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arXiv:2012.04666 · Physical review. B./Physical review. B

Criticality and entanglement in nonunitary quantum circuits and tensor networks of noninteracting fermions

Chao‐Ming Jian, Bela Bauer, Anna Keselman, Andreas W. W. Ludwig

Published Oct 21, 2022• 107 citations
arXivHTMLPDF

Abstract

Entanglement dynamics in nonunitary quantum systems have been found to exhibit novel nonequilibrium quantum phase transitions and have attracted tremendous attention with connections to various aspects of quantum information theory. This work establishes, for systems of noninteracting fermions, an exact three-way correspondence between (i) nonunitary (as well as unitary) quantum circuits dynamics in $d$ spatial dimensions; (ii) ($d$+1)-dimensional Gaussian tensor networks, and (iii) fermion systems in ($d$+1) spatial dimensions subject to unitary evolution with static Hamiltonians. Through this correspondence, $d$-dimensional random quantum circuits are connected with the classic subject area of Anderson localization in ($d$+1) spatial dimensions and consequently follow the ``tenfold way'' Altland-Zirnbauer classification. Selected examples of quantum circuits exhibiting entanglement phases and criticalities are used to illustrate these general principles.

DOI
10.1103/physrevb.106.134206

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Entanglement dynamics in nonunitary quantum systems have been found to exhibit novel nonequilibrium quantum phase transitions and have attracted tremendous attention with connections to various aspects of quantum information theory. This work establishes, for systems of noninteracting fermions, an exact three-way correspondence between (i) nonunitary (as well as unitary) quantum circuits dynamics in ddd spatial dimensions; (ii) (ddd+1)-dimensional Gaussian tensor networks, and (iii) fermion systems in (ddd+1) spatial dimensions subject to unitary evolution with static Hamiltonians. Through this correspondence, ddd-dimensional random quantum circuits are connected with the classic subject area of Anderson localization in (ddd+1) spatial dimensions and consequently follow the ``tenfold way'' Altland-Zirnbauer classification. Selected examples of quantum circuits exhibiting entanglement phases and criticalities are used to illustrate these general principles.
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 ddd spatial dimensions and classical statistical mechanical models in d+1d+1d+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 ddd-dimensional percolation and (d+1)(d+1)(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.
Tensor network quantum states are powerful tools for strongly correlated systems, tailored to capture local correlations such as in ground states with entanglement area laws. When applying tensor network states to interacting fermionic systems, a proper choice of the basis or orbitals can reduce the bond dimension of tensors and provide physically relevant orbitals. We introduce such a change of basis with unitary gates obtained from compressing fermionic Gaussian states into quantum circuits corresponding to various tensor networks. These circuits can reduce the ground-state entanglement entropy and improve the performance of algorithms such as the density matrix renormalization group. We study the Anderson impurity model with one and two impurities to show the potential of the method for improving computational efficiency and interpreting impurity physics. Furthermore, fermionic Gaussian circuits can also suppress entanglement during the time evolution out of low-energy state. Last, we consider Gaussian multiscale entanglement renormalization ansatz (GMERA) circuits which compress fermionic Gaussian states hierarchically. The emergent coarse-grained physical models from these GMERA circuits are studied in terms of their entanglement properties and suitability for performing time evolution.
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.
We present random quantum circuit models for nonunitary quantum dynamics of free fermions in one spatial dimension.Numerical simulations reveal that the dynamics tends toward steady states with logarithmic violations of the entanglement area law and power law correlation functions.Moreover, starting with a short-range entangled many-body state, the dynamical evolution of entanglement and correlations quantitatively agrees with the predictions of two-dimensional conformal field theory with a spacelike time direction.We argue that this behavior is generic in nonunitary free quantum dynamics with time-dependent randomness, and we show that the emergent conformal dynamics of two-point functions arises out of a simple "nonlinear master equation."
Time evolution of quantum many-body systems typically leads to a state with maximal entanglement allowed by symmetries.Two distinct routes to impede entanglement growth are inducing localization via spatial disorder, or subjecting the system to nonunitary evolution, e.g., via projective measurements.Here we employ the idea of spacetime rotation of a circuit to explore the relation between systems that fall into these two classes.In particular, by spacetime rotating unitary Floquet circuits that display a localization transition, we construct nonunitary circuits that display a rich variety of entanglement scaling and phase transitions.One outcome of our approach is a nonunitary circuit for free fermions in one dimension that exhibits an entanglement transition from logarithmic scaling to volume-law scaling.This transition is accompanied by a "purification transition" analogous to that seen in hybrid projective-unitary circuits.We follow a similar strategy to construct a nonunitary two-dimensional (2D) Clifford circuit that shows a transition from area to volume-law entanglement scaling.Similarly, we spacetime rotate a 1D spin chain that hosts many-body localization to obtain a nonunitary circuit that exhibits an entanglement transition.Finally, we introduce an unconventional correlator and argue that if a unitary circuit hosts a many-body localization transition then the correlator is expected to be singular in its nonunitary counterpart as well.
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.