Liouvillian Tomography in Noisy Intermediate-scale Quantum Computers

In the past four decades, many exciting applications of quantum computers have been proposed. However, the current noisy intermediate-scale quantum computers (NISQ) are severely limited by faulty gates and environmental interactions. To assess the role these factors play in qubit dynamics, Liouvillian tomography protocols were introduced as means of estimating the generator of the qubits’ Markovian evolution. In this work, we propose a novel Liouvillian tomography technique that eliminates the common Markovian assumption by considering a regression problem over derivatives of Pauli string probability distributions.

Additionally, two companion algorithms for process and self-consistent state preparation and measurement (SPAM) tomography are presented. All three tomographic procedures are benchmarked with simulated data for systems of 1,2 and 3 qubits. SPAM and process tomography were demonstrably capable of producing high-fidelity estimates sufficient for the application of Liouvillian tomography.

The retrieved Liouvillians were shown to match the target values, at the level of the Hamiltonian, dissipative rates, and spectrum. Conversely, the jump operators were observed to be more susceptible to noise, resulting in estimates that deviated from the target values. Liouvillian tomography is then applied to two-qubit circuits in the Helmi quantum computer, to retrieve the generator of the dynamics obtained when qubits are left to idle. Despite some limitations, we believe this work marks an advancement in the field of Liouvillian tomography by providing the first protocol that captures non-Markovian dynamics.