Many-Body Anti-Zeno Thermalization and Zeno Determinism in Monitored Hamiltonian Dynamics

TL;DR

The paper addresses generating Haar-random pure states for a subsystem while keeping the bath small by using holographic deep thermalization (HDT) under Hamiltonian dynamics with mid-circuit measurements. It develops a framework based on the frame potential $F^{(K)}$ to quantify randomness, analyzes the interplay between anti-Zeno effects that accelerate thermalization and quantum Zeno effects that suppress it, and provides analytical bounds on saturation and Zeno thresholds ($n_{ m sat}$ and $n_{ m Zeno}$) for a fixed evolution time $T$. The authors show that a local ETH-consistent Hamiltonian can approach Haar randomness for modest measurement counts, while locality imposes fundamental limits; they validate the theory with numerical simulations and a proof-of-principle experiment on IBMQ demonstrating the AZE–ZE dynamics. The work offers a resource-efficient route to generate genuinely random states on NISQ devices by reducing bath size and exploiting controlled mid-circuit measurements, with implications for quantum information processing and benchmarking.

Abstract

Random quantum states are essential for quantum information science, with applications ranging from quantum computing to cryptography. Prior approaches for generating these states often rely on using a large bath to thermalize a smaller system, with a subsequent measurement on the bath used to post-select a random state. To reduce the required size of the bath, we propose a resource-efficient scheme using holographic deep thermalization driven by Hamiltonian evolution, combined with mid-circuit measurements. This scheme relies on dynamical circuits, enabling a trade-off between spatial and temporal resources and allowing the generation of genuinely random states with only a constant-size bath. We quantify the randomness using the frame potential and derive its asymptotic behavior, which shows good agreement with our numerical simulations and experimental results on IBM quantum devices. For a fixed total evolution time, increasing the number of mid-circuit measurements initially produces an exponential decrease in the frame potential -- a quantum anti-Zeno behavior arising from holographic deep thermalization. Past a critical number of mid-circuit measurements, the frame potential rises again, signaling the onset of the quantum Zeno effect.

Figures (8)

• Figure 1: Schematic of the quantum circuit implementing holographic deep thermalization via Hamiltonian dynamics with multiple mid-circuit measurements.
• Figure 2: Frame potential as a function of the number of measurements. The blue lines represent numerical results, while the orange and green lines correspond to analytical results for the asymptotic region of HDT in Eq. \ref{['eq:HDT1']} and the ZE in Eq. \ref{['eq:zeno']}, respectively. Other parameters are: (a),(d) $n_{\rm s}=5$, $n_{\rm b}=3$; (b),(c),(e),(f) $n_{\rm s}=7$, $n_{\rm b}=1$. Total evolution time is $T = 15/J_{zz}$ for (a),(b),(d),(e) and $T = 5/J_{zz}$ for (c),(f). The orange, green, and blue dashed lines indicate the positions of $n_{\rm sat}$, $n_{\rm Zeno}$, and $n_{\gamma}$, respectively, with $r=0.1$. Other parameters are $J_z=0.9J_{zz}$ and $J_x=1.4J_{zz}$.
• Figure 3: The convergence dynamics of frame potential as a function of the number of measurements for (a) $H_{\rm Ising}$, (b) $H_{YY}$, and (c) $H_{XXX}$. Blue lines represent numerical results, while orange lines correspond to analytical predictions for HDT. Other parameters are $n_{\rm s}=7$, $n_{\rm b}=1$, $J_z=0.9J_{zz}$, $J_x=1.3J_{zz}$, and $T=15/J_{zz}$.
• Figure 4: Frame potential for $K$-designs corresponding to (a) $K=1$, (b) $K=3$, and (c) $K=10$. The black solid line represents the analytical approach. Blue dots indicate results from the non-integrable Hamiltonian ($J_z=0.9J_{zz}$), while green dots correspond to the integrable Hamiltonian ($J_z = 0$). Other parameters are $n_{\rm s}=7$, $n_{\rm b}=1$, $J_x=1.3J_{zz}$, and $T=15/J_{zz}$ .
• Figure 5: Frame potential as a function of the number of measurements for (a) $H_{\rm Ising}$, (b) $H_{YY}$, and (c) $H_{XXX}$. Blue lines represent numerical results, while green lines correspond to analytical predictions for ZE with (a) $\alpha=3$, and (b)(c) $\alpha=1$. The total evolution time is $T=15/J_{zz}$. The system consists of 5 qubits, with 3 bath qubit. Other parameters are $J_z=0.9J_{zz}$ and $J_x=1.3J_{zz}$.

Theorems & Definitions (1)

• Lemma 1