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arXiv:2310.01699 · Physical review. B./Physical review. B

Quantum entanglement phase transitions and computational complexity: Insights from Ising models

Hanchen Liu, Vikram Ravindranath, Xiao Chen

Published Jan 23, 2025• 3 citations
arXivHTMLPDF

Abstract

In this paper, we construct two-dimensional bipartite cluster states and perform single-qubit measurements on the bulk qubits. We explore the entanglement scaling of the unmeasured one-dimensional boundary state and show that under certain conditions, the boundary state can undergo a volume-law to an area-law entanglement transition driven by variations in the measurement angle. We bridge this boundary state entanglement transition and the measurement-induced phase transition in the nonunitary ($1+1$)D circuit via the transfer matrix method. We also explore the application of this entanglement transition on the computational complexity problems. Specifically, we establish a relation between the boundary state entanglement transition and the sampling complexity of the bipartite 2D cluster state, which is directly related to the computational complexity of the corresponding Ising partition function with complex parameters. By examining the boundary state entanglement scaling, we numerically identify the parameter regime for which the 2D quantum state can be efficiently sampled, which indicates that the Ising partition function can be evaluated efficiently in such a region.

DOI
10.1103/physrevb.111.024312

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In this paper, we construct two-dimensional bipartite cluster states and perform single-qubit measurements on the bulk qubits. We explore the entanglement scaling of the unmeasured one-dimensional boundary state and show that under certain conditions, the boundary state can undergo a volume-law to an area-law entanglement transition driven by variations in the measurement angle. We bridge this boundary state entanglement transition and the measurement-induced phase transition in the nonunitary (1+11+11+1)D circuit via the transfer matrix method. We also explore the application of this entanglement transition on the computational complexity problems. Specifically, we establish a relation between the boundary state entanglement transition and the sampling complexity of the bipartite 2D cluster state, which is directly related to the computational complexity of the corresponding Ising partition function with complex parameters. By examining the boundary state entanglement scaling, we numerically identify the parameter regime for which the 2D quantum state can be efficiently sampled, which indicates that the Ising partition function can be evaluated efficiently in such a region.
We investigate the phase diagram at the boundary of an infinite two-dimensional cluster state subject to bulk measurements using tensor network methods. The state is subjected to uniform measurements $M=cos\ensuremath{\theta}Z+sin\ensuremath{\theta}X$ on the lower boundary qubits and in all bulk qubits. Our results show that the boundary of the system exhibits volume-law entanglement at the measurement angle $\ensuremath{\theta}=\ensuremath{\pi}/2$ and area-law entanglement for any $\ensuremath{\theta}<\ensuremath{\pi}/2$. Within the area-law phase, a phase transition occurs at ${\ensuremath{\theta}}_{c}=1.371$. The phase with $\ensuremath{\theta}\ensuremath{\in}({\ensuremath{\theta}}_{c},\ensuremath{\pi}/2)$ is characterized by a noninjective matrix product state, which cannot be realized as the unique ground state of a one-dimensional local, gapped Hamiltonian. Instead, it resembles a cat state with spontaneous symmetry breaking. These findings demonstrate that the phase diagram of the boundary of a two-dimensional system can be more intricate than that of a standard one-dimensional system.
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.
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.
We prepare two-dimensional states generated by shallow circuits composed of (1) one layer of the two-qubit controlled-Z (CZ) gate or (2) a few layers of the two-qubit random Clifford gate. After measuring all of the bulk qubits, we study the entanglement structure of the remaining qubits on the one-dimensional boundary. In the first model, we observe that the competition between the bulk XXX and ZZZ measurements can lead to an entanglement phase transition between an entangled volume law phase and a disentangled area law phase. We numerically evaluate the critical exponents and generalize this idea to other qudit systems with a local Hilbert space dimension larger than 2. In the second model, we observe the entanglement transition by varying the density of the two-qubit gate in each layer. We give an interpretation of this transition in terms of the random bond Ising model in a similar shallow circuit composed of random Haar gates.
In quantum field theory, entanglement entropy under spatial bipartitioning serves as a powerful information-theoretic probe of quantum correlations. In this work, we present the first comprehensive numerical study of the dynamical evolution and geometric scaling of entanglement entropy in a nonperturbative, strong-field QED setting -- specifically, in the context of the Sauter-Schwinger effect. While the weak-field regime is dominated by area-law states, we show that the entanglement entropy undergoes a transition from area-law to a volume-law scaling for certain strong-field regimes in the pulse-profile parameter space -- signaling a fundamental shift in the underlying correlation structure induced by nonperturbative pair production. For intermediate regimes, the scaling is a power-law that interpolates between area- and volume-law behavior. Finally, we provide interpretations based on the behavior of the low-energy pair-creation spectrum and discuss how these insights could inform future investigations of related phenomena.
The presence of quantum noises inherent to real physical systems can strongly impact the physics in hybrid quantum circuits with local random unitaries and midcircuit measurements. The quantum noises with a size-independent occurring probability can lead to the disappearance of a measurement-induced entanglement phase transition and the emergence of a single area-law phase. In this work, we investigate the effects of quantum noises with size-dependent probabilities $q=p/{L}^{\ensuremath{\alpha}}$, where $\ensuremath{\alpha}$ represents the scaling exponent. We have identified a noise-induced entanglement phase transition from a volume law to a power (area) law in the presence (absence) of measurements as ppp increases when $\ensuremath{\alpha}=1$. With the help of an effective statistical model, we reveal that the phase transition is of first order arising from the competition between two types of spin configurations and shares the same analytical understanding as the noise-induced coding transition. This unified picture further deepens the understanding of the connection between entanglement behavior and the capacity of information protection. When $\ensuremath{\alpha}\ensuremath{\ne}1$, one spin configuration always dominates regardless of ppp and thus the phase transition disappears. Moreover, we highlight the difference between the effects of size-dependent bulk noise and boundary noises. We validate our analytical predictions with extensive numerical results from stabilizer circuit simulations.