Observing measurement induced entanglement transitions via a tensor network based hybrid quantum algorithm

Monitored random circuits, consisting of alternating layers of entangling two-qubit gates and projective single-qubit measurements applied to some fraction $p$ of the qubits, have been a topic of recent interest. In particular, the resulting steady state exhibits a phase transition from highly correlated, “volume-law” entanglement states at $p<p_{c}$ to localized states with “area-law” entanglement at $p>p_{c}$.

It is hard to access this transition experimentally, as it cannot be seen at the ensemble level. Naively, to observe it one must repeat the experiment until the set of measurement results repeats itself, with likelihood that is exponentially small in the number of measurements.

To overcome this issue, we present a hybrid quantum-classical algorithm based on the use of matrix product states (MPS), which use polynomial-sized tensor networks to represent quantum states with area-law entanglement. An MPS can thus well-approximate the experimental state above $p_{c}$ but fail exponentially below it. We propose using the breakdown of this approximation to pinpoint the critical point.