Disordered purification phase transition in hybrid random circuits

This paper investigates how spatial modulation of noise and gate parameters in hybrid random Clifford circuits alters purification phase transitions, revealing that such non-uniformity changes the criticality exponents via the Harris criterion and can induce a distinct short-range entangled pure phase.

The Gist

Imagine you have a giant, invisible web made of quantum threads connecting a row of people (qubits). In a perfect world, this web is strong and complex, linking everyone together in a deep, mysterious way called entanglement. This is the "pure" state of the system.

However, in the real world, things get messy. Imagine someone randomly poking holes in the web or cutting threads. In the world of quantum circuits, these "pokes" are measurements or noise. If you poke too many holes, the web collapses, and the people become isolated individuals again. This is the "mixed" state.

The paper you provided is a study of exactly when and how this web collapses, and what happens if the "poking" isn't random but follows a specific, uneven pattern.

Here is the breakdown of their findings using everyday analogies:

1. The Game: Cutting the Quantum Web

The researchers set up a game with a line of quantum bits. Every round, they do two things:

• The Weaver: They twist the threads between neighbors, making the web stronger and more complex (this is the "random unitary gate").
• The Cutter: They randomly cut some threads (this is the "measurement").

If the Cutter is too aggressive, the web falls apart (the system becomes "mixed" or noisy). If the Weaver is strong enough, the web stays intact (the system stays "pure"). There is a tipping point—a specific rate of cutting—where the system suddenly switches from a tangled web to isolated threads. This is called a phase transition.

2. The Problem: Measuring the Invisible

Usually, scientists look at how "pure" the system is by checking if the whole thing is clean or dirty. But the researchers wanted a better tool to see the structure of the web, especially when it's already a bit dirty (mixed).

They used a special magnifying glass called Many-Body Negativity (MBN).

• Analogy: Imagine you have a tangled ball of yarn. A standard purity check just tells you if the ball is wet or dry. MBN is like a tool that counts exactly how many strands are actually knotted together, ignoring the loose, non-knotted fluff. It helps them see the "quantum knots" even in a messy state.

3. Experiment A: The Random Pokes (Uniform Noise)

First, they simulated a scenario where the "Cutter" pokes holes randomly but evenly across the whole line.

• Result: They found the exact moment the web collapses. They measured how "sensitive" the system is to the cutting. In physics, this sensitivity is called the correlation length exponent (let's call it the "wobble factor").
• Finding: In this uniform world, the "wobble factor" was relatively low (around 1.5). This means the system reacts in a predictable, standard way to the noise.

4. Experiment B: The Uneven Pokes (Disordered Noise)

Next, they changed the rules. Instead of poking evenly, they made the Cutter's behavior spatially modulated.

• Analogy: Imagine the Cutter has a bad mood swing. Some days, they are very gentle; other days, they are very aggressive. Or, imagine the Cutter only pokes the people on the left side of the room, leaving the right side alone. The "noise" is now messy and uneven.
• The Theory: There is an old rule in physics called the Harris Criterion. It basically says: "If a system is already very sensitive (wobbly), adding messy, uneven noise will break the rules and change how the system behaves entirely."
• Result: The researchers found that because the system was sensitive, the uneven noise did break the rules.
  • The "wobble factor" jumped up significantly (to around 3.0).
  • The system didn't just collapse; it collapsed in a completely different way than before. It entered a new "universality class" (a new category of behavior).

5. Experiment C: The Uneven Weaving

Finally, they tried something different. They kept the cutting even, but they made the Weaver uneven.

• Analogy: Imagine the person twisting the threads is only good at their job in some spots and bad in others, following a strange, repeating pattern (like a rhythm that never quite repeats perfectly).
• Result: This also caused a phase transition! But here, the web didn't just collapse into isolated threads. It settled into a "Pure-Like" state.
• The Twist: In this new state, the threads weren't connected all the way across the room (long-range entanglement). Instead, they formed tight, short little knots between immediate neighbors. It was a "pure" state, but a very local, short-range one.

The Big Takeaway

The paper proves that where the noise happens matters just as much as how much noise there is.

1. MBN is a Great Tool: The "Many-Body Negativity" tool they used is excellent at spotting these transitions and measuring the "wobble factor" in messy, mixed states.
2. Unevenness Changes Everything: When the noise is uneven (disordered), it doesn't just shift the tipping point; it fundamentally changes the laws of how the system collapses. The system becomes much more sensitive to the noise.
3. New States Exist: By messing with the pattern of the quantum operations, you can create new types of "pure" states that are different from the standard ones, characterized by short-range connections rather than long-range ones.

In short: If you want to understand how a quantum computer loses its magic, you can't just look at the average amount of noise. You have to look at the pattern of the noise, because a messy, uneven pattern changes the game entirely.

