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Register a new userMeasurement and entanglement phase transitions in all-to-all quantum circuits, on quantum trees, and in Landau-Ginsburg theory
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A quantum many-body system whose dynamics includes local measurements at a nonzero rate can be in distinct dynamical phases, with differing entanglement properties. We introduce theoretical approaches to measurement-induced phase transitions (MPT) and also to entanglement transitions in random tensor networks. Many of our results are for all-to-all quantum circuits with unitaries and measurements, in which any qubit can couple to any other, and related settings where some of the complications of low-dimensional models are reduced. We also propose field theory descriptions for spatially local systems of any finite dimensionality. To build intuition, we first solve the simplest minimal cut toy model for entanglement dynamics in all-to-all circuits, finding scaling forms and exponents within this approximation. We then show that certain all-to-all measurement circuits allow exact results by exploiting local tree-like structure in the circuit geometry. For this reason, we make a detour to give general universal results for entanglement phase transitions random tree tensor networks, making a connection with classical directed polymers on a tree. We then compare these results with numerics in all-to-all circuits, both for the MPT and for the simpler Forced Measurement Phase Transition (FMPT). We characterize the two different phases in all-to-all circuits using observables sensitive to the amount of information propagated between initial and final time. We demonstrate signatures of the two phases that can be understood from simple models. Finally we propose Landau-Ginsburg-Wilson-like field theories for the MPT, the FMPT, and entanglement transitions in random tensor networks. This analysis shows a surprising difference between the MPT and the other cases. We discuss measurement dynamics with additional structure (e.g. free-fermion structure), and questions for the future.
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We study the level-spacing statistics in the entanglement spectrum of output states of random universal quantum circuits where qubits are subject to a finite probability of projection to the computational basis at each time step. We encounter two phase transitions with increasing projection rate: The first is the volume-to-area law transition observed in quantum circuits with projective measurements; The second separates the pure Poisson level statistics phase at large projective measurement rates from a regime of residual level repulsion in the entanglement spectrum within the area-law phase, characterized by non-universal level spacing statistics that interpolates between the Wigner-Dyson and Poisson distributions. By applying a tensor network contraction algorithm introduced in Ref. [1] to the circuit spacetime, we identify this second projective-measurement-driven transition as a percolation transition of entangled bonds. The same behavior is observed in both circuits of random two-qubit unitaries and circuits of universal gate sets, including the set implemented by Google in its Sycamore circuits.
Models for non-unitary quantum dynamics, such as quantum circuits that include projective measurements, have been shown to exhibit rich quantum critical behavior. There are many complementary perspectives on this behavior. For example, there is a known correspondence between d-dimensional local non-unitary quantum circuits and tensor networks on a D=(d+1)-dimensional lattice. Here, we show that in the case of systems of non-interacting fermions, there is furthermore a full correspondence between non-unitary circuits in d spatial dimensions and unitary non-interacting fermion problems with static Hermitian Hamiltonians in D=(d+1) spatial dimensions. This provides a powerful new perspective for understanding entanglement phases and critical behavior exhibited by non-interacting circuits. Classifying the symmetries of the corresponding non-interacting Hamiltonian, we show that a large class of random circuits, including the most generic circuits with randomness in space and time, are in correspondence with Hamiltonians with static spatial disorder in the ten Altland-Zirnbauer symmetry classes. We find the criticality that is known to occur in all of these classes to be the origin of the critical entanglement properties of the corresponding random non-unitary circuit. To exemplify this, we numerically study the quantum states at the boundary of Haar-random Gaussian fermionic tensor networks of dimension D=2 and D=3. We show that the most general such tensor network ensemble corresponds to a unitary problem of non-interacting fermions with static disorder in Altland-Zirnbauer symmetry class DIII, which for both D=2 and D=3 is known to exhibit a stable critical metallic phase. Tensor networks and corresponding random non-unitary circuits in the other nine Altland-Zirnbauer symmetry classes can be obtained from the DIII case by implementing Clifford algebra extensions for classifying spaces.
Measurement-driven transitions between extensive and sub-extensive scaling of the entanglement entropy receive interest as they illuminate the intricate physics of thermalization and control in open interacting quantum systems. Whilst this transition is well established for stroboscopic measurements in random quantum circuits, a crucial link to physical settings is its extension to continuous observations, where for an integrable model it has been shown that the transition changes its nature and becomes immediate. Here, we demonstrate that the entanglement transition at finite coupling persists if the continuously measured system is randomly nonintegrable, and show that it is smoothly connected to the transition in the stroboscopic models. This provides a bridge between a wide range of experimental settings and the wealth of knowledge accumulated for the latter systems.
Random measurements have been shown to induce a phase transition in an extended quantum system evolving under chaotic unitary dynamics, when the strength of measurements exceeds a threshold value. Below this threshold, a steady state with a sub-thermal volume law entanglement emerges, which is resistant to the disentangling action of measurements, suggesting a connection to quantum error-correcting codes. Here we quantify these notions by identifying a universal, subleading logarithmic contribution to the volume law entanglement entropy: $S^{(2)}(A)=kappa L_A+frac{3}{2}log L_A$ which bounds the mutual information between a qudit inside region $A$ and the rest of the system. Specifically, we find the power law decay of the mutual information $I({x}:bar{A})propto x^{-3/2}$ with distance $x$ from the regions boundary, which implies that measuring a qudit deep inside $A$ will have negligible effect on the entanglement of $A$. We obtain these results by mapping the entanglement dynamics to the imaginary time evolution of an Ising model, to which we can apply field-theoretic and matrix-product-state techniques. Finally, exploiting the error-correction viewpoint, we assume that the volume-law state is an encoding of a Page state in a quantum error-correcting code to obtain a bound on the critical measurement strength $p_{c}$ as a function of the qudit dimension $d$: $p_{c}log[(d^{2}-1)({p_{c}^{-1}-1})]le log[(1-p_{c})d]$. The bound is saturated at $p_c(drightarrowinfty)=1/2$ and provides a reasonable estimate for the qubit transition: $p_c(d=2) le 0.1893$.
We numerically investigate the structure of many-body wave functions of 1D random quantum circuits with local measurements employing the participation entropies. The leading term in system size dependence of participation entropies indicates a multifractal scaling of the wave-functions at any non-zero measurement rate. The sub-leading term contains universal information about measurement--induced phase transitions and plays the role of an order parameter, being non-zero in the error-correcting phase and vanishing in the quantum Zeno phase. We provide an analytical interpretation of this behavior expressing the participation entropy in terms of partition functions of classical statistical models in 2D.
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