From Classical Stochastic to Monitored Quantum Dynamics: Dynamical Phase Coexistence in East Circuit Models

This paper investigates monitored quantum East circuit models to demonstrate that dynamical phase coexistence between active and inactive states, originally observed in classical kinetically constrained systems, persists in the quantum regime and can be experimentally probed via measurement records interpreted as microstates of a fictitious 1+1D spin system.

The Gist

The Big Idea: From Traffic Jams to Quantum Traffic

Imagine a giant city where every house has a light switch. In this city, there is a very strange rule: You can only flip your light switch on or off if your neighbor to the left has their light ON.

This is the "East Model." It's a simple rule, but it creates a chaotic, unpredictable dance of lights. Sometimes, the whole city is buzzing with activity (lights flipping everywhere). Other times, the city freezes into a deep sleep (no lights flip).

In the world of classical physics (our everyday world), scientists have known for a long time that this city can exist in two states at once: a "busy" state and a "frozen" state. They call this Dynamical Phase Coexistence. It's like a traffic jam where half the cars are speeding and half are parked, and they can switch back and forth.

The Big Question: What happens if we replace these classical light switches with Quantum Light Switches?

Quantum switches are weird. They can be on, off, or a fuzzy mix of both at the same time. They can also be "entangled," meaning the state of one switch instantly affects another far away. The big mystery was: Does this "busy vs. frozen" coexistence survive when we go quantum? Or does the weirdness of quantum mechanics wash it all away?

The Experiment: The "Glass Wall" City

The researchers built a digital simulation of this city, but with a twist. They didn't just let the lights flip randomly; they watched them through a "glass wall."

• The Setup: They created a grid of quantum bits (qubits) acting like the light switches.
• The Rule: They kept the "East Rule" (you need a neighbor to be active to move).
• The Twist (The Glass Wall): They added a "measurement" step. Imagine a security guard checking every light switch every few seconds.
  • If the guard looks hard (strong measurement), the quantum switches act like normal classical switches. The city behaves predictably.
  • If the guard looks softly (weak measurement), the switches stay quantum. They remain fuzzy and entangled.
  • The researchers could slide a dial to change how hard the guard looked, moving smoothly from a classical city to a quantum city.

The Discovery: The Ghost of Coexistence

The team wanted to see if the "busy" and "frozen" states could still coexist in the quantum world.

1. The Classical Limit (The Guard Looks Hard):
When the measurement was strong, the results matched what we already knew. The city clearly showed two phases: a bustling active zone and a quiet inactive zone. They could see them coexisting in the same timeline, like a split screen showing a party and a library happening in the same building.

2. The Quantum Limit (The Guard Looks Soft):
This is where it got exciting. Even when they turned the measurement dial down to make the system fully quantum, the coexistence didn't disappear.

The quantum city still managed to have "active islands" (where lights were flipping wildly) and "inactive islands" (where everything was frozen) existing side-by-side in the same simulation.

The Metaphor:
Imagine a crowded dance floor.

• Classical: Half the people are dancing wildly, and half are standing still. They are distinct groups.
• Quantum: The dancers are now ghosts. They can be dancing and standing still at the same time. The researchers found that even in this ghostly, fuzzy state, you could still see a "ghostly dance floor" where the energy of the dance and the stillness of the rest coexisted.

How They Saw It: The "Space-Time Map"

How do you see something that is fuzzy and quantum? You can't just look at the average. Instead, the researchers looked at the history.

They treated the entire timeline of the simulation as a map.

• Red dots on the map meant a "measurement event" happened (a light flipped).
• White space meant nothing happened.

They found that these red dots didn't scatter randomly. They clumped together into clusters.

• Active Clusters: Big, dense blobs of red dots (the party).
• Inactive Clusters: Large white patches (the library).

In the quantum regime, these clusters were still there. The researchers proved that as the city got bigger, these two types of clusters became more distinct, just like in the classical world.

Why This Matters: The "Hydrophobic" Effect

The paper uses a clever analogy from chemistry to explain what they saw.

• Think of the "inactive" (frozen) region as an oil drop in water.
• In normal water, oil drops try to be small spheres to minimize their surface area.
• But near a phase transition, the "oil" (the frozen region) starts behaving strangely. It stops caring about its surface area and starts caring about its volume.

The researchers found that in the quantum system, the "frozen" regions grew in a way that suggested they were trying to become a massive, solid block of silence. This "shape-shifting" behavior is a signature that the system is teetering on the edge of a phase transition, even in the quantum world.

