Giuseppe Carleo

A central challenge in quantum many-body physics is to understand how operators evolve in time. Key phenomena like operator growth, transport, and scrambling are most naturally framed in the Heisenberg picture, but classical approaches (e.g., tensor networks or Pauli propagation) run into limits set by entanglement or magic. In the first paper of this joint submission, we develop a general Heisenberg-picture simulation framework for quantum computers based on vectorization and transfer-matrix ideas. This lets us represent time-evolved Heisenberg operators as quantum states in a structure-preserving way, so a wide range of Heisenberg-native tasks can be recast as standard state-based procedures. As a result, we obtain new quantum algorithms for computing many operator diagnostics, including Pauli statistics, OTOCs, superoperator moments, two-point correlators, and operator entanglement/stabilizer entanglement measures. In the second paper, we target the problem of learning many correlators simultaneously. We introduce “shadows of Heisenberg operators,” obtained via randomized measurements with local or global Clifford schemes, enabling simultaneous estimation of broad families of OTOCs or two-point correlators from shared data. We also prove information-theoretic lower bounds for multi-OTOC estimation across different learning models, yielding exponential separations that formalize when and why the vectorized approach provides genuine measurement-efficiency advantages.