Chao Yin

Lieb-Robinson bounds demonstrate the emergence of locality in many-body quantum systems. Intuitively, Lieb-Robinson bounds state that with local or exponentially decaying interactions, the correlation that can be built up between two sites separated by distance $r$ after a time $t$ decays as $\exp(vt-r)$, where $v$ is the emergent Lieb-Robinson velocity. In many problems, it is important to also capture how much of an operator grows to act on $r^d$ sites in $d$ spatial dimensions. Perturbation theory and cluster expansion methods suggest that at short times, these volume-filling operators are suppressed as $\exp(-r^d)$ at short times. We confirm this intuition, showing that for $r > vt$, the volume-filling operator is suppressed by $\exp(-(r-vt)^d/(vt)^{d-1})$. This closes a conceptual and practical gap between the cluster expansion and the Lieb-Robinson bound. We then present two very different applications of this new bound. Firstly, we obtain improved bounds on the classical computational resources necessary to simulate many-body dynamics with error tolerance $\epsilon$ for any finite time $t$: as $\epsilon$ becomes sufficiently small, only $\epsilon^{-\mathrm{O}(t^{d-1})}$ resources are needed. A protocol that likely saturates this bound is given. Secondly, we prove that disorder operators have volume-law suppression near the "solvable (Ising) point" in quantum phases with spontaneous symmetry breaking, which implies a new diagnostic for distinguishing many-body phases of quantum matter.

All-to-all interactions arise naturally in many areas of theoretical physics and across diverse experimental quantum platforms, motivating a systematic study of their information-processing power. Assuming each pair of qubits interacts with $\mathcal{O}(1)$ strength, time-dependent all-to-all Hamiltonians can simulate arbitrary all-to-all quantum circuits, performing quantum computation in time proportional to the circuit depth. In this work, we show that this naive correspondence is far from optimal: all-to-all Hamiltonians can process information on much faster timescales. Our first main result establishes that any two-qubit gate can be simulated by all-to-all Hamiltonians on $N$ qubits in time $\mathrm{O}(1/N)$ (up to factor $N^{\delta}$ with an arbitrarily small constant $\delta>0$), with polynomially small error $1/\mathrm{poly}(N)$. Immediate consequences include: 1) Certain $\mathrm{O}(N)$-qubit unitaries, such as the multiply-controlled Toffoli gate, can be realized in $\mathrm{O}(1/N)$ time. 2) Globally entangled states such as the GHZ and W states of $\mathrm{O}(N)$ qubits can be prepared in $\mathrm{O}(1/N)$ time. 3) Any depth-$D$ circuit on $N_{\rm d}$ qubits can be simulated in arbitrarily short time $T=\mathrm{O}(DN_{\rm d}/N)$, inversely proportional to the space overhead $N/N_{\rm d}$. 4) The existing Lieb-Robinson bound for strong power-law interactions $H_{ij}\sim r_{ij}^{-\gamma}$ in spatial dimension $\mathsf{d}>\gamma$ is tight, requiring time $T=\Omega(N^{\frac{\gamma}{\mathsf{d}}-1})$ for information to propagate. Our second main result shows that any depth-$D$ quantum circuit can be simulated in time $T=\mathrm{O}(D/\sqrt{N})$ with constant space overhead, with error $1/\mathrm{poly}(N)$ for almost all inputs. This “optimistic” simulation may suffice for practical purposes: for instance, building on prior work, we demonstrate that Shor’s factoring algorithm can be implemented by $\mathrm{O}(\sqrt{N})$-time Hamiltonian evolution with constant space overhead. The techniques underlying our results depart fundamentally from the existing literature on parallelizing commuting gates: We rely crucially on non-commuting Hamiltonians and draw on diverse ideas from semiclassical quantum mechanics, (spin) squeezing, Mølmer–Sørensen gates, and spin-wave physics.