Gapless Symmetry-Protected Topological States in Measurement-Only Circuits

TL;DR

The work demonstrates that gapless symmetry-protected topological (gSPT) physics can emerge in measurement-only quantum circuits. By combining large-scale Clifford-circuit simulations with observables like $S_{ m Half}$, $S_{ m topo}$ and nonlocal string operators, the authors uncover a symmetry-enriched percolation critical point and a steady-state gSPT phase in a $ ext{Z}_4$ circuit, both mapped to a Majorana loop framework. Key results include a Bond Percolation universality class at the SSB–SPT transition, a nonunitary symmetry-enriched critical point, and robust edge modes persisting through steady-state criticality. The Majorana loop model provides a unified lens to understand these non-equilibrium topological phenomena and the interplay between topology, symmetry, and criticality in measurement-driven dynamics.

Abstract

Measurement-only quantum circuits offer a versatile platform for realizing intriguing quantum phases of matter. However, gapless symmetry-protected topological (gSPT) states remain insufficiently explored in these settings. In this Letter, we generalize the notion of gSPT to the critical steady state by investigating measurement-only circuits. Using large-scale Clifford circuit simulations, we investigate the steady-state phase diagram across several families of measurement-only circuits that exhibit topological nontrivial edge states at criticality. In the Ising cluster circuits, we uncover a symmetry-enriched non-unitary critical point, termed symmetry-enriched percolation, characterized by both topologically nontrivial edge states and string operator. Additionally, we demonstrate the realization of a steady-state gSPT phase in a $\mathbb Z_4$ circuit model. This phase features topological edge modes and persists within steady-state critical phases under symmetry-preserving perturbations. Furthermore, we provide a unified theoretical framework by mapping the system to the Majorana loop model, offering deeper insights into the underlying mechanisms.

Figures (16)

• Figure 1: (a) Circuit diagram of the Ising cluster circuit model with $L=8$ qubits and $8$ measurements (i.e., a single time step): blue, green, and orange rectangles represent the projective measurements $X_{i}$, $Z_{i}Z_{i+1}$, and $Z_{i-1}X_{i}Z_{i+1}$, respectively. (b) The steady-state phase diagram of the cluster circuit as a function of the probabilities $p_{\text{x}}$, $p_{\text{zz}}$, and $p_{\text{zxz}}$. The blue, green, and orange filled regions exhibit PM, SSB, and SPT orders, respectively. The red circles are critical points obtained by the data collapse of the generalized topological entanglement entropy [see Fig. \ref{['fig:fig2']}(a) and Sec. II of the SM] and the red dashed line is a guide to the eye. Percolation$^*$ means the symmetry-enriched percolation universality. The blue circles and the corresponding dashed line are obtained by $p_{\rm x} \leftrightarrow p_{\rm zxz}$.
• Figure 2: (a) The generalized topological entanglement entropy $S_{\rm topo}$ versus $p_{\rm zxz}$ for different system sizes with fixed $p_{\text{x}} = 0$. $A|B|D|C$ represents the equal partition used in the definition of $S_{\rm topo}$. The inset shows the data collapse of $S_{\text{topo}}$ with $\nu = 4/3$ and $p_{\text{zxz},c} = 1/2$ . (b) The half-chain entanglement entropy $S_{\text{Half}}$ grows logarithmically as the system size increases at the critical points for the case of $p_{\text{x}} = 0.0$, $0.2$, and $0.4$, respectively. The effective central charge $c_\text{eff}$ are obtained from least-squares fittings. (c) The edge magnetization, $M_{\rm b} \equiv (\overline{|\langle Z_{1} \rangle|} + \overline{|\langle Z_{L} \rangle|}) / 2$, in the presence of a small probability $p_{\rm h}$ of the boundary measurement $Z_{1/L}$ for the topologically trivial and nontrivial critical points, respectively. The inset shows the corresponding purification dynamics with an initial maximally mixed state for the same two critical points. (d) The string operators $O_{\rm SPT}(r)$ (square markers) and $O_{\rm PM}(r)$ (diamond markers) as a function of the lattice distance $r$ at the critical points for the case of $p_{\text{x}} = 0.0$, $0.2$, and $0.4$ .
• Figure 3: (a1) Steady state phase diagram of the $\mathbb Z_4$ symmetric circuits. The BKT (percolation) transition line is $4p_{\delta} + p_{h} = 1$ ($4p_{h} + p_{\delta} = 1$) (see Sec. III B of the SM for additional numerical results). (a2) The evolution of the full system entropy, $S$, starting from a maximally mixed state for four representative points in different phases. $S_\text{topo}$ as a function of $p_{\delta}$ at $p_{h} = 0$ (b1) and as a function of $p_{h}$ at $p_{\delta} = 0.08$ (b2). The inset gives the data collapse of $S_\text{topo}$. (c1) The string operator $O_{\tau}(r)$ exhibits a power-law decay when $p_{\delta} \in [0, 0.25)$ for $p_{h} = 0$. Data within the interval $r \in (L/8, L/2)$ are used in the fittings and one end of the string operator was fixed at the boundary. The inset shows $S_{\rm Half}$ as a function of $L$. (c2) The string operator $O_{\tau}(r)$ exhibits a power-law dependence on the lattice distance in the Phase I and Phase II, while an exponential decay in the Phase IV. The inset shows the logarithmic behavior of $S_{\rm Half}$ as a function of $L$ for three representative points. The effective central charge $c_\text{eff}$ is obtained from the least-squares fitting. The simulated system size is $L = 512$ in (a2) and $L = 256$ in (c1) and (c2).
• Figure 4: Majorana representation of the $\mathbb{Z}_{4}$ symmetric circuit. (a) When $p_{\delta} = p_{h} = 0$, there are only three-site measurements {$\tau_{2i-1}^{z}\sigma_{2i}^{x}\tau_{2i+1}^{z}$, $\tau_{2i-1}^{y}\sigma_{2i}^{x}\tau_{2i+1}^{y}$, $\sigma_{2i}^{z}\tau_{2i+1}^{x}\sigma_{2i+2}^{z}$}; this model can be seen as two decoupled Majorana chains exhibited by (b) and (c). (b) The two-site perturbation $\tau_{2i-1}^{x}\tau_{2i+1}^{x}$ is added on the $\tau$ degrees of freedom, which is a four-Majorana operator. (c) The single-site perturbation $\sigma_{2i}^{x}$ is added on the $\sigma$ degrees of freedom. (d) gives the table of the associated projective measurements and their corresponding probabilities and Majorana representations.
• Figure 5: Majorana representation of the cluster circuit model composed of $X_{i}$, $Z_{i}Z_{i+1}$, and $Z_{i-1}X_{i}Z_{i+1}$ measurements. (a) Each spin on site $i$ is represented by two Majorana modes $\gamma_{2i-1}$ and $\gamma_{2i}$. The green solid line, the red double solid line, and the blue dotted line represent the one-site $X_{i}$, the two-site $Z_{i}Z_{i+1}$, and the three-site $Z_{i-1}X_{i}Z_{i+1}$ measurements in the Majorana representation respectively. (b) A rearrangement of (a) preserving the linking structure under periodic boundary conditions. It is easy to see that there is a symmetry $p_{\text{x}} \leftrightarrow p_{\text{zxz}}$. (c) gives the table of the associated projective measurements and their corresponding probabilities and Majorana representations.