Neural network enhanced cross entropy benchmark for monitored circuits

TL;DR

This work investigates observing measurement-induced phase transitions (MIPT) in trapped-ion monitored circuits using the cross entropy benchmark ($χ$), aiming to overcome post-selection and sample-complexity bottlenecks. It validates the XEB in this setting and introduces an autoregressive recurrent neural network (RNN) framework to learn the distribution of measurement records, enabling a data-efficient estimation of the circuit-averaged cross entropy $χ$ via two RNNs for initial states $\rho$ and $\sigma$. For $L=8$ with $p=0.1$ and $p=0.2$, the RNN-enhanced protocol significantly reduces the required number of measurement runs $M$ to achieve a target accuracy, by factors up to $10^3$–$10^4$, while stabilizing estimates of $χ_C$. The results illustrate a practical route to integrate quantum experiments with autoregressive generative models to improve benchmarking and control, with clear paths to scalability using transformers and extensions to noisy, experimental data.

Abstract

We explore the interplay of quantum computing and machine learning to advance experimental protocols for observing measurement-induced phase transitions (MIPT) in quantum devices. In particular, we focus on trapped ion monitored circuits and apply the cross entropy benchmark recently introduced by [Li et al., Phys. Rev. Lett. 130, 220404 (2023)], which can mitigate the post-selection problem. By doing so, we reduce the number of projective measurements -- the sample complexity -- required per random circuit realization, which is a critical limiting resource in real devices. Since these projective measurement outcomes form a classical probability distribution, they are suitable for learning with a standard machine learning generative model. In this paper, we use a recurrent neural network (RNN) to learn a representation of the measurement record for a native trapped-ion MIPT, and show that using this generative model can substantially reduce the number of measurements required to accurately estimate the cross entropy. This illustrates the potential of combining quantum computing and machine learning to overcome practical challenges in realizing quantum experiments.

Figures (7)

• Figure 1: Schematic diagram of the trapped ion monitored circuit with open boundary condition. Rectangles with $U$ represent random unitary gates defined in Eq. (\ref{['equ:hybridU']}) acting on adjacent qubits. Cyan circles denote local projective measurements along the $Z$-basis with a measurement rate $p$. The size of the whole spin chain is $L$.
• Figure 2: For two given circuits with $L=8$ and different $p$'s, the evolution of the cross entropy $\chi_C$ estimated using Eq. (\ref{['equ:estimate_chi']}) as a function of the number of measurement runs $M$. For $p=0.1$, the exact value of $\chi_C$ is plotted in red. For $p=0.2$, the estimation of $\chi_C$ at $M=6\times 10^5$ is plotted in green for reference.
• Figure 3: Numerical results of the circuit-averaged cross entropy $\chi$ for trapped ion hybrid circuits with initial states $\rho=|+\rangle^{\otimes L}$ and $\sigma=|0\rangle^{\otimes L}$. For each circuit, we estimate the cross entropy $\chi_C$ following Eq. (\ref{['equ:estimate_chi']}) and using $M=5\times 10^3$ measurement runs. This value of $M$ is selected based on convergence trends. Then we average over $M_C=100$ circuits to estimate $\chi$. Error bars represent one standard deviation of the cross entropy, averaged over 100 independent circuits.
• Figure 4: (a): A schematic diagram of the RNN model used in this work. The gray-shaded block represents the RNN cell, which processes a sequence of inputs $\{\vec{\mathbf{m}}\}$, where each input vector has a dimension of $N_i=2$. Here $\vec{m}_0=(0,0)$ is an initial vector to kick off the model. Within the RNN cell, the GRU layer generates a hidden state with a dimension of $N_h$. After passing through a Softmax activation layer, the output vector $\vec{y}_i$ (dimension $N_o=2$) corresponds to the conditional probability distribution, as described in Eq. (\ref{['equ:cond_prob']}). (b): The RNN estimation procedure for the numerator and denominator in Eq. (\ref{['equ:chiC']}). The autoregressive property of the model yields a recursive sampling process as follows. An initial zero vector $\vec{m}_0$ is fed into the RNN, producing the conditional probability $\vec{y}_1$ for the first site. Based on this probability, a configuration is sampled at the first site. This outcome is then used to sample the second site based on $\vec{y}_2$, and so on.
• Figure 5: The performance of the $\chi_C$ estimation for the circuit with $L=8$ and $p=0.1$ using the histogram approach versus vanilla RNNs. (a): the evolution of $\chi_C$ as a function of $M$. The exact result of $\chi_C$ is plotted in red. (b): $\Delta M$ defined in Eq. (\ref{['equ:DeltaM']}) as a function of the accuracy $\varepsilon$ defined in Eq. (\ref{['equ:accuracy']}).