arXiv:1908.08051 · Physical review. B./Physical review. B

Measurement-induced criticality in random quantum circuits

Chao‐Ming Jian, Yi‐Zhuang You, Romain Vasseur, Andreas W. W. Ludwig

Published Mar 3, 2020• 452 citations

Abstract

A new class of quantum entanglement transitions separating phases with different entanglement entropy scaling has been observed in recent numerical studies. Despite the numerical efforts, an analytical understanding of such transitions has remained elusive. Here, the authors propose a theory for the area-law to volume-law entanglement transition in many-body systems that undergo both random unitary evolutions and projective measurements. Using the replica method, the authors map analytically this entanglement transition to an ordering transition in a classical statistical mechanics model. They derive the general entanglement scaling properties at the transition and show a solvable limit where this transition can be mapped onto two-dimensional percolation.

DOI: 10.1103/physrevb.101.104302

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arXiv:1908.08051 · Physical review. B./Physical review. B

Measurement-induced criticality in random quantum circuits

Chao‐Ming Jian, Yi‐Zhuang You, Romain Vasseur, Andreas W. W. Ludwig

Published Mar 3, 2020• 452 citations

arXiv HTML PDF

Abstract

DOI: 10.1103/physrevb.101.104302

Related papers

Dimensional hybridity in measurement-induced criticality

Oliver Lunt, Marcin Szyniszewski, Arijeet Pal

Entanglement transitions in quantum dynamics present a novel class of phase transitions in non-equilibrium systems. When a many-body quantum system undergoes hybrid quantum dynamics, consisting of unitary evolution interspersed with monitored random measurements, the steady-state can exhibit a phase transition between volume- and area-law entanglement. The role of dimension in the nature of these transitions is an open problem. There is a dimensional correspondence between measurement-induced transitions in non-unitary quantum circuits in $d$ spatial dimensions and classical statistical mechanical models in $d+1$ dimensions, where the time dimension in the quantum problem is mapped to a spatial dimension in the classical model. In this work we show that the role of dimension is considerably richer by unveiling a form of `dimensional hybridity': critical properties of the steady-state entanglement are governed by a combination of exponents consistent with $d$ -dimensional percolation and $(d+1)$ -dimensional percolation. We uncover this dimensional hybridity in 1+1D and 2+1D circuits using a graph-state based simulation algorithm where the entanglement structure is encoded in an underlying graph, providing access to the geometric structure of entanglement. We locate the critical point using the tripartite information, revealing area-law entanglement scaling at criticality, and showing that the entanglement transition coincides with the purification transition. The emergence of this `dimensional hybridity' in these non-unitary quantum circuits sheds new light on the universality of measurement-induced transitions, and opens the way for analyzing the quantum error correcting properties of random unitary circuits in higher dimensions.Entanglement transitions in quantum dynamics present a novel class of phase transitions in non-equilibrium systems. When a many-body quantum system undergoes hybrid quantum dynamics, consisting of unitary evolution interspersed with monitored random measurements, the steady-state can exhibit a phase transition between volume- and area-law entanglement. The role of dimension in the nature of these transitions is an open problem. There is a dimensional correspondence between measurement-induced transitions in non-unitary quantum circuits in $d$ spatial dimensions and classical statistical mechanical models in $d+1$ dimensions, where the time dimension in the quantum problem is mapped to a spatial dimension in the classical model. In this work we show that the role of dimension is considerably richer by unveiling a form of `dimensional hybridity': critical properties of the steady-state entanglement are governed by a combination of exponents consistent with $d$-dimensional percolation and $(d+1)$-dimensional percolation. We uncover this dimensional hybridity in 1+1D and 2+1D circuits using a graph-state based simulation algorithm where the entanglement structure is encoded in an underlying graph, providing access to the geometric structure of entanglement. We locate the critical point using the tripartite information, revealing area-law entanglement scaling at criticality, and showing that the entanglement transition coincides with the purification transition. The emergence of this `dimensional hybridity' in these non-unitary quantum circuits sheds new light on the universality of measurement-induced transitions, and opens the way for analyzing the quantum error correcting properties of random unitary circuits in higher dimensions.

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Measurement-Induced Phase Transitions: Universal Entanglement Scaling at Quantum Criticality — E8 Intelligence Research

Andrew Stewart Caldin

FINDING: Measurement-induced phase transitions (MIPT) in quantum systems reveal a sharp entanglement phase boundary driven by the competition between unitary evolution and projective measurements, with critical points often characterized by universal exponents and effective central charges. | MATH: The MIPT is typically studied via the entanglement entropy $S = -\text{Tr}(\rho_A \ln \rho_A)$ scaling as $S \sim \frac{c}{6} \ln L$ at criticality (for 1+1D conformal field theory), where $c$ is the effective central charge. The critical measurement rate $p_c$ separates volume-law ( $S \propto L$ ) from area-law ( $S \propto \text{const}$ ) phases. For Clifford circuits, the transition is in the percolation universality class with $c = 0$ (or $c = \ln 2$ for Haar random circuits). The "1/(2π)" appears in the context of the quantum Zeno effect: for a system measured at rate $\gamma$ , the effective Hamiltonian is $H_{\text{eff}} = H - i \frac{\gamma}{2} \sum_m |m\rangle\langle Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.comFINDING: Measurement-induced phase transitions (MIPT) in quantum systems reveal a sharp entanglement phase boundary driven by the competition between unitary evolution and projective measurements, with critical points often characterized by universal exponents and effective central charges. | MATH: The MIPT is typically studied via the entanglement entropy \(S = -\text{Tr}(\rho_A \ln \rho_A)$ scaling as $S \sim \frac{c}{6} \ln L$ at criticality (for 1+1D conformal field theory), where $c$ is the effective central charge. The critical measurement rate $p_c$ separates volume-law ($S \propto L$) from area-law ($S \propto \text{const}$) phases. For Clifford circuits, the transition is in the percolation universality class with $c = 0$ (or $c = \ln 2$ for Haar random circuits). The "1/(2π)" appears in the context of the quantum Zeno effect: for a system measured at rate $\gamma$, the effective Hamiltonian is \(H_{\text{eff}} = H - i \frac{\gamma}{2} \sum_m |m\rangle\langle Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com

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d

spatial dimensions and classical statistical mechanical models in

d+1

dimensions, where the time dimension in the quantum problem is mapped to a spatial dimension in the classical model. In this work we show that the role of dimension is considerably richer by unveiling a form of `dimensional hybridity': critical properties of the steady-state entanglement are governed by a combination of exponents consistent with

d

-dimensional percolation and

(d+1)

-dimensional percolation. We uncover this dimensional hybridity in 1+1D and 2+1D circuits using a graph-state based simulation algorithm where the entanglement structure is encoded in an underlying graph, providing access to the geometric structure of entanglement. We locate the critical point using the tripartite information, revealing area-law entanglement scaling at criticality, and showing that the entanglement transition coincides with the purification transition. The emergence of this `dimensional hybridity' in these non-unitary quantum circuits sheds new light on the universality of measurement-induced transitions, and opens the way for analyzing the quantum error correcting properties of random unitary circuits in higher dimensions.

L=48

for two distinct measurement protocols. For both protocols, we observe a transition point where the dominant contribution in the entanglement entropy displays multiplicative logarithmic violations to the area law. We obtain estimates of the correlation length critical exponent at the percent level; these estimates suggest universal behavior and are incompatible with the universality class of 3D percolation.

S = -\text{Tr}(\rho_A \ln \rho_A)

scaling as

S \sim \frac{c}{6} \ln L

at criticality (for 1+1D conformal field theory), where

c

is the effective central charge. The critical measurement rate

p_c

separates volume-law (

S \propto L

) from area-law (

S \propto \text{const}

) phases. For Clifford circuits, the transition is in the percolation universality class with

c = 0

(or

c = \ln 2

for Haar random circuits). The "1/(2π)" appears in the context of the quantum Zeno effect: for a system measured at rate

\gamma

, the effective Hamiltonian is \(H_{\text{eff}} = H - i \frac{\gamma}{2} \sum_m |m\rangle\langle Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com