The Conclusion: A New Tool for Quantum Computers

The most important takeaway is this: Quantum mechanics doesn't destroy complex, glassy behavior; it preserves it.

This is huge news for the future of technology.

• The Problem: Simulating these complex quantum systems on a normal computer is incredibly hard. It's like trying to count every grain of sand on a beach while the tide is coming in.
• The Solution: The researchers showed that we don't need to simulate the whole quantum wave. We can just look at the measurement records (the "red dots" on our map). These records are easy to read and directly tell us about the hidden phases.

In Simple Terms:
The paper proves that even in the weird, fuzzy world of quantum mechanics, systems can still get "stuck" in different modes of behavior (like a traffic jam vs. free-flowing traffic). Furthermore, we can detect these jams by simply watching the "traffic logs" (measurement records) without needing to solve the impossible math of the whole quantum system.

This opens the door for using future Quantum Simulators (computers that mimic nature) to study complex materials, like the glass in your window or the strange behavior of superconductors, by simply watching how they "measure" themselves over time.

Technical Summary

1. Problem Statement

The paper addresses the fundamental question of how dynamical phase coexistence, a phenomenon well-established in classical kinetically constrained models (KCMs) like the East model, persists when transitioning to monitored quantum dynamics.

• Context: In classical non-equilibrium statistical mechanics, KCMs exhibit "glassy" behavior where dynamics are constrained by local rules (e.g., a spin can only flip if a neighbor is in a specific state). These models display a first-order dynamical phase transition between an "active" phase (high mobility) and an "inactive" phase (glassy, low mobility).
• The Gap: While classical stochastic dynamics are well-understood, simulating the long-time dynamics of open, monitored quantum systems is computationally prohibitive. It remains unclear whether the sharp dynamical phase transitions observed in classical trajectories survive in the quantum regime, where unitary evolution and quantum coherence compete with measurement-induced decoherence.
• Objective: To investigate the emergence and persistence of dynamical phase coexistence in a class of monitored quantum circuit models that interpolate between classical stochastic dynamics and unitary quantum dynamics.

2. Methodology

The authors introduce and analyze a Monitored Quantum East Circuit Model.

A. Model Definition

• System: A chain of $L$ qubits with open boundary conditions.
• Dynamics: Discrete-time evolution consisting of alternating layers:
  1. Unitary Gates: A "brickwork" structure of conditioned Quantum-East gates. The gate $u_{i-1,i} = e^{-i\omega n_{i-1}\sigma_x^i}$ flips qubit $i$ only if the control qubit $i-1$ is in state $|1\rangle$ (the East constraint).
  2. Weak Monitoring: At each timestep, every qubit is coupled to an ancilla via a Quantum-East gate controlled by a parameter $\gamma$. The ancilla is measured, yielding outcomes $k_i(t) \in \{0, 1\}$.
• Interpolation: The measurement strength $\gamma$ acts as the control parameter:
  • $\gamma = \pi/2$: Projective measurements reduce the system to the classical stochastic Floquet-East model.
  • $\gamma = 0$: No measurement; the system evolves as a purely unitary Floquet-Quantum East model.
  • $0 < \gamma < \pi/2$: The system exhibits monitored quantum dynamics with weak measurements.

B. Theoretical Framework

The authors treat space-time measurement records as microstates of a fictitious 1+1D spin system.

• Order Parameter: The time-integrated activity $A_{L,T} = \sum_{i,t} k_i(t)$, representing the total number of "active" measurement outcomes ($k=1$).
• Dynamical Partition Function: Defined as $Z_{L,T}(s) = \sum_{\eta} \pi(\eta) e^{-s A_{L,T}(\eta)}$, where $s$ is a counting field (analogous to inverse temperature).
• Scaled Cumulant Generating Function (SCGF): $\theta(s) = \lim_{L,T \to \infty} \frac{1}{LT} \log Z_{L,T}(s)$. Non-analyticities in $\theta(s)$ (specifically at $s=0$) signal a dynamical phase transition.
• Large Deviation Theory: The probability of observing a specific activity density follows a large deviation principle $P(a) \sim e^{-LT \phi(a)}$. A discontinuity in the derivative of $\theta(s)$ implies a first-order transition and phase coexistence.

