N_s}$ scaling, where $N_s$ is the number of samples. The function being fitted is $\tilde{f}(x)=(1+i)e^{(0.3)i\pi x}+e^{-(0.19)i\pi x}$ with $K=3$.