Technical Summary

Technical Summary: Disordered Purification Phase Transition in Hybrid Random Circuits

Problem Statement
Measurement-induced phase transitions (MIPT) and purification phase transitions in hybrid random quantum circuits are typically studied under the assumption of spatially uniform noise or measurement probabilities. However, in realistic quantum systems, noise and measurement errors often arise with spatial non-uniformity. While the stability of critical points against disorder is a well-established concept in statistical mechanics (governed by the Harris criterion), its specific application to purification phase transitions in mixed states within random quantum circuits remains underexplored. This paper investigates how spatial modulation in measurement probabilities and unitary gate applications affects the criticality and universality class of purification phase transitions.

Methodology
The authors study a one-dimensional hybrid random Clifford circuit with periodic boundary conditions consisting of $L$ qubits. The circuit evolution comprises alternating layers of:

1. Random Unitary Gates: Two-site random Clifford gates acting on neighboring qubits.
2. Projective Measurements: On-site $Z_i$ measurements.

The study focuses on two distinct scenarios:

• Case I (Uniform): All sites have an identical measurement probability $p$.
• Case II (Spatially Modulated): Measurement probabilities $p_i$ vary spatially according to a random distribution $p_i \equiv w_i^n$, where $w_i$ is a uniform random value in $[0,1]$ and $n$ controls the average probability $\bar{p}$.
• Extension (Modulated Unitaries): A third scenario where the application of two-site random Clifford gates is modulated quasi-periodically, while the measurement probability remains uniform.

To analyze the late-time steady states, the authors employ two primary observables:

1. Logarithmic Purity ($S_P$): Defined as $-\log \text{Tr}[\rho^2]$, serving as an indicator of the purification transition (volume law vs. area law).
2. Many-Body Negativity (MBN): Defined as $E_A = \log_2 ||\rho^{\Gamma_A}||_1$, where $\Gamma_A$ is the partial transpose over a half-chain subsystem. MBN is utilized specifically to quantify quantum entanglement in mixed states, filtering out classical correlations.

The numerical simulations utilize the stabilizer formalism for efficient large-scale calculations. The system is evolved for $O(L)$ time steps for logarithmic purity and $O(L^2)$ steps for MBN to ensure saturation. Finite-size scaling analysis is performed to extract critical probabilities ($p_c$) and correlation length critical exponents ($\nu$).

Key Results

1. Validity of MBN in Mixed States: The study demonstrates that MBN effectively characterizes the purification phase transition in mixed states. In the mixed phase, MBN exhibits a power-law scaling with system size ($\langle E_A \rangle \propto L^b$), while in the pure phase, it follows an area law.
2. Criticality in Uniform Circuits (Case I): For uniform measurement probability, the authors find a critical probability $p_c \approx 0.167$ and a correlation length exponent $\nu \approx 1.56$ (via MBN) and $\nu \approx 1.22$ (via logarithmic purity). These values are consistent with previous studies on MIPT but show slight deviations, potentially converging to the expected $4/3$ with larger system sizes.
3. Disorder-Induced Universality Change (Case II): When measurement probabilities are spatially modulated, the critical behavior changes significantly. The critical average probability shifts to $\bar{p}_c \approx 0.220$, and the correlation length exponent increases to $\nu \approx 3.06$ (via MBN) and $\nu \approx 3.38$ (via logarithmic purity).
4. Harris Criterion Verification: The change in the critical exponent from $\nu < 2$ (Case I) to $\nu > 2$ (Case II) aligns with the Harris criterion. Since the original system (Case I) has $\nu < 2/d$ (with spatial dimension $d=1$), the introduction of disorder (spatial modulation) is predicted to alter the universality class of the phase transition, which the numerical results confirm.
5. Modulated Unitary Gates: Introducing quasi-periodic modulation to the unitary gates (rather than measurements) induces a distinct phase transition at a critical modulation strength $A_J \approx 0.35$. This leads to a "pure-like" phase where MBN exhibits short-range entanglement and oscillatory behavior dependent on the modulation period, differing from the standard area-law pure phase observed in uniform circuits.

Significance and Claims
The paper claims that spatial non-uniformity in measurement probabilities fundamentally alters the criticality of purification phase transitions. Specifically:

• MBN as a Diagnostic Tool: The work establishes MBN as a robust observable for detecting critical exponents and phase transitions in mixed-state dynamics, extending its utility beyond pure-state MIPT.
• Universality Class Shift: The study provides numerical evidence that disorder in measurement probabilities changes the universality class of the purification transition, consistent with the theoretical predictions of the Harris criterion for quantum phase transitions.
• Effective Hamiltonian Connection: The results support the mapping of these circuit dynamics to disordered effective spin Hamiltonians (specifically a disordered $ZY$ model with random transverse fields), where the spatial modulation acts as a random field affecting the critical behavior.
• Distinct Phases: The identification of a "pure-like" phase with short-range entanglement under modulated unitary gates suggests that spatial modulation can create unique entanglement structures distinct from standard pure or mixed phases.

The authors conclude that spatial modulation is a critical factor in determining the nature of steady states in quantum circuits and that the Harris criterion remains a valid framework for understanding disorder effects in these non-equilibrium quantum systems.