In circuit-based quantum state preparation, qubit loss and coherent errors are circuit imperfections that imperil the formation of long-range entanglement beyond a certain threshold. The critical theory at the threshold is a continuous entanglement transition known to be described by a ( 2 + 0 ) -dimensional nonunitary conformal field theory which, for the two types of imperfections of certain circuits, is described by either percolation or Nishimori criticality, respectively. Here we study the threshold behavior when the two types of errors occur and show that, when moving away from the Clifford regime of projective stabilizer measurements, the percolation critical point becomes unstable and the critical theory flows to Nishimori universality. We track this critical renormalization group (RG) crossover flow by mapping out the entanglement phase diagrams, parametrized by the probability and strength of random weak measurements, of two dual protocols preparing surface code or GHZ-class cat states from a parent cluster state via constant-depth circuits. Extensive numerical simulations, using hybrid Gaussian fermion and tensor network/Monte Carlo sampling techniques on systems with more than a million qubits, demonstrate that an infinitesimal deviation from the Clifford regime leads to a sudden, strongly nonmonotonic entanglement growth at the incipient nonunitary RG flow. We argue that spectra of scaling dimensions of both the percolation and Nishimori fixed points exhibit . For percolation, we provide the exact (nonquadratic) multifractal spectrum of exponents, while for the Nishimori fixed point we show high-precision numerical results for five leading exponents characterizing multifractality.
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.
Random projective measurements are an effective tool to model the interlinked phenomenology of monitoring, inferring, and decoding highly entangled states such as quantum error-correcting codes. At the same time, the study of such monitored dynamics has opened a new arena for critical phenomena, renewing longstanding questions in disordered statistical mechanics, including percolation, random-bond models, and Nishimori physics. Here we investigate the robustness of stabilizer quantum error-correcting codes under a single round of independently sampled single-qubit Pauli measurements and demonstrate that their threshold behavior falls into two broad categories - geometric and homological percolation transitions, with the toric code and color code being the principal examples for each class. We establish the distinct character of these two threshold theories by analyzing the logical-support criteria of these codes and by providing numerical estimates of their respective critical exponents obtained from optimized stabilizer simulations of large-scale systems with over one million qubits. Our framework extends to non-CSS, subsystem, higher-dimensional, and Floquet stabilizer codes and allows us to determine their respective threshold phase diagrams. We exemplify how the latter provide quantitative and qualitative guidance for the computationally more demanding problems of tunable learning and maximum-likelihood decoding. As a practical application of our framework, we discuss the robustness of the gross code.
We study boundary criticality at the Nishimori multicritical point of the two-dimensional (2D) random-bond Ising model. Using tensor-network methods, we construct a family of microscopic boundary conditions that incorporates both boundary-spin rotation and boundary disorder. We identify three conformal boundary fixed points, corresponding to free, fixed, and random boundary conditions, and map out the boundary renormalization group flows among them. We extract the corresponding boundary conformal data, including the boundary entropies and the scaling dimensions of boundary primary operators, which characterize the boundary universality class. At the free boundary fixed point, we uncover the multifractal scaling of boundary-spin fields. We further complement the numerical results with a controlled renormalization group analysis. Finally, we connect the boundary conformal data to quantum error-correcting codes, establishing a bridge between boundary universality class and boundary decoding threshold.
The authors explore critical properties of entanglement phase transitions in random tensor networks and monitored quantum circuits with Clifford tensors/gates for on-site Hilbert space dimensions which are powers of a prime number
p. Exact mappings to replica spin models and characterizations of their symmetry groups predict that all universal properties will depend only on
p. These predictions are confirmed with extensive numerical simulations. The authors also establish multifractal scaling of the purity, reflected in a continuous spectrum of critical exponents, while the typical exponent is the prefactor of the logarithm in the entanglement entropy.
Motivated by a previous Ising study, we identify a
higher Nishimori line in the learning phase diagram of the
2D q-state Potts model
(2<q≤4) under bond-energy measurements. This
higher Nishimori line meets the critical temperature line of the Potts model, in a
higher Nishimori critical point -- a tricritical point at finite inference strength that separates a paramagnetic, a ferromagnetic and a 'spin-glass' phase. With analytical tools, we discuss the general structure of the rich phase diagram, which contains two unstable and three stable fixed points, and obtain a number of exact results for universal quantities, including the decay exponent of the Edwards-Anderson correlator, using a Gaussian measurement protocol which allows for exact calculations. Using extensive numerical tools, we confirm these statements for a generic, discrete
q-state measurement protocol and determine precise numerical estimates for the location of higher and ordinary Nishimori critical points as well as RG flows between the various fixed points. We also discuss the Casimir effective central charges of the critical points in the learning phase diagram, and their monotonic
decrease along measurement-induced RG flows, as established non-perturbatively by the c-effective theorem and its extensions, and contrast it to the monotonic increase along the corresponding RG flows in the random-bond Potts model. Finally, we discuss a general argument based on
Elitzur′stheorem that establishes stability of the ordinary Nishimori critical points in their respective learning phase diagrams. Equivalently, our results describe a monitored deformed
Zq toric code where the tricritical
higher Nishimori point is an 'information' critical point that separates stable quantum, classical, and no memory phases.
Measurement-induced entanglement transitions in quantum dynamics represent a new class of nonequilibrium transitions, akin to thresholds in quantum error correction. In this work, the authors explore the universal properties of entanglement transitions in one- and two-dimensional monitored Clifford circuits, using a graph-state-based algorithm to unravel geometric properties of entanglement. A study of entanglement clusters in the steady state reveals that, despite similarities in the bulk, the surface critical exponents show strong deviations from classical percolation.