Nishimori's Cat: Stable Long-Range Entanglement from Finite-Depth Unitaries and Weak Measurements

In the field of monitored quantum circuits, it has remained an open question whether finite-time protocols for preparing long-range entangled states lead to phases of matter that are stable to gate imperfections, that can convert projective into weak measurements. Here, we show that in certain cases, long-range entanglement persists in the presence of weak measurements, and gives rise to novel forms of quantum criticality. We demonstrate this explicitly for preparing the two-dimensional Greenberger-Horne-Zeilinger cat state and the three-dimensional toric code as minimal instances. In contrast to random monitored circuits, our circuit of gates and measurements is deterministic; the only randomness is in the measurement outcomes. We show how the randomness in these weak measurements allows us to track the solvable Nishimori line of the random-bond Ising model, rigorously establishing the stability of the glassy long-range entangled states in two and three spatial dimensions. Away from this exactly solvable construction, we use hybrid tensor network and Monte Carlo simulations to obtain a nonzero Edwards-Anderson order parameter as an indicator of long-range entanglement in the two-dimensional scenario. We argue that our protocol admits a natural implementation in existing quantum computing architectures, requiring only a depth-3 circuit on IBM's heavy-hexagon transmon chips.

Universal dependence on disorder of two-dimensional randomly diluted and random-bond +/-J Ising models

We consider the two-dimensional randomly site diluted Ising model and the random-bond +/-J Ising model (also called the Edwards-Anderson model), and study their critical behavior at the paramagnetic-ferromagnetic transition. The critical behavior of thermodynamic quantities can be derived from a set of renormalization-group equations, in which disorder is a marginally irrelevant perturbation at the two-dimensional Ising fixed point. We discuss their solutions, focusing in particular on the universality of the logarithmic corrections arising from the presence of disorder. Then, we present a finite-size scaling analysis of high-statistics Monte Carlo simulations. The numerical results confirm the renormalization-group predictions, and in particular the universality of the logarithmic corrections to the Ising behavior due to quenched dilution.

Multicritical Nishimori point in the phase diagram of the +/-J Ising model on a square lattice

We investigate the critical behavior of the random-bond +/-J Ising model on a square lattice at the multicritical Nishimori point in the T-p phase diagram, where T is the temperature and p is the disorder parameter ( p=1 corresponds to the pure Ising model). We perform a finite-size scaling analysis of high-statistics Monte Carlo simulations along the Nishimori line defined by 2p-1=tanh(1/T) , along which the multicritical point lies. The multicritical Nishimori point is located at p;{ *}=0.890 81(7) , T;{ *}=0.9528(4) , and the renormalization-group dimensions of the operators that control the multicritical behavior are y_{1}=0.655(15) and y_{2}=0.250(2) ; they correspond to the thermal exponent nu identical with1/y_{2}=4.00(3) and to the crossover exponent phi identical withy_{1}/y_{2}=2.62(6) .