Flow to Nishimori universality in weakly monitored quantum circuits with qubit loss

Abstract

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 non-unitary 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 simultaneously 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 non-monotonic entanglement growth at the incipient non-unitary RG flow. We argue that spectra of scaling dimensions of both the percolation and Nishimori fixed points exhibit multifractality. For percolation, we provide the exact (non-quadratic) multifractal spectrum of exponents, while for the Nishimori fixed point we show high-precision numerical results for five leading exponents characterizing multifractality.

Figures (12)

• Figure 1: Entanglement phase diagram, protocols and tensor network. (a) Long-range entanglement phase diagram of GHZ-class cat state or topological surface code, driven by random and weak measurements. Here "LRE" stands for long-range entangled phase, and "SRE" stands for short-range entangled phase. The defining character of LRE surface code phase is the anyon excitations $e$ and $m$ particles which pick up a $\pi$ phase when winding around each other Kitaev2003. The condensation of $e$ particles leads to the SRE "Higgs" phase Zhu19. Wegner's duality relates LRE surface code to SRE paramagnet, and SRE Higgs phase to a LRE cat phase. (b) Protocol starting from a parent cluster state. With respect to a square lattice, the orange spheres are denoted as "site" qubits, while the blue spheres are denoted as "bond" qubits. Together, they form a 2D Lieb lattice. The first protocol projectively measures out the bond qubits in a rotated basis, which effectively induces two-body weak parity measurements for the adjacent site qubits (orange spheres): $\exp{-\beta s_{ij} Z_i Z_j/2}$, resulting in a cast sate for the site qubits NishimoriCat. For the second protocol, projectively measuring out the site qubits results in a surface code for the bond qubits (blue spheres), which can be further subject to weak measurements $\exp{\beta s_{ij} Z_{ij}/2}$ with a probability teleportcode, denoted by the white hollow arrow in the top right "surface code" lattice of panel (b). (c) Tensor network, sliced into a sequence of transfer matrix product operators. Each circle on the bond denotes a matrix, which is chosen to capture the Ising interaction or null interaction depending on the weak or null measurement. Smooth boundary condition is shown, but rough boundary condition for surface code can be implemented by sending the left and right boundary white circles to be identity matrices.
• Figure 2: Effective central charge on the critical line. Inset: The angle $\theta$ describes the position on the critical line. The Nishimori point corresponds to $\theta = 0$, whereas the percolation point corresponds to $\theta = \pi/2$. The effective central charge governing the average entanglement entropy at Nishimori criticality is numerically computed to be $c_\text{ent} = 0.41956(3)$ (orange diamond), see Fig. \ref{['fig:fig5multifractal']}. For reference we also show the effective central charge determined by the Casimir energy at the Nishimori point $c_\text{Casimir} = 0.464(4)$Pujol2001 (orange star). At the percolation critical point, our numerically determined effective central charge agrees with the analytically known $c_\text{ent} = \frac{3\sqrt{3}\ln(2)}{2\pi}=0.573\ldots$Lang_2020 (red diamond), distinct from the Casimir energy estimate $c_{\rm Casimir} = \frac{5 \sqrt{3}}{4 \pi} \ln 2 = 0.478\ldots$ (red star).
• Figure 3: Critical exponent $\nu$ on the critical line. We calculate the critical exponent $\nu$ for different angles $\theta$ on the critical line by finite-size scaling for the coherent information of the surface code (with code distance $L=L_x-1= L_y$ up to $L = 512$ on the Nishimori line and up to $L=256$ elsewhere). At the Nishimori point we obtain a value of $\nu = 1.532(4)$ (orange diamond). At the percolation point we determine $\nu = 1.337(6)$, which is consistent with the analytical value Stauffer_1979$\nu = 4/3$ (red diamond). Inset (a) shows the raw data for different system sizes and inset (b) shows their data collapse for the Nishimori point ($\theta=0$). These numerical results clearly indicate that the entire line $\theta<\pi/2$ flows toward Nishimori criticality, in agreement with our theory prediction.
• Figure 4: Multifractality of percolation and Nishimori criticality, which are computed for the representative fixed points to increase numerical accuracy. Shown are the first four cumulants, $\kappa_1, \kappa_2, \kappa_3$ and $\kappa_4$ of the von Neumann entropy as a function of chord distance. Note that the fourth-order cumulant $\kappa_4$ shows a sign-flip with an inverted 'entanglement arc' [cf. Eq. \ref{['UnitaryCFTEntropy']}] as shown in the inset.
• Figure 5: Cumulant fits for Nishimori universality. Shown is a subset of the data of Fig. \ref{['fig:fig5multifractal']}(b), with the color-highlighted data used in the fits indicated by the dashed lines over the remaining data (gray). The data is fitted using Eq. \ref{['LabelEqCumulants']} to obtain the numerical estimates indicated in the figure (and listed in Table \ref{['tab:criticalpoints']}) To minimize boundary effects we fit only the data in an intermediate window of large cut length $L_x/8 < l< 7L_x/8$, see Appendix \ref{['app:cumulants']} for supplemental data and a detailed discussion.