Department of Physics, Boston College, Chestnut Hill, MA 02467, USA
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Abstract
We study the entanglement dynamics of quantum automaton (QA) circuits in the presence of U(1) symmetry. We find that the second Rényi entropy grows diffusively with a logarithmic correction as $sqrt{tln{t}}$, saturating the bound established by Huang [1]. Thanks to the special feature of QA circuits, we understand the entanglement dynamics in terms of a classical bit string model. Specifically, we argue that the diffusive dynamics stems from the rare slow modes containing extensively long domains of spin 0s or 1s. Additionally, we investigate the entanglement dynamics of monitored QA circuits by introducing a composite measurement that preserves both the U(1) symmetry and properties of QA circuits. We find that as the measurement rate increases, there is a transition from a volume-law phase where the second Rényi entropy persists the diffusive growth (up to a logarithmic correction) to a critical phase where it grows logarithmically in time. This interesting phenomenon distinguishes QA circuits from non-automaton circuits such as U(1)-symmetric Haar random circuits, where a volume-law to an area-law phase transition exists, and any non-zero rate of projective measurements in the volume-law phase leads to a ballistic growth of the Rényi entropy.
Featured image: The entanglement dynamics of quantum automaton circuits can be mapped to a classical bit string model. More specifically, the second Renyi entropy can be approximated by the dynamics of particles representing the difference of a bit string pair whose spins perform simple symmetric exclusion random walks under the circuit dynamics. Under U(1) symmetry, there are two distinct modes of particle dynamics, namely, the fast mode and the slow mode. In the example illustrated in the cartoon, although both modes have the same particle configuration, the bit string configuration of the slow mode consists of extensively long domains of spin 0s or 1s, resulting in a diffusive (with a logarithmic correction) expansion of the region occupied by particles.
Popular summary
In this work, we use random circuit models to study U(1)-symmetric quantum systems. Specifically, we focus on quantum automaton (QA) circuits, one of the few circuit models that allow an analytic understanding of the entanglement dynamics, and demonstrate that the second Renyi entropy scales as $sqrt{tln{t}}$, saturating the bound mentioned above. By mapping the second Renyi entropy to the quantity of a classical particle model, we show that this diffusive dynamics is the consequence of the emergence of rare slow modes under U(1) symmetry.
In addition, we introduce measurements into QA circuits and examine the monitored entanglement dynamics. Interestingly, as we manipulate the measurement rate, we observe a phase transition from a volume-law phase where the second Renyi entropy persists the diffusive growth, to a critical phase where it grows logarithmically. This is different from non-automaton U(1)-symmetric hybrid quantum circuits where a volume-law to area-law entanglement phase transition exists, and any non-zero rate of measurements below the critical point induces a linear growth of the Renyi entropy.
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► References
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