Free fermions under adaptive quantum dynamics

Vikram Ravindranath; Zhi-Cheng Yang; Xiao Chen

doi:10.22331/q-2025-04-03-1685

Free fermions under adaptive quantum dynamics

Vikram Ravindranath¹, Zhi-Cheng Yang^2,3, and Xiao Chen¹

¹Department of Physics, Boston College, Chestnut Hill, MA 02467, USA
²School of Physics, Peking University, Beijing 100871, China
³Center for High Energy Physics, Peking University, Beijing 100871, China

Published:	2025-04-03, volume 9, page 1685
Editor:	Abhinav Deshpande
Eprint:	arXiv:2306.16595v3
Doi:	https://doi.org/10.22331/q-2025-04-03-1685
Citation:	Quantum 9, 1685 (2025).

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Abstract

We study free fermion systems under adaptive quantum dynamics consisting of unitary gates and projective measurements followed by corrective unitary operations. We further introduce a classical flag for each site, allowing for an active or inactive status which determines whether or not the unitary gates are allowed to apply. In this dynamics, the individual quantum trajectories exhibit a measurement-induced entanglement transition from critical to area-law scaling above a critical measurement rate, similar to previously studied models of free fermions under continuous monitoring. Furthermore, we find that the corrective unitary operations can steer the system into a state characterized by charge-density-wave order. Consequently, an additional phase transition occurs, which can be observed at both the level of the quantum trajectory and the quantum channel. We establish that the entanglement transition and the steering transition are fundamentally distinct. The latter transition belongs to the parity-conserving (PC) universality class, arising from the interplay between the inherent fermionic parity and classical labelling. We demonstrate both the entanglement and the steering transitions via efficient numerical simulations of free fermion systems, which confirm the PC universality class of the latter.

Featured image: (Left, Fig. 6 in text) The phase diagram exhibited by the family of circuits considered in this paper. $p$ refers to the rate at which measurements are made on each pair of qubits, while $r$ is the rate at which feedback is applied. The MIPT is insensitive to feedback. In the "absorbing" phase, the final state is $\left|\psi_{\rm targ}\right\rangle$.
(Right, Fig. 8 in text) An illustration of the classical mapping between occupation numbers and fictitious $\bullet$ particles, along with the flags that deem a site to be (in)active. Measurements and feedback impose restrictions on the random walkers (which are annihilated or created in pairs), so that once the system is in an "empty" state that is also inactive, it remains in that state for the remainder of the protocol.

Popular summary

Entanglement is a fundamentally quantum property which was thought to be susceptible to measurements, until the recent discovery of the measurement-induced phase transition (MIPT). Quantum systems, under a variety of scenarios, can retain entanglement through measurements. This leads to a transition as the frequency of measurements is increased. Parallely, measurements and feedback – operations that alter the system based on measurement outcomes – can also efficiently prepare a plethora of quantum states of interest, by using measurements to detect "errors" and feedback to "correct" them.

How, then, do these processes interact with each other? In this work, we considered a system of free fermions subject to unitary gates, randomly placed particle number measurements, and feedback. We proposed a hybrid classical-quantum scheme to prepare a state with staggered occupations $\left|\psi_{\rm targ}\right\rangle = \left|1010\dots\right\rangle$; this scheme augments each site with a classical flag that determines whether that site is affected by quantum gates ("active") or not ("inactive").

We showed that a minimum rate of measurements and feedback is needed in order to prepare $\left|\psi\right\rangle$. We also found that feedback does not alter the MIPT. Despite the innate quantumness of entanglement, we uncovered a mapping between the quantum circuit and branching-annihilating random walks (BARW) – paradigmatic models of nonequilibrium phase transitions. This mapping provides a physical understanding of the phases that this model can exhibit, as well as the quantitative critical properties of the transitions between those phases.

► BibTeX data

@article{Ravindranath2025freefermionsunder,
  doi = {10.22331/q-2025-04-03-1685},
  url = {https://doi.org/10.22331/q-2025-04-03-1685},
  title = {Free fermions under adaptive quantum dynamics},
  author = {Ravindranath, Vikram and Yang, Zhi-Cheng and Chen, Xiao},
  journal = {{Quantum}},
  issn = {2521-327X},
  publisher = {{Verein zur F{\"{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}},
  volume = {9},
  pages = {1685},
  month = apr,
  year = {2025}
}

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