Error correction phase transition in noisy random quantum circuits

TL;DR

The paper investigates how depth and mid-circuit noise affect the ability to preserve quantum information encoded by random circuits. It combines numerical experiments on 2D Clifford brickwork circuits with mid-circuit depolarizing noise and all-to-all Haar random circuits (via a second-moment log-average-purity proxy) and a rigorous worst-case upper bound to map a depth-noise phase diagram, finding a depth threshold $d^* = \Theta(1/p)$ separating recoverable from unrecoverable information. Across Clifford and Haar ensembles, it reports consistent evidence of a sharp transition in the coherent information $I_c$, with detailed scaling behavior $I_c = f(n^{\lambda}(p-p^*))$ and distinct critical exponents for different noise models, while showing that random encoders are optimal in this asymptotic sense. The results distinguish depolarizing and amplitude-damping noise and link the phase transition to a fundamental limit on information protection under noise, informing both the design of robust quantum encoders and fault-tolerance perspectives. The work also connects to broader themes in quantum information, including channel capacity and percolation-like transformations of entanglement under noise.

Abstract

In this work, we study the task of encoding logical information via a noisy quantum circuit. It is known that at superlogarithmic depth, the output of any noisy circuit without reset gates or intermediate measurements becomes indistinguishable from the maximally mixed state, implying that all input information is destroyed. This raises the question of whether there is a low-depth regime where information is preserved as it is encoded into an error-correcting codespace by the circuit. When considering noisy random encoding circuits, our numerical simulations show that there is a sharp phase transition at a critical depth of order $p^{-1}$, where $p$ is the noise rate, such that below this depth threshold quantum information is preserved, whereas after this threshold it is lost. Furthermore, we rigorously prove that this is the best achievable trade-off between depth and noise rate for any noisy circuit encoding a constant rate of information. Thus, random circuits are optimal noisy encoders in this sense.

Figures (9)

• Figure 1: Depiction of a general encoding circuit. In this case $\ket{\psi}$ is the $k$-qubit unencoded logical state. $\ket{\overline{\psi}}$ is the $n$-qubit encoded logical state.
• Figure 2: Diagram for the Stinespring dilation of the noisy circuit acting on one half of the maximally entangled state. Importantly, the coherent information of this operation characterizes how well quantum information can be recovered from the output of the noisy circuit.
• Figure 3: Diagram of our numerical simulation. Qubit 1 represents the reference qubit and is prepared in a Bell pair with qubit 2, which represents the logical qubit (here, there is only one logical qubit and so $k=1$). Qubits 3-5 represent the ancilla, or stabilizer, qubits. The noisy Clifford circuit is applied to the data qubits (qubits 2-5) which is denoted by $\tilde{\mathcal{C}}$. Next, the stabilizer generators of the code are measured perfectly which is denoted by $\mathcal{S}$. Finally, the coherent information of this channel is calculated.
• Figure 4: Numerical results for 2D random brickwork Clifford circuits with periodic boundary conditions subjected to mid-circuit depolarizing noise. In all cases an encoding rate of $r = 1/8$ is used. (a) Coherent information per logical qubit is displayed with respect to a range of error rates $p$ between $0.005$ and $0.1$. System size is scaled from $n=16$ and $n=400$. The circuit depth is fixed at $d=12$. The critical error rate is depicted by the vertical line which occurs at $p^* = 0.0472$. (b) A scaling collapse for the data in (a) is plotted for error rates near the critical point. The scaling ansatz uses the estimated critical error rate $p^* = 0.0472$ and critical exponent $\lambda = 0.616$. (c) The plots in (a) and (b) are repeated for circuit depths $d$ ranging from $10$ to $24$. Linear regression on the inverse depth is used to estimate the relationship that $p^* = 0.627d^{-1} -0.005$. (d) Finally, the critical error rate $p^*$ is plotted with respect to the inverse depth.
• Figure 5: Numerical results for all-to-all Haar random circuits subjected to mid-circuit depolarizing noise. In all cases an encoding rate of $r = 1/8$ is used. (a) Log-average purity per logical qubit is displayed with respect to a range of error rates $p$ between $0.002$ and $0.1$. System size is scaled from $n=10$ and $n=50$. The circuit depth is fixed at $d=12$. The critical error rate is depicted by the vertical line which occurs at $p^* = 0.0304$. (b) A scaling collapse for the data in (a) is plotted for error rates near the critical point. The scaling ansatz uses the estimated critical error rate $p^* = 0.0304$ and critical exponent $\lambda = 0.467$. (c) The plots in (a) and (b) are repeated for circuit depths $d$ ranging from $5$ to $25$. Linear regression on the inverse depth is used to estimate the relationship that $p^* = 0.387d^{-1} -0.0015$. (d) Finally, the critical error rate $p^*$ is plotted with respect to the inverse depth.

Theorems & Definitions (8)

• theorem 6.1
• proof
• proof
• proof
• proof : Proof sketch
• proof : Proof sketch
• theorem A.7: Restatement of \ref{['theorem:worst-case']}
• proof