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arXiv:2305.14231 · Physical Review Research

Triggering boundary phase transitions through bulk measurements in two-dimensional cluster states

Yuchen Guo, Jian-Hao Zhang, Zhen Bi, Shuo Yang

Published Oct 20, 2023• 5 citations
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Abstract

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.

DOI
10.1103/physrevresearch.5.043069

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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\ensuremathθZ+sin\ensuremathθXM=cos\ensuremath{\theta}Z+sin\ensuremath{\theta}XM=cos\ensuremathθZ+sin\ensuremathθX 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 \ensuremathθ=\ensuremathπ/2\ensuremath{\theta}=\ensuremath{\pi}/2\ensuremathθ=\ensuremathπ/2 and area-law entanglement for any \ensuremathθ<\ensuremathπ/2\ensuremath{\theta}<\ensuremath{\pi}/2\ensuremathθ<\ensuremathπ/2. Within the area-law phase, a phase transition occurs at \ensuremathθc=1.371{\ensuremath{\theta}}_{c}=1.371\ensuremathθc​=1.371. The phase with \ensuremathθ\ensuremath∈(\ensuremathθc,\ensuremathπ/2)\ensuremath{\theta}\ensuremath{\in}({\ensuremath{\theta}}_{c},\ensuremath{\pi}/2)\ensuremathθ\ensuremath∈(\ensuremathθc​,\ensuremathπ/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.
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 propose a method, based on matrix product states, for studying the time evolution of many-body quantum lattice systems under continuous and site-resolved measurement. Both the frequency and the strength of generalized measurements can be varied within our scheme, thus allowing us to explore the corresponding two-dimensional phase diagram. The method is applied to one-dimensional chains of nearest-neighbor interacting hard-core bosons. A transition from an entangling to a disentangling (area-law) phase is found. However, by resolving time-dependent density correlations in the monitored system, we find important differences between different regions at the phase boundary. In particular, we observe a peculiar phenomenon of measurement-induced particle clusterization that takes place only for frequent moderately strong measurements, but not for strong infrequent measurements.
We discuss the boundary critical behaviors of two-dimensional (2D) quantum phase transitions with fractionalized degrees of freedom in the bulk, motivated by the fact that usually it is the one-dimensional boundary that is exposed and can be conveniently probed in many experimental platforms. In particular, we mainly discuss boundary criticality of two examples: (i) the quantum phase transition between a 2D Z2{Z}_{2}Z2​ topological order and an ordered phase with spontaneous symmetry breaking; (ii) the continuous quantum phase transition between metal and a particular type of Mott insulator [U(1)\mathrm{U}(1)U(1) spin liquid]. In particular, we obtain the critical exponents and scaling laws of these exotic quantum phase transitions when the systems are probed from the boundary in proposed experimental setup. These critical exponents obtained are significantly different from those one would see through bulk measurements. This theoretical study could be relevant to many purely 2D systems, where recent experiments have found correlated insulator, superconductor, and metal in the same phase diagram.
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.
Monitored random circuits, consisting of alternating layers of entangling two-qubit gates and projective single-qubit measurements applied to some fraction p of the qubits, have been a topic of recent interest. In particular, the resulting steady state exhibits a phase transition from highly correlated states with "volume-law" entanglement at p<p_{c} to localized states with "area-law" entanglement at p>p_{c}. It is hard to access this transition experimentally, as it cannot be seen at the ensemble level. Naively, to observe it one must repeat the experiment until the set of measurement results repeats itself, with likelihood that is exponentially small in the number of measurements. To overcome this issue, we present a hybrid quantum-classical algorithm which creates a matrix product state (MPS) based "unitary mirror" of the projected circuit. Polynomial-sized tensor networks can represent quantum states with area-law entanglement, and so the unitary mirror can well approximate the experimental state above p_{c} but fails exponentially below it. The breaking of this mirror can thus pinpoint the critical point. We outline the algorithm and how such results would be obtained. We present a bound on the maximum entanglement entropy of any given state that is well represented by an MPS, and from the bound suggest how the volume-law phase can be bounded. We consider whether the entanglement could similarly be bounded from below where the MPS fails. Finally, we present numerical results for small qubit numbers and for monitored circuits with random Clifford gates.
Competition among repetitive measurements of noncommuting observables and unitary dynamics can give rise to a wide variety of entanglement phases. Here, we propose a general framework based on Lyapunov analysis to characterize topological properties in monitored quantum systems through their spectrum and many-body topological invariants. We illustrate this framework by analyzing (1+1)-dimensional monitored circuits for Majorana fermions, which are known to exhibit topological and trivial area-law entangled phases as well as a critical phase with sub-volume-law entanglement. Through the Lyapunov analysis, we identify the presence (absence) of edge-localized zero modes inside the bulk gap in the topological (trivial) phase and a bulk gapless spectrum in the critical phase. Furthermore, by suitably exploiting the fermion parity with twisted measurement outcomes at the boundary, we construct a topological invariant that distinguishes the two area-law phases and dynamically characterizes the critical phase. Our framework thus provides a general route to extend the notion of bulk-edge correspondence to monitored quantum dynamics.