Scaling Laws of Quantum Information Lifetime in Monitored Quantum Dynamics

Abstract

Quantum information is typically fragile under measurements and environmental coupling. Remarkably, we find that its lifetime can scale exponentially with system size when the environment is continuously monitored via mid-circuit measurements -- regardless of bath size. Starting from a maximally entangled state with a reference, we analytically prove this exponential scaling for typical Haar random unitaries and confirm it through numerical simulations in both random unitary circuits and chaotic Hamiltonian systems. In the absence of bath monitoring, the lifetime exhibits a markedly different scaling: it grows at most linearly -- or remains constant -- with system size and decays inversely with the bath size. We further extend our findings numerically to a broad class of initial states. In the intermediate regime of partial monitoring, we identify and prove a two-scale transition, where the QMI decays logarithmically at microscopic time scales but linearly at macroscopic time scales.} We discuss implications for {monitored quantum circuits in the weak measurement limit, quantum algorithms such as quantum diffusion models and quantum reservoir computing, and quantum communication. Finally, we experimentally verify the gap of persisted information on IBM Quantum hardwares.

Figures (29)

• Figure 1: Schematic plot of quantum dynamics with mid-circuit measurements. In (a), initial input $A_0$ ($N_A$ qubits) goes through interactions with bath via unitaries $U_1,U_2,\cdots,U_t$ to produce the final output $A_t$. Between each unitary, bath qubits $B$ ($N_B$ qubits) are measured and reset to $\ket{0}$'s in most of our discussions, except in Section \ref{['sec:reset']}. Equivalently, one can regard the measurement as applied on bath qubits $B_1,B_2,\cdots, B_t$, each initialized in $\ket{0}^{\otimes N_B}$. To track the evolution of quantum information, we introduce the reference system $R$ initially maximally entangled with $A_0$. The corresponding quantum mutual information (QMI) dynamics is sketched in (b), which decays logarithmically in monitored dynamics but linearly in time with an initial perfect information protection and a late exponential tail with unmonitored bath. Panel (c) highlights connections between the model above and other models and algorithmic frameworks, including quantum denoising diffusion probabilistic model (QuDDPM) zhang2024generative, quantum reservoir computing (QRC) chen_temporal_2020hu2024overcoming, monitored quantum circuits li2019measurementskinner2019measurement, and entanglement-assisted communication bennett2002entanglement. The reference system surrounded by the blue dashed box in (a) indicates that it is not included in the original quantum circuit models except for communication setting.
• Figure 2: Measurement-conditioned QMI $\overline{I(R:A_t|\mathbf{z})}$ in monitored dynamics with reset. In (a) we plot numerical simulation results of (von Neumann entropy version) $\overline{I(R:A_t|\mathbf{z})}$ with Haar-random unitaries (dark blue dots) and fixed Ising Hamiltonian evolution (light blue dots) in a system of $N_R =N_A=5, N_B=1$ qubits. Errorbars represent sample fluctuations of Haar unitary implementations. Red dashed line represents the theoretical asymptotic lower bound of Eq. \ref{['eq:avgMI_lb_simplify']} in Theorem \ref{['theorem_avgMI']}. Horizontal black dashed line indicate the threshold of $\overline{I(R:A_t|\mathbf{z})}=\overline{I(R:A_0|\mathbf{z})}/4$. The inset shows the growth of lifetime versus system size $N_A$ of Haar unitary and Hamiltonian evolution. Dots are and red dashed line represent the numerical simulation results and theoretical result in Eq. \ref{['eq:avgMI_time']} with $\epsilon = 1/4$. In (b), we plot QMI versus a rescaled time in the dynamics with random Clifford unitaries (dots) in a system of $N_R = N_A = 32, 64, 128, 256$ qubits (light to dark) and $N_B = 16$ qubits. The solid lines represent the theory of Eq. \ref{['eq:avgMI_lb_simplify']}. In (c) and (d), we plot decay of normalized QMI in numerical simulations with initial states of perturbed Haar states $\phi_\delta$ and CQ states $\rho_\delta$ versus a shifted time separately. Blue lines are Bell states result with Haar unitary in (a) for reference. The color bar shows the value of $\delta$.
• Figure 3: Normalized measurement-conditioned QMI $\overline{I(R:A_t|\mathbf{z})}$ for $N_R < N_A$ in monitored dynamics with reset. In (a) we plot numerical simulation results with random Clifford unitaries (dots) in a system of $N_A = 128, N_B=16$ qubits and different $N_R$ reference qubits. Dashed line represents the theoretical asymptotic lower bound of Eq. \ref{['eq:avgMI_lb_simplify_lessR']} in Theorem \ref{['theorem_avgMI_lessR']} for the corresponding $N_R$. In (b), we plot the asymptotic dynamics of $\overline{I(R:A_t|\mathbf{z})}/I(R:A_0) = 1 - \log_2(d_Rt/d_A + 1)/N_R$ with $N_R = 20$ and different $N_A$ (shown by the colorbar).
• Figure 4: Measurement-unconditioned QMI $I(R:A_t)$ in unmonitored dynamics with reset. In (a) we plot numerical simulation results of $I(R:A_t)$ of a Bell initial state with random Haar unitary (dark blue dots) and fixed Ising Hamiltonian evolution (light blue dots) in a system of $N_A=5, N_B=1$ qubits. Errorbars represent sample fluctuations of Haar unitary implementations. Red and orange lines represent the exact and asymptotic theory of Eq. \ref{['eq:MI_traceout']} and Eq. \ref{['eq:MI_traceout_asymp']} in Theorem \ref{['MI_traceout_theorem']}. Horizontal black dashed line indicate the threshold of $\epsilon=1/4$. Late-time dynamics is presented in (b) with orange line showing asymptotic theory of Eq. \ref{['eq:MI_traceout_asymp_late']} in Theorem \ref{['MI_traceout_theorem']}. In (c) and (e), we plot numerical simulation of $I(R:A_t)$ under Haar unitary dynamics with initial perturbed Haar states and CQ states separately. Dark to light green lines represent different perturbation $\delta$ indicated by the colorbar. Blue lines represent Haar result from (a) for reference. In both cases, we have 5 reference and data qubits. (d) and (f) are corresponding late-time dynamics, and blue lines is Haar result from (b). In (g), we show the eigen-spectrum of ${\cal P}$ implemented by Haar unitary from $N_A=2$ to $6$ (from left to right) with $N_B=2$ bath qubits. The red dashed circle indicates the radius of $1/\sqrt{d_B}$.
• Figure 5: Measurement-unconditioned QMI $I(R:A_t)$ in unmonitored Clifford dynamics with reset. The initial state is the Bell state with $N_R = N_A$. In subplot (a), we plot the numerical simulations of the normalized QMI $I(R:A_t)/I(R:A_0)$ (dots) in a system of various $N_A$ and $N_B = 16$. The solid lines represent the asymptotic result of Eq. \ref{['eq:MI_traceout_asymp']} in Theorem \ref{['MI_traceout_theorem']}. In subplot (b), we show the QMI lifetime. Blue dots are numerical results obtained from (a) and the red dashed line is the theoretical prediction of Eq. \ref{['eq:MI_traceout_time']} with $\epsilon = 1/4$.

Theorems & Definitions (7)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6
• Theorem 7