Hamiltonian-Driven Shadow Tomography of Quantum States
This addresses the problem of efficient quantum state tomography for researchers in quantum computing, offering a potentially more practical approach for near-term devices, though it appears incremental as it builds on existing shadow tomography methods.
The paper tackles the challenge of realizing deep unitary circuits for classical shadow tomography on near-term quantum devices by proposing a method using shallow unitary channels generated by quantum chaotic Hamiltonians, showing it can be more efficient for predicting Pauli observables in certain time windows, with improvements such as a factor of D for diagonal observables without sacrificing off-diagonal ones.
Classical shadow tomography provides an efficient method for predicting functions of an unknown quantum state from a few measurements of the state. It relies on a unitary channel that efficiently scrambles the quantum information of the state to the measurement basis. Facing the challenge of realizing deep unitary circuits on near-term quantum devices, we explore the scenario in which the unitary channel can be shallow and is generated by a quantum chaotic Hamiltonian via time evolution. We provide an unbiased estimator of the density matrix for all ranges of the evolution time. We analyze the sample complexity of the Hamiltonian-driven shadow tomography. For Pauli observables, we find that it can be more efficient than the unitary-2-design-based shadow tomography in a sequence of intermediate time windows that range from an order-1 scrambling time to a time scale of $D^{1/6}$, given the Hilbert space dimension $D$. In particular, the efficiency of predicting diagonal Pauli observables is improved by a factor of $D$ without sacrificing the efficiency of predicting off-diagonal Pauli observables.