Coherent Information Phase Transition in a Noisy Quantum Circuit

Abstract

Coherent information quantifies the transmittable quantum information through a channel and is directly linked to the channel's quantum capacity. In a monitored quantum circuit, regarded as a quantum channel, extensive and positive coherent information is sustained at low measurement rates, protected by the scrambling dynamics. However, noise suppresses coherent information, driving it to zero or negative values. Here, we show that incorporating quantum-enhanced operations facilitates reliable quantum information transmission even in the presence of noise, as evidenced by a phase transition in coherent information from a recoverable phase with positive values to an irrecoverable phase with negative values. We provide both analytical understanding and numerical evidence demonstrating this transition, which is modulated by the relative frequencies of noise and quantum-enhanced operations. Additionally, we propose a resource-efficient protocol to characterize this phase transition in experiments, effectively avoiding post-selection by utilizing every run of the quantum circuit. This approach bridges the gap between theoretical insights and practical implementation, making the phase transition feasible to demonstrate on realistic noisy intermediate-scale quantum devices.

Figures (4)

• Figure 1: Circuit structure and phase diagram. (a) Circuit structure. Orange, green, yellow, and blue rectangles represent unitary gates, measurements, noise, and QE operations, respectively. The initial state is entangled with reference qubits $R$. We primarily consider QE operations where $U_{SA} = \text{SWAP}$. (b) Phase diagram. We focus on the coherent information phase transition from recoverable to irrecoverable phase, tuned by the relative frequency of noise and QE operations. This phase transition manifests irrespective of the measurement rate $p$.
• Figure 2: Numerical simulation for coherent information phase transition. $L$ denotes number of qubits. (a) Resetting noise. (b) Depolarizing noise. (c) Dephasing noise. (d) Depolarizing noise with measurement probablility $p=0.1$. Insets show the data collapse results. Every data point is averaged over $6\times10^3$ realizations. The circuit is evolved for $5L$ time steps.
• Figure 3: Numerical evidence of critical slowing down. We consider resetting noise with $p=0$ as an example. (a) Temporal evolution of $I_C$ for varying $q$. We take system size to be $L=256$. (b) Convergence time vs. $q$ for multiple system sizes. (c), (d) analogous to (a), (b), but in absence of random unitary gates.
• Figure 4: Efficient protocol. (a) Schematic workflow of the protocol. An important additional ingredient, compared to the approach in Ref. li2023, is the incorporation of projective measurements on the ancilla qubits. (b) Take depolarizing noise as an example. We choose $\rho = (\left | + \right \rangle \left \langle + \right \rangle)^{\otimes L/2}\otimes(\left | 0 \right \rangle \left \langle 0 \right \rangle)^{\otimes L/2}$ and $\sigma = (\left | 0 \right \rangle \left \langle 0 \right \rangle)^{\otimes L}$. Every data point is averaged over $3\times10^3$ realizations. The inset shows the data collapse result. (c) Analogous to (b), with measurement probability $p=0.1$.