Theory of the phase transition in random unitary circuits with measurements

By Yimu Bao, Soonwon Choi, E. Altman. Physical review B, 2019

399 citations recorded by the source corpus.

We present a theory of the entanglement transition tuned by measurement strength in qudit chains evolved by random unitary circuits and subject to either weak or random projective measurements. The transition can be understood as a nonanalytic change in the amount of information extracted by the measurements about the initial state of the system, quantified by the Fisher information. To compute the von~Neumann entanglement entropy $S$ and the Fisher information $\mathcal{F}$, we apply a replica method based on a sequence of quantities $\tilde{S}^{(n)}$ and $\mathcal{F}^{(n)}$ that depend on the $n$-th moments of density matrices and reduce to $S$ and $\mathcal{F}$ in the limit $n\to 1$. These quantities with $n\ge 2$ are mapped to free energies of a classical spin model with $n!$ internal states in two dimensions with specific boundary conditions. In particular, $\tilde{S}^{(n)}$ is the excess free energy of a domain wall terminating on the top boundary, and $\mathcal{F}^{(n)}$ is related to the magnetization on the bottom boundary. Phase transitions occur as the spin models undergo ordering transitions in the bulk. Taking the limit of large local Hilbert space dimension $q$ followed by the replica limit $n\to 1$, we obtain the critical measurement probability $p_c=1/2$ and identify the transition as a bond percolation in the 2D square lattice in this limit. Finally, we show there is no phase transition if the measurements are allowed in an arbitrary nonlocal basis, thereby highlighting the relation between the phase transition and information scrambling. We establish an explicit connection between the entanglement phase transition and the purification dynamics of a mixed state evolution and discuss implications of our results to experimental observations of the transition and simulability of quantum dynamics.

Subfields: Artificial Intelligence, Atomic and Molecular Physics, and Optics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/91c6aa1d16275418e6253f0b26f514bf8157917d

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Artificial Intelligence

1. A. Nahum
2. Matthew Fisher
3. V. Khemani
4. B. Skinner
5. D. Huse
6. S. Vijay
7. J. Ruhman
8. Yaodong Li

In Atomic and Molecular Physics, and Optics

1. A. Nahum
2. Matthew Fisher
3. V. Khemani
4. D. Huse
5. B. Skinner
6. Yaodong Li
7. J. Ruhman
8. S. Vijay

In Statistical and Nonlinear Physics

1. A. Nahum
2. V. Khemani
3. J. Ruhman
4. B. Skinner
5. S. Gopalakrishnan
6. Tianci Zhou
7. Sthitadhi Roy
8. Romain Vasseur

Citations in the corpus

• Entanglement Membrane in Chaotic Many-Body Systems (2019, background): “[91] Yimu Bao, Soonwon Choi, and Ehud Altman.”
• Dynamical Purification Phase Transition Induced by Quantum Measurements (2019, background): “Interestingly, Bao, Choi, and Altman found a whole family of critical points at intermediate values of p between pce and pcc [41]; however, these critical points emerged only after weighting the trajectories by powers of their Born probability, which leads to the averages being dominated by rare tra”
• Effective Theory for the Measurement-Induced Phase Transition of Dirac Fermions (2021, methodology): “Previous work on measurement-induced phase transitions has mainly focused on the case of projective (discrete) measurements [1, 2, 4, 32, 33]; however, the continuous measurement scenario is smoothly connected to the latter.”
• Entanglement and Charge-Sharpening Transitions in U(1) Symmetric Monitored Quantum Circuits (2021, methodology): “Our conclusions are supported by exact numerics on Haar random U(1) monitored circuits, and by a replica statistical mechanics model [19, 20] obtained from the study of a U(1) symmetric qubit degree of freedom coupled to a d-dimensional qudit.”
• Measurement-induced entanglement phase transition on a superconducting quantum processor with mid-circuit readout (2022, methodology): “In a recent work on a trapped-ion quantum computer, a purification phase transition, expected to occur concurrently with entanglement phase transitions [33, 37], was observed using Clifford circuits and deferred measurements [38].”
• Measurement-protected quantum phases (2020, background): “[15] Yimu Bao, Soonwon Choi, and Ehud Altman, “Theory of the phase transition in random unitary circuits with measurements,” Phys.”
• Measurement and Entanglement Phase Transitions in All-To-All Quantum Circuits, on Quantum Trees, and in Landau-Ginsburg Theory (2020, methodology): “In the many-body case this random dynamics admits a “measurement phase transition” (MPT) between two qualitatively different, stable dynamical phases, with distinct entanglement properties [6–27].”
• Statistical mechanics of quantum error correcting codes (2020, methodology): “As for random Haar circuits [19, 20] and random tensor networks [32], the identification in Eq.”
• Random Quantum Circuits (2022, methodology): “15In the min cut/domain wall picture, this is related to the cost of a horizontal domain wall that separates the initial from the final time (55, 56, 57), and which measures the “entanglement” between the initial and final time boundaries of the non-unitary circuit.”