Error tomography in many-body quantum simulation

Quantum simulation of many-body Hamiltonians constitutes early advances in multi-

qubit quantum control. This has been demonstrated in linear ion traps and in ultracold atoms

trapped in optical lattices and tweezer arrays. An important next step is to characterize the

accuracy of this quantum control in terms of a figure-of-merit. This problem is challenging due to

the large Hilbert space dimension and well-known techniques such as randomized

benchmarking are not effective for a specific multi-qubit gate. Here, we develop an experimental

protocol to characterize the errors in many-body Hamiltonians in trapped ultracold atom

experiments [1]

We consider two forms of errors: (i) unitary errors arising out of systematic errors in the applied

Hamiltonian and (ii) canonical non-Markovian errors arising out of random shot-to-shot

fluctuations in the applied Hamiltonian. We show that the dynamics of the expectation value of

the target Hamiltonian itself, which is ideally constant in time, can be used to characterize these

errors. In the presence of errors, the expectation value of the target Hamiltonian shows a

characteristic thermalization dynamics, when it satisfies the operator thermalization hypothesis

(OTH). That is, an oscillation in the short time followed by relaxation to a steady-state value in

the long time limit. We show that while the steady-state value can be used to characterize the

coherent errors, the amplitude of the oscillations can be used to estimate the non-Markovian

errors.

[1] Aditya Prakash and Bharath Hebbe Madhusudhana, A symmetry-based protocol to

benchmark quantum simulation of many-body physics , arXiv: 2311.03452