Measurement Induced Entanglement Phase Transition

Fig. 1: Random unitary circuit with measurements. The initial state is assumed to be a product state and the unitary layers act on all the odd and even bonds in an interleaving fashion. The total time steps T is chosen to be large enough so that steady state is reached and random local projective measurements in the Z basis could independently happen at any site in space and time with probability p. (Source: Z. Wang)

Quantum entanglement has been the essential concept behind many modern understanding of condensed matter physics, especially for characterizing many-body ground states. Time evolution of entanglement entropy also reveals universal behavior and provides new insights into many-body dynamics out of equilibrium. For thermalizing phases the entanglement entropy grows linearly in time and saturates with a volume law, while for many-body localized systems it shows a slow logarithmic growth. Practically, entanglement growth typically determines the complexity of simulating many-body quantum dynamics classically, while the ability to generate and manipulate entanglement is also crucial for quantum computation.

Unitary dynamics typically drives a closed many-body system towards larger entanglement, eventually saturating at a volume-law scaling for the entanglement entropies of subsystems. On the other hand, local measurements could disentangle local degrees of freedom from the rest of the system and therefore reduce the amount of entanglement. Recently it has been demonstrated that an entanglement phase transition exists at some critical measurement rate. [1-3] Below that rate, the system exhibits volume-law entanglement and above that rate the system becomes area-law entangled. From a quantum computing perspective, such a competition widely exists in the near-term noisy quantum devices where the depth of quantum circuits are limited by the amount of noise in the system. [4] Studying such measurement induced entanglement phase transition may also shed light on new schemes of quantum error correction as well as future supremacy experiments. [5]

Model Description

Fig. 2: (a) Time evolution of the entanglement entropy SA in the middle of the system, with subsystem A consisting of sites 1∼L/2. (b) Rescaled steady state entanglement entropy SA as a function of the subsystem size |A| for different measurement rates p. For better visualization, each curve is rescaled linearly by the maximum entropy at that measurement rate, so that after rescaling the maximum entropy for different p will always be 1. In both plots, the legend gives the measurement rate p for each line and the black dashed line represent the critical rate pc≈0.16. The simulations are done with L=256 qubits and total time steps T=150 for reaching steady state. (Source: Z. Wang, after Li et al. [1])

Consider a random circuit model with both local unitaries and measurements (Fig. 1). The unitary evolution is decomposed into multiple layers with each layer formed by a sequence of non-overlapping two-qubit gates acting on either all odd bonds or all even bonds. Every two adjacent layers are defined as one discrete time step and the total time steps T will be large enough so that steady state could be reached.

The measurements happen between unitary layers and each qubit is measured independently with probability p. Since all two-qubit gate are random, the measurements are assumed to be performed along Z axis without loss of generality. In other words, with probability 1-p nothing happens and with probability p a single qubit will be projected to |0⟩ or |1⟩ and gets disentangled from the rest of the system.

All unitaries are sampled independently from the uniform distribution over the two-qubit Clifford group, instead of from the Haar measure on U(4). Due to the efficient classical simulatibility of Clifford circuits, we could track the dynamics of L=256 qubits, including both unitary evolution and projective measurements, as well as calculating the entanglement entropy for any bipartition of the system.

Entanglement Phase Transition

The initial product state is evolved with the hybrid random Clifford circuit and average over many circuit instances for convergence. Fig. 2 (a) plots the time evolution of the entanglement entropy in the middle of the L qubits, with subsystem A consisting of sites 1∼L/2. From this entanglement dynamics, we could identify features that correspond to an entanglement phase transition at critical measurement rate p_c≈0.16 (black dashed line). Below the critical rate p_c the entanglement entropy grows linearly in time until saturation, while above the critical rate entanglement growth is slower than linear and also saturates at much smaller values.

To study the entanglement scaling, we calculate and plot the rescaled steady state entanglement entropy S_A between subsystems 1∼|A| and |A|+1∼L for 1≤|A|≤128 at different measurement rates p (Fig. 2 (b)). Below p_c, the entanglement entropy scales linearly with subsystem size which corresponds to a volume-law scaling while above p_c the entanglement entropy saturates to a constant regardless what the subsystem size |A| is, which is a clear feature of 1D area law scaling. A schematic for the phase diagram and entanglement scaling is shown in Fig. 3.

Higher Dimensions

Here we would like to extend the 1D results above to higher dimensions and investigate how the entanglement phase transition behaves. Intuitively, at higher dimensions, the connectivity between qubits is higher and therefore the random unitary circuit could generate entanglement more efficiently than 1D. For example, in the absence of projective measurements, 2D random unitary circuit only takes O(L^1/2) time to fully scramble and reach the maximal

Fig. 3: Phase diagram and the leading order scaling of steady state entanglement as a function of the measurement rate p. (Source: Z. Wang, after Li et al. [1])

entanglement volume law scaling, while in 1D the time required is O(L). Therefore we would expect the critical measurement rate p_c to increase at higher dimensions. Indeed the 2D simulation results confirm this intuition (see the full paper for more details) In the extreme case with all-to-all coupling, every qubit could interact with every other qubit randomly and therefore the connectivity is much higher. Our numerical simulation suggests that an entanglement phase transition from volume law to area law scaling probably doesn't exist with all-to-all coupling, which is not too surprising because the notion of "boundary" as well as "area" of any subsystem A is not really well defined without the locality of the interaction.

Interestingly, the entanglement phase transition was recently studied from a different perspective as a purification transition, where the random unitary circuits with measurements takes a fully mixed state as input and the purity of the output state determines different phases of the transition. [5] Basically above a critical measurement rate, the output state will be pure and below that rate, the output state will stay mixed for an exponentially long time. With this new measure, even all-to-all coupled systems exhibit phase transition behavior. [5]

© Zhaoyou Wang. The author warrants that the work is the author's own and that Stanford University provided no input other than typesetting and referencing guidelines. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.

References

[1] Y. Li, X. Chen, and M. P. A. Fisher, "Measurement-Driven Entanglement Transition in Hybrid Quantum Circuits," Phys. Rev. B 100, 134306 (2019).

[2] B. Skinner, J. Ruhman, and A. Nahum, "Measurement-Induced Phase Transitions in the Dynamics of Entanglement," Phys. Rev. X 9, 031009 (2019).

[3] Y. Li, X. Chen, and M. P. A. Fisher, "Quantum Zeno Effect and the Many-Body Entanglement Transition," Phys. Rev. B 98, 205136 (2018).

[4] F. Arute et al., "Quantum Supremacy Using a Programmable Superconducting Processor," Nature 574, 505 (2019).

[5] M. J. Gullans and D. A Huse, "Dynamical Purification Phase Transitions Induced by Quantum Measurements," Phys. Rev. X 10, 041020 (2020).