Simulating a measurement-induced phase transition for trapped-ion circuits

The rise of programmable quantum devices has motivated the exploration of circuit models which could realize novel physics. A promising candidate is a class of hybrid circuits, where entangling unitary dynamics compete with disentangling measurements. Novel phase transitions between different entanglement regimes have been identified in their dynamical states, with universal properties hinting at unexplored critical phenomena. Trapped-ion hardware is a leading contender for the experimental realization of such physics, which requires not only traditional two-qubit entangling gates, but also a constant rate of local measurements accurately addressed throughout the circuit. Recent progress in engineering high-precision optical addressing of individual ions makes preparing a constant rate of measurements throughout a unitary circuit feasible. Using tensor network simulations, we show that the resulting class of hybrid circuits, prepared with native gates, exhibits a volume-law to area-law transition in the entanglement entropy. This displays universal hallmarks of a measurement-induced phase transition. Our simulations are able to characterize the critical exponents using circuit sizes with tens of qubits and thousands of gates. We argue that this transition should be robust against additional sources of experimental noise expected in modern trapped-ion hardware and will rather be limited by statistical requirements on postselection. Our work highlights the powerful role that tensor network simulations can play in advancing the theoretical and experimental frontiers of critical phenomena.

Figure 1

One time cycle realization of a quantum circuit setup to probe measurement-induced phase transitions, with vertical lines indicating the $N=6$ evolved qubits. Squares denote single-qubit unitary rotation gates $R(\theta ,\phi )$, rectangles correspond to two-qubit Mølmer-Sørensen gates $M\left(\mathrm{\Theta }\right)$, and circles show single-qubit projective measurements in the $Z$-basis, which are injected uniformly in space and time at a rate $p$. The time cycle consists of an even (upper, $U_{\mathrm{e}}$) and an odd (lower, $U_{\mathrm{o}}$) unitary layer, indicated in gray, each followed by a measurement layer. In the even unitary layer the outer two qubits are not affected by the Mølmer-Sørensen gates due to open boundary conditions.

Figure 2

(a) The maximum bond dimension $\mathcal{D}$ observed in the long-time states of 500 random circuits as a function of error rate $p$ for tensor network simulations with a truncation cutoff of $\varepsilon ={10}^{-10}$. Different system sizes $N$ are considered, where for $N=28$ no data exist for $p\le 0.09$ due to exceeding computation times for large bond dimensions. (b) Illustration of the entanglement phase diagram. A measurement-induced phase transition separates the two phases at the critical measurement rate $p_{\mathrm{c}}$; see Sec. 4.

Figure 3

The evolution of the first and second half-chain entanglement Rényi entropies under the quantum circuit are plotted as functions of time cycles $t$ for different system sizes $N$ and measurement rates $p=0.1$ (left) and $p=0.2$ (right). Convergence to a constant value for large $t$ indicates a saturated long-time entanglement entropy. The dependence on $N$ for $p=0.1$ for both measures indicates a volume-law entanglement, while the right column shows area-law entanglement since the entropies converge to similar values independent of the system size. All data points result from averaging 500 circuit runs, except for sizes $N=30$ and $N=36$ with measurement rate $p=0.1$, where only 85 and 19 circuit runs are averaged, respectively, as the large entanglement entropy leads to excessive computation times. Shaded areas show the standard mean error, which is barely visible when averaging 500 runs. Entanglement entropies for subsystems with an odd number $N_A=N/2$ of spins are smaller than those for subsystems with even $N_A$ due to open boundary conditions [60, 61].

Figure 4

The scaling behavior of the half-chain von Neumann entanglement entropy [Eq. (13)] for different system sizes $N\in \{6,8,\cdots ,36\}$, for circuits with projective measurements $P_i^{\pm }$. The critical measurement density $p_{\mathrm{c}}$ and scaling exponent $\nu$ are adjusted to produce the best collapse, resulting in $p_{\mathrm{c}}=0.17\pm 0.02$ and $\nu =1.4\pm 0.2$. All data points result from averaging the entanglement entropy after $T=2N$ time cycles of 500 circuit runs, and error bars are smaller than data points. Inset: The scaling behavior for additional applications of ${\tilde{P}}_i=|0\rangle \langle 0|+|0\rangle \langle 1|$ after each measurement, with $N\in \{6,8,\cdots ,36\}$. An optimal collapse gives $p_{\mathrm{c}}=0.17\pm 0.02$, $\nu =1.4\pm 0.2$ in this case; see Sec. 5a.

Figure 5

Scaling function of the entanglement entropy close to the critical measurement rate $p_{\mathrm{c}}$. $S_1(N,0.17,t)$ is plotted as function of $\ln \left(t\right)$ for fixed system sizes $N=34$ (triangles) and $N=36$ (circles), and as a function of $\ln \left(x\right)$, $x=2N/\pi$, for system sizes $N\in \{6,8,\cdots ,36\}$ after $t=80$ time cycles, distinguishing even (circles) and odd (triangles) system sizes due to open boundary conditions [60, 61]. All data points are averaged over 500 circuit runs, and error bars are smaller than the data points. Black lines show functions $g\left(N\right)=\alpha \left(0.17\right)\ln \left(x\right)+\tilde{b}$ fitted to the data points, giving fit parameters $\alpha \left(0.17\right)=0.26\pm 0.01$, $\tilde{b}=0.33\pm 0.03$ for even and $\alpha \left(0.17\right)=0.28\pm 0.01$, $\tilde{b}=0.1\pm 0.03$ for odd subsystem sizes. Shaded regions show fit uncertainties. Inset: Mean-squared error, Eq. (14), of fitted scaling forms according to Eq. (12) vs measurement rate $p$ for odd and even subsystem sizes. A clear minimum around $p=0.15$ to $p=0.17$ indicates the critical measurement rate. Same data as in Fig. 4.