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Shadow tomography aims to build a classical description of a quantum state from a sequence of simple random measurements. Physical observables are then reconstructed from the resulting classical shadow. Shadow protocols which use single-body random measurements are simple to implement and capture few-body observables efficiently, but do not apply to systems with fundamental number conservation laws, such as ultracold atoms. We address this shortcoming by proposing and analyzing a new local shadow protocol adapted to such systems. The "All-Pairs" protocol requires one layer of two-body gates and only $\textrm{poly}(V)$ samples to reconstruct arbitrary few body observables. Moreover, by exploiting the permutation symmetry of the protocol, we derive a linear time post-processing algorithm. We provide a proof-of-principle reference implementation and demonstrate the reconstruction of 2- and 4-point functions in a paired Luttinger liquid of hardcore bosons., Comment: 6 pages 3 figures

Local random circuits scramble efficiently and accordingly have a range of applications in quantum information and quantum dynamics. With a global $U(1)$ charge however, the scrambling ability is reduced; for example, such random circuits do not generate the entire group of number-conserving unitaries. We establish two results using the statistical mechanics of $k$-fold replicated circuits. First, we show that finite moments cannot distinguish the ensemble that local random circuits generate from the Haar ensemble on the entire group of number-conserving unitaries. Specifically, the circuits form a $k_c$-design with $k_c = O(L^d)$ for a system in $d$ spatial dimensions with linear dimension $L$. Second, for $k < k_c$, we derive bounds on the depth $\tau$ required for the circuit to converge to an approximate $k$-design. The depth is lower bounded by diffusion $k L^2 \ln(L) \lesssim \tau$. In contrast, without number conservation $\tau \sim \text{poly}(k) L$. The convergence of the circuit ensemble is controlled by the low-energy properties of a frustration-free quantum statistical model which spontaneously breaks $k$ $U(1)$ symmetries. We conjecture that the associated Goldstone modes set the spectral gap for arbitrary spatial and qudit dimensions, leading to an upper bound $\tau \lesssim k L^{d+2}$., Comment: 18 pages, 2 figures

We determine the zero and finite temperature phase diagram of the fully frustrated quantum Ising model on the bathroom tile (4-8) lattice. The phase diagram exhibits a wealth of 2+1d physics, including 1. classical Coulomb dimer liquids of both square and triangular lattice types; 2. quantum order-by-disorder induced phases breaking $\mathbb{Z}_4$, $\mathbb{Z}_6$, and $\mathbb{Z}_8$ symmetries; 3. finite temperature Kosterlitz-Thouless (KT) phases floating over the $\mathbb{Z}_6$ and $\mathbb{Z}_8$ orders; and, 4. staircases of (in)-commensurate symmetry breaking phases at intermediate coupling. We establish this elaborate phase diagram using a combination of dimer model mapping, perturbation theory, Landau analysis and Stochastic Series Expansion Quantum Monte Carlo (QMC-SSE). Our results provide a baseline for studying frustrated magnetism with D-Wave architecture annealers, where the 4-8 lattice can be embedded naturally without `cloning', reducing the number of competing energy scales. Simulations with the D-Wave 2000Q demonstrate qualitative agreement with the high temperature portion of the phase diagram, but are unable to access the low temperature phases., Comment: 14 pages, 20 figures