Les Houches lectures on random quantum circuits and monitored quantum dynamics

These lecture notes apply statistical mechanics principles to analyze the dynamics of quantum information in ideal and monitored random quantum circuits, addressing systems where exact descriptions of individual realizations are typically intractable.

The Gist

The Big Picture: A Quantum Game of "Hide and Seek"

Imagine you have a giant, chaotic room full of people (these are quantum particles). You want to know how information spreads through this room.

In the world of quantum physics, there are two main forces at play:

1. The Mixer (Unitary Evolution): This is like a DJ spinning records and shuffling people around. It scrambles information, mixing everyone together so thoroughly that you can't tell who was talking to whom. This creates entanglement (a deep, spooky connection between particles).
2. The Spy (Measurements): This is like a security guard walking around taking photos of specific people. Every time the guard takes a photo, they "collapse" the quantum state of that person, revealing their secrets and breaking the deep connections they had with others.

The Core Question: What happens if you have a room where a DJ is constantly mixing people, but a spy is also taking photos at a certain rate?

• If the DJ is faster than the spy, the room stays a giant, messy, connected blob (High Entanglement).
• If the spy is faster than the DJ, the room gets broken up into small, isolated groups (Low Entanglement).

The paper explains that there is a sharp tipping point—a phase transition—where the system suddenly switches from being a "messy blob" to "isolated islands." This is called a Measurement-Induced Phase Transition (MIPT).

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Part 1: The "Brick Wall" of Chaos

The author starts by looking at a simplified version of this room: a Random Quantum Circuit.

• The Analogy: Imagine a wall made of bricks. Each brick is a "gate" that swaps two people. The gates are chosen randomly.
• The Result: If you just let these gates run without anyone watching, the information spreads incredibly fast. It's like dropping a drop of ink in water; it spreads out until the whole glass is purple. In physics terms, the "entanglement" grows linearly with time until it fills the whole system. This is called Volume Law (the more space you have, the more "messy" the connection is).

Part 2: The "Spy" Enters (Measurements)

Now, we add the measurements. Every time a gate happens, there's a chance a "spy" (a measurement) happens instead.

• The Game: Imagine you are trying to guess which of two secret states the room started in.
  • Low Spy Rate: The DJ mixes everything so well that by the time the spies take a few photos, the clues are gone. You can't tell the difference between the two starting states. You are just guessing (50/50).
  • High Spy Rate: