Efficient tensor-network simulations of weakly-measured quantum circuits

TL;DR

The paper addresses simulating weak mid-circuit measurements in monitored quantum circuits using a Markov-chain tensor-network algorithm, enabling efficient sampling and forward propagation of measurement effects. By applying the method to a (1+1)-D brickwall circuit with Haar-random unitaries and variable-strength measurements, the authors demonstrate scalable simulations for tens to hundreds of qubits and reveal a measurement-induced phase transition between area-law and volume-law entanglement as a function of measurement strength. The approach yields sampled measurement outcomes akin to hardware runs and shows potential for validating quantum devices and enabling generative-machine-learning tasks that rely on sampling from complex stochastic processes. The work broadens the applicability of tensor-network methods to monitored quantum dynamics and temporal stochastic modeling, highlighting practical pathways for hybrid quantum-classical algorithms and future extensions to higher dimensions and other measurement protocols.

Abstract

We present a tensor-network-based method for simulating a weakly-measured quantum circuit. In particular, we use a Markov chain to efficiently sample measurements and contract the tensor network, propagating their effect forward along the spatial direction. Applications of our algorithm include validating quantum computers (capable of mid-circuit measurements) in regimes of easy classical simulability, and studying generative-machine-learning applications, where sampling from complex stochastic processes is the main task. As a demonstration of our algorithm, we consider a (1+1)-dimensional brickwall circuit of Haar-random unitaries, interspersed with generalized single-qubit measurements of variable strength. We simulate the dynamics for tens to hundreds of qubits if the circuit exhibits area-law entanglement (under strong measurements), and tens of qubits if it exhibits volume-law entanglement (under weak measurements). We observe signatures of a measurement-induced phase transition between the two regimes as a function of measurement strength.

Figures (5)

• Figure 1: (a) The (1+1)-dimensional random unitary circuit studied in this work. The spatial direction consists of $N$ qubits (orange circles), with $N=6$ in this example. The temporal direction consists of Haar-random two-qubit unitaries $U$ in a brickwall structure (purple rectangles), interspersed with single-qubit weak measurements (red diamonds) with probability $p$ on a given qubit. The four layers shown here (two of random unitaries, two of weak measurements) constitute one time step. The unitaries $U$ are applied with periodic boundary conditions. (b) Explicit depiction of the weak measurement procedure of qubit $q_i$, where $M_{i\bar{i}}$ is given by Eq. \ref{['eq:MeasGate']}. Here, $\bar{i}$ is an ancilla qubit that is initialized in the $\ket{0}$ state and projectively measured after the application of $M_{i\bar{i}}$. $A_i$ is the tensor corresponding to the matrix-product state of qubit $q_i$, with link indices not shown.
• Figure 2: The Markov chain approach to efficiently performing the weak measurements Eq. \ref{['eq:MeasGate']}. See the main text for full details. (a) Start with the first qubit, $q_1$. Form the tensor corresponding to the physical qubit's tensor $A_1$ and the ancilla qubit's initial state, $\ket{0}_{\bar{1}}$. Call this tensor $A_{1\bar{1}}$. (b) Apply the measurement operator Eq. \ref{['eq:MeasGate']}. Call the measurement-modified tensor $\tilde{A}_{1\bar{1}}$. (c) Contract the physical indices of the real qubit between $\tilde{A}_{1\bar{1}}$ and its Hermitian conjugate. Call this tensor $B_{\bar{1}}$ or $P_{\bar{1}}$. (d) Contract the right link indices of $P_{\bar{1}}$ and normalize it to form the one-particle density matrix for the ancilla qubit, $\rho_{\bar{1}}$. Sample a measurement outcome $s_{\bar{1}}$. (e) Return to the tensor $P_{\bar{1}}$. Take the component of this tensor corresponding to the measurement outcome $s_{\bar{1}}$, such that only right link indices remain. Call the resulting tensor $R_{\bar{1}}$. (f) For the next qubit, $q_2$, the procedure is similar. However, the tensor $B_{\bar{2}}$ will now possess left link indices. To form $P_{\bar{2}}$, contract $B_{\bar{2}}$ with $R_{\bar{1}}$ of the previous qubit. This propagates the previous measurement outcome forward in the Markov chain and along the matrix-product state.
• Figure 3: (a) Time series of $S(t)$ within the quantum circuit, for $p=1$, $N=20$, and various measurement strengths: $\theta=\pi/2$ (red downwards triangles), $\theta=\pi/3$ (blue squares), $\theta=\pi/4$ (orange circles), and $\theta=\pi/6$ (green upwards triangles). The averaging procedure over $N_{\rm} = 1000$ quantum trajectories is described in the main text. Error bars represent one standard deviation. (b) Long-time entanglement entropy $S(t\to\infty)$ for the same measurement rate $p$ and measurement strengths $\theta$ as in (a), but as a function of system size $N$. Error bars correspond to the standard error of the mean.
• Figure 4: Long-time entanglement entropy $S(t\to\infty)$ for $p=1$ and measurement strengths $\theta = \pi/4, \pi/3$. The data is the same as in Fig. \ref{['fig:Transition']}(b), with the inclusion of larger system sizes: $N = 50$ ($\theta = \pi/4$) and $N=100,\ 500$ ($\theta=\pi/3$). Error bars correspond to the standard error of the mean. The dashed lines show lines of best fit for each choice of $\theta$, corresponding to a $\log(N)$ scaling of the long-time entanglement entropy.
• Figure 5: Long-time entanglement entropy $S(t\to\infty)$ for $p=1$, $N=20$, and $\theta=\pi/6$, as a function of bond dimension $\chi$ in the MPS simulations. Error bars correspond to the standard error of the mean.