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arXiv:2302.09094

Measurement-induced entanglement transitions in quantum circuits of non-interacting fermions: Born-rule versus forced measurements

Chao‐Ming Jian, Hassan Shapourian, Bela Bauer, Andreas W. W. Ludwig

Published Feb 17, 2023• 7 citations
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Abstract

We address entanglement transitions in monitored random quantum circuits of non-interacting fermions, in particular, the question of whether Born-rule and forced measurements yield the same universality class. For a generic circuit with no symmetry other than fermion parity, acting on a one-dimensional Majorana chain, we numerically obtain several critical exponents, providing clear evidence that the two transitions with Born-rule and forced measurements are in different universality classes. We provide a theoretical understanding for our numerical results by identifying the underlying statistical mechanics model which follows from the general correspondence, established in Jian et al., Phys. Rev. B 106, 134206, between non-unitary circuits of non-interacting fermions and the ten-fold Altland-Zirnbauer (AZ) symmetry classes. The AZ class is the same for Born-rule and forced measurements of the circuits. For the circuit under consideration (in AZ class DIII), the statistical mechanics model describing the transition is the principal chiral non-linear sigma model whose field variable is an ${\rm SO}(n)$ matrix in the replica limits $n\to 0$ and $n\to 1$ for forced and Born-rule measurements, respectively. The former is in an Anderson localization universality class while we show that the latter is in a novel universality class beyond Anderson localization. Both entanglement transitions are driven by proliferation of $\mathbb{Z}_2$ topological defects. The different replica limits account for the difference in the universality classes. Furthermore, we provide numerical and symmetry-based arguments that the entanglement transition in the previously-studied monitored circuit of Majorana fermions based on the loop model with crossings, a highly fine-tuned circuit, belongs to a universality class different from both transitions in the generic circuits discussed in this paper.

DOI
10.48550/arxiv.2302.09094

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We address entanglement transitions in monitored random quantum circuits of non-interacting fermions, in particular, the question of whether Born-rule and forced measurements yield the same universality class. For a generic circuit with no symmetry other than fermion parity, acting on a one-dimensional Majorana chain, we numerically obtain several critical exponents, providing clear evidence that the two transitions with Born-rule and forced measurements are in different universality classes. We provide a theoretical understanding for our numerical results by identifying the underlying statistical mechanics model which follows from the general correspondence, established in Jian et al., Phys. Rev. B 106, 134206, between non-unitary circuits of non-interacting fermions and the ten-fold Altland-Zirnbauer (AZ) symmetry classes. The AZ class is the same for Born-rule and forced measurements of the circuits. For the circuit under consideration (in AZ class DIII), the statistical mechanics model describing the transition is the principal chiral non-linear sigma model whose field variable is an SO(n){\rm SO}(n)SO(n) matrix in the replica limits n→0n\to 0n→0 and n→1n\to 1n→1 for forced and Born-rule measurements, respectively. The former is in an Anderson localization universality class while we show that the latter is in a novel universality class beyond Anderson localization. Both entanglement transitions are driven by proliferation of Z2\mathbb{Z}_2Z2​ topological defects. The different replica limits account for the difference in the universality classes. Furthermore, we provide numerical and symmetry-based arguments that the entanglement transition in the previously-studied monitored circuit of Majorana fermions based on the loop model with crossings, a highly fine-tuned circuit, belongs to a universality class different from both transitions in the generic circuits discussed in this paper.
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We study the critical properties of random quantum circuits with a U(1)U(1)U(1) symmetry subject to local projective measurements that explicitly break this symmetry. We find that, at the measurement-induced phase transition, symmetry-breaking measurements act as a relevant perturbation at large scales, leading to the same universal critical properties as the corresponding monitored random circuit with non-symmetric unitary dynamics. In particular, we consider monitored U(1)U(1)U(1)-symmetric Haar-random circuits in the limit of large local Hilbert-space dimension, where the trajectory-averaged entanglement entropy can be exactly obtained in terms of a classical statistical mechanics model. In this model, the charge associated with the conservation law follows a symmetric simple exclusion process, in which symmetry-breaking measurements correspond to disordered defects that create and destroy charges. We prove that the charge correlation length remains finite for any measurement rate, ruling out a charge-sharpening transition, in contrast to the case of symmetry-preserving measurements. We further support our predictions at finite local Hilbert-space dimension through numerical finite-size scaling analyses of the entanglement transition in monitored U(1)U(1)U(1)-symmetric Haar and stabilizer random circuits.
Quantum circuits offer a versatile platform for simulating digital quantum dynamics and uncovering novel states of nonequilibrium quantum matter. One principal example are measurement-induced phase transitions arising from nonunitary dynamics in monitored circuits, which employ midcircuit measurements as an essential building block next to standard unitary gates. Although a comprehensive understanding of the dynamics in generic circuits is still evolving, we contend that monitored quantum circuits give rise to robust phases of dynamic matter, which---akin to Hamiltonian ground-state phases---yield emergent universal behavior, which can be categorized based on circuit symmetries and spatial dimensionality. To illustrate this concept, we focus on measurement-only quantum circuits within symmetry classes BDI and D, which are measurement-only circuit adaptations of the paradigmatic Kitaev and Yao-Kivelson models, embodying particle-hole-symmetric Majorana fermions with or without time reversal. We establish a general framework (Majorana loop models) for both symmetry classes (in arbitrary spatial dimensions) to provide access to the phenomenology of the entanglement dynamics in these circuits, displaying both an area-law phase of localized Majorana loops and a delocalized, highly entangled Majorana liquid phase. The two phases are separated by a continuous transition displaying quantum Lifshitz scaling, albeit with critical exponents of two distinct, non-Hamiltonian universality classes. The loop model framework provides not only analytical understanding of these universality classes in terms of nonlinear sigma models but also allows for highly efficient numerical techniques capable of simulating excessively large circuits with up to 108{10}^{8}108 qubits. We utilize this framework to accurately determine universal probes that distinguish both the entangled phases and the critical points of the two symmetry classes. Our work thereby further solidifies the concept of emergent circuit phases and their phase transitions.
Projective measurements in random quantum circuits lead to a rich breadth of entanglement phases and extend the realm of nonunitary quantum dynamics.Here, we explore the connection between measurement-only quantum circuits in one spatial dimension and the statistical mechanics of loop models in two dimensions.While Gaussian Majorana circuits admit a microscopic mapping to loop models, for non-Gaussian, i.e., generic Clifford, circuits, a corresponding mapping may emerge only on a coarse-grained scale.We then focus on a fundamental symmetry of loop models: the orientability of worldlines.We discuss how orientability enters in the measurement framework, acting as a separatrix for the universal long-wavelength behavior in a circuit.When orientability is broken, the circuit falls into the universality class of closely packed loops with crossings (CPLC) and features a Goldstone phase with a peculiar, universal log 2 ðLÞ scaling of the entanglement entropy.In turn, when orientability is preserved, the longwavelength behavior of the circuit mimics that of (coupled) two-dimensional Potts models.We demonstrate the strength of the loop model approach by numerically simulating a variety of measurement-only Clifford circuits.Upon varying the set of measured operators, a rich circuit dynamics is observed, ranging from CPLC to the one-state Potts model (percolation), the two-state Potts model (Ising), and coupled Potts models (Berezhinskii-Kosterlitz-Thouless) universality class.Loop models, thus, provide a handle to access a large class of measurement-only circuits and yield a blueprint on how to realize desired entanglement phases by measurement.
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We use Majorana operators to study entanglement dynamics under random free fermion unitary evolution and projective measurements in one dimension. For certain choices of unitary evolution, namely, those which swap neighboring Majorana operators, and measurements of neighboring Majorana bilinears, one can map the evolution to the statistical model of completely packed loops with crossings (CPLC) and study the corresponding phase diagram. We generalize this model using the language of fermionic Gaussian states to a general free fermion unitary evolution acting on neighboring Majorana operators and numerically compute its phase diagram. We find that both the Goldstone and area-law phases persist in this new phase diagram, but with a shifted phase boundary. One important qualitative aspect of the new phase boundary is that even for the case of commuting measurements, the Goldstone phase persists up to a finite nonzero measurement rate. This is in contrast with the CPLC, in which noncommuting measurements are necessary for realizing the Goldstone phase. We also numerically compute the correlation length critical exponent at the transition, which we find to be near to that of the CPLC, and give a tentative symmetry-based explanation for some differences in the phase transition line between the CPLC and generalized models.