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Register a new userMeasurement-induced purification in large-N hybrid Brownian circuits
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Competition between unitary dynamics that scrambles quantum information non-locally and local measurements that probe and collapse the quantum state can result in a measurement-induced entanglement phase transition. Here we study this phenomenon in an analytically tractable all-to-all Brownian hybrid circuit model composed of qubits. The system is initially entangled with an equal sized reference, and the subsequent hybrid system dynamics either partially preserves or totally destroys this entanglement depending on the measurement rate. Our approach can access a variety of entropic observables which are distinguished by the averaging procedure, and for concreteness we focus on a particular purity quantity for which the averaging is particularly simple. We represent the purity as a path integral coupling four replicas with twisted boundary conditions. Saddle-point analysis reveals a second-order phase transition corresponding to replica permutation symmetry breaking below a critical measurement rate. The transition is mean-field-like and we characterize the critical properties near the transition in terms of a simple Ising field theory in 0+1 dimensions. In addition to studying the purity of the entire system, we study subsystem purities and relate these results to manifestations of quantum error correction in the model. We also comment on the experimental feasibility for simulating this averaged purity, and corroborate our results with exact diagonalization for modest system sizes.
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We study the effects of power-law long-range couplings on measurement-induced phases and transitions in tractable large-$N$ models, including a Brownian qubit model and a Brownian SYK model. In one dimension, the long-range coupling is irrelevant for $alpha>3/2$, with $alpha$ being the power-law exponent, thus the volume-law and area-law entanglement phases and the phase transition remain intact. For $alpha<3/2$ the long-range coupling becomes relevant, leading to a nontrivial dynamical exponent at the measurement-induced phase transition. More interestingly, for $alpha<1$ the entanglement pattern receives a sub-volume correction for both area-law and volume-law phases. The volume-law phase with such a sub-volume correction realizes a novel quantum error correcting code whose code distance scales as $L^{2-2alpha}$. We further extend the calculation to a quadratic SYK model, where two distinct fractal entangled phases emerge, leading to a complete phase diagram of the long-range free fermion model under monitoring.
Entanglement renormalization is a unitary real-space renormalization scheme. The corresponding quantum circuits or tensor networks are known as MERA, and they are particularly well-suited to describing quantum systems at criticality. In this work we show how to construct Gaussian bosonic quantum circuits that implement entanglement renormalization for ground states of arbitrary free bosonic chains. The construction is based on wavelet theory, and the dispersion relation of the Hamiltonian is translated into a filter design problem. We give a general algorithm that approximately solves this design problem and provide an approximation theory that relates the properties of the filters to the accuracy of the corresponding quantum circuits. Finally, we explain how the continuum limit (a free bosonic quantum field) emerges naturally from the wavelet construction.
We numerically study the measurement-driven quantum phase transition of Haar-random quantum circuits in $1+1$ dimensions. By analyzing the tripartite mutual information we are able to make a precise estimate of the critical measurement rate $p_c = 0.17(1)$. We extract estimates for the associated bulk critical exponents that are consistent with the values for percolation, as well as those for stabilizer circuits, but differ from previous estimates for the Haar-random case. Our estimates of the surface order parameter exponent appear different from that for stabilizer circuits or percolation, but we are unable to definitively rule out the scenario where all exponents in the three cases match. Moreover, in the Haar case the prefactor for the entanglement entropies $S_n$ depends strongly on the Renyi index $n$; for stabilizer circuits and percolation this dependence is absent. Results on stabilizer circuits are used to guide our study and identify measures with weak finite-size effects. We discuss how our numerical estimates constrain theories of the transition.
The competition between scrambling unitary evolution and projective measurements leads to a phase transition in the dynamics of quantum entanglement. Here, we demonstrate that the nature of this transition is fundamentally altered by the presence of long-range, power-law interactions. For sufficiently weak power-laws, the measurement-induced transition is described by conformal field theory, analogous to short-range-interacting hybrid circuits. However, beyond a critical power-law, we demonstrate that long-range interactions give rise to a continuum of non-conformal universality classes, with continuously varying critical exponents. We numerically determine the phase diagram for a one-dimensional, long-range-interacting hybrid circuit model as a function of the power-law exponent and the measurement rate. Finally, by using an analytic mapping to a long-range quantum Ising model, we provide a theoretical understanding for the critical power-law.
We compute analytically and in closed form the four-point correlation function in the plane, and the two-point correlation function in the upper half-plane, of layering vertex operators in the two dimensional conformally invariant system known as the Brownian Loop Soup. These correlation functions depend on multiple continuous parameters: the insertion points of the operators, the intensity of the soup, and the charges of the operators. In the case of the four-point function there is non-trivial dependence on five continuous parameters: the cross-ratio, the intensity, and three real charges. The four-point function is crossing symmetric. We analyze its conformal block expansion and discover a previously unknown set of new conformal primary operators.
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