Fast quantum computation with all-to-all Hamiltonians

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 $\mathrm{O}(1)$ strength, programmable time-dependent all-to-all Hamiltonians can simulate arbitrary all-to-all quantum circuits, performing quantum computation in time proportional to the circuit depth. We show that this naive correspondence is far from optimal: all-to-all Hamiltonians can process information on much shorter timescales.
First, we prove 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 and entangled states, such as the multiply-controlled Toffoli gate and the GHZ and W states, can be generated in $\mathrm{O}(1/N)$ time; 2) Information could propagate in a fast way that saturates known Lieb-Robinson bounds in strongly power-law interacting systems.
Our second main result proves that any depth-$D$ quantum circuit can be simulated by a randomized Hamiltonian protocol in time $T=\mathrm{O}(D/\sqrt{N})$, with constant space overhead and polynomially small error. Applied to circuit ensembles forming unitary designs and pseudorandom unitaries, this simulation gives an operational proof of the fast scrambling conjecture for dense Hamiltonians.
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 physical ideas.