Tomotaka Kuwahara

Abstract We study the problem of learning the Hamiltonian of a quantum many-body system given samples from its Gibbs (thermal) state. The classical analog of this problem, known as learning graphical models or Boltzmann machines, is a well-studied question in machine learning and statistics. In this work, we give the first sample-efficient algorithm for the quantum Hamiltonian learning problem. In particular, we prove that polynomially many samples in the number of particles (qudits) are necessary and sufficient for learning the parameters of a spatially local Hamiltonian in l2-norm. Our main contribution is in establishing the strong convexity of the log-partition function of quantum many-body systems, which along with the maximum entropy estimation yields our sample-efficient algorithm. Classically, the strong convexity for partition functions follows from the Markov property of Gibbs distributions. This is, however, known to be violated in its exact form in the quantum case. We introduce several new ideas to obtain an unconditional result that avoids relying on the Markov property of quantum systems, at the cost of a slightly weaker bound. In particular, we prove a lower bound on the variance of quasi-local operators with respect to the Gibbs state, which might be of independent interest. Our work paves the way toward a more rigorous application of machine learning techniques to quantum many-body problems.

Abstract One of the most fundamental problems in quantum many-body physics is the characterization of correlations among thermal states. Of particular relevance is the thermal area law, which justifies the tensor network approximations to thermal states with a bond dimension growing polynomially with the system size. In the regime of sufficiently low temperatures, which is particularly important for practical applications, the existing techniques do not yield optimal bounds. Here, we propose a new thermal area law that holds for generic many-body systems on lattices. We improve the temperature dependence from the original O(β)to ̃O(β^2/3), thereby suggesting diffusive propagation of entanglement by imaginary time evolution. This qualitatively differs from the real-time evolution which usually induces linear growth of entanglement. We also prove analogous bounds for the Rényi entanglement of purification and the entanglement of formation. Our analysis is based on a polynomial approximation to the exponential function which provides a relationship between the imaginary-time evolution and random walks. Moreover, for one-dimensional (1D) systems with n spins, we prove that the Gibbs state is well-approximated by a matrix product operator with a sublinear bond dimension of exp( ̃O(βlog(n))). This allows us to rigorously establish, for the first time, a quasi-linear time classical algorithm for constructing an MPS representation of 1D quantum Gibbs states at arbitrary temperatures ofβ=o(log(n)). Our new technical ingredient is a block decomposition of the Gibbs state, that bears resemblance to the decomposition of real-time evolution given by Haah et al., FOCS’18.

In locally interacting quantum many-body systems, a velocity of the information propagation is finitely bounded and the linear light cone can be defined. Outside the light cone, amount of the information propagation rapidly decays with the distance. When systems have long-range interactions, it is highly nontrivial whether such a linear light cone exists or not. We herein consider generic long-range interacting systems with decaying interactions as $R^{-\alpha}$ with the distance $R$. We rigorously prove the existence of the linear light cone for $\alpha>2D+1$ ($D$: the spatial dimension), where we obtain the Lieb-Robinson bound as $\|[O_i(t),O_j]\| < t^{2D+1}(R-\bar{v}t)^{-\alpha}$ with $\bar{v}=O(1)$ for arbitrary two operators $O_i$ and $O_j$ separated by a distance $R$. Moreover, we give an explicit quantum-state transfer protocol that achieves the above bound up to a constant coefficient and violates the linear light cone for $\alpha<2D+1$. This implies that our Lieb-Robinson bound is the best general upper bound in the regime of $\alpha>2D+1$.