C. Numerical Techniques

• Classical Limit: Exact diagonalization of the tilted transition matrix for intermediate system sizes ($L \le 48$).
• Quantum Regime:
  • Matrix Product States (MPS): Used to simulate individual stochastic trajectories (Quantum-Jump Monte Carlo) to generate space-time records.
  • Matrix Product Operators (MPO): Used to evolve the density matrix (or tilted operator) to compute the dynamical partition function $Z_{L,T}(s)$ and the probability of inactive clusters.
  • Tensor Network Contraction: Implemented using the ITensor library to handle large system sizes ($L$ up to 48) and long times ($T$ up to 10,000).

3. Key Contributions

1. Interpolation Model: The construction of a unified framework where the measurement strength $\gamma$ continuously tunes the system from classical stochasticity to unitary quantum dynamics, allowing for a direct comparison of dynamical phases.
2. Proof of Persistence: Demonstration that the dynamical phase transition (coexistence of active and inactive phases) observed in the classical limit ($\gamma = \pi/2$) persists into the quantum regime ($\gamma < \pi/2$).
3. Experimental Accessibility: Identification of space-time measurement records (ancilla outcomes) as the experimentally accessible proxy for dynamical phases. Unlike quantum expectation values $\langle n_i(t) \rangle$, which suffer from post-selection overheads, measurement records can be directly accessed in single experimental runs.
4. Finite-Size Scaling Analysis: Detailed analysis showing that the crossover between active and inactive phases sharpens as system size $L$ increases, converging to a discontinuity at $s=0$ in the thermodynamic limit, even for finite measurement strengths.

4. Key Results

• Classical Limit ($\gamma = \pi/2$):
  • The activity density $a(s)$ exhibits a sharp crossover at $s^* > 0$ for finite $L$.
  • As $L \to \infty$, $s^* \to 0$, and the activity density becomes discontinuous: $a(0) = 1/2$ (active) vs. $a(0^+) = 0$ (inactive). This confirms a first-order dynamical phase transition.
• Quantum Regime ($\gamma < \pi/2$):
  • Phase Coexistence: For finite measurement strengths, the system still exhibits distinct active and inactive phases separated by a sharp crossover in the activity density $a_{L,T}(s)$.
  • Scaling: The crossover position $s^*$ shifts toward $s=0$ as $L$ increases, confirming the persistence of the transition in the thermodynamic limit.
  • Measurement Strength Dependence: As $\gamma$ decreases (weaker measurement), the crossover becomes harder to resolve (requires larger $L$ and $T$) because the average activity density decreases, making it difficult to distinguish collective inactive domains from random fluctuations.
• Statistics of Inactive Clusters:
  • The authors analyze the dynamical free energy $F_{\ell \times \tau} = -\log p_{\ell \times \tau}$ of inactive clusters of size $\ell \times \tau$.
  • Hydrophobic Crossover: A transition is observed from area-scaling ($F \propto \ell \tau$, uncorrelated outcomes) to perimeter-scaling ($F \propto \ell + \tau$, collective behavior) as the cluster size increases.
  • This perimeter-scaling is a signature of the dynamical phase transition, indicating that large inactive regions are collective states rather than random fluctuations. The crossover time $\tau^*$ increases as $\gamma$ decreases.

5. Significance

• Bridging Classical and Quantum: The work provides strong evidence that dynamical phase transitions, previously thought to be a feature of classical stochastic systems, are robust features of monitored quantum many-body systems.
• Experimental Relevance: The findings are directly applicable to current quantum simulation platforms (e.g., superconducting circuits, Rydberg atoms) that possess mid-circuit measurement capabilities. The paper suggests that dynamical phase coexistence can be detected experimentally by analyzing the statistics of space-time measurement records without needing full state tomography.
• Theoretical Insight: It highlights the role of weak measurements in preserving quantum coherence while still allowing for the emergence of classical-like dynamical phases. The "hydrophobic crossover" in space-time clusters serves as a universal precursor to dynamical phase transitions in kinetically constrained systems.
• Future Directions: The paper suggests that characterizing these phases in the deep quantum regime ($\gamma \to 0$) will require larger system sizes and longer times, making it a prime candidate for investigation using quantum computers and simulators that can overcome classical computational limits.

In summary, the paper establishes that the rich phenomenology of dynamical phase coexistence in glassy systems is not lost in the quantum realm but is instead accessible through the statistical analysis of monitored trajectories, offering a new pathway to study non-equilibrium quantum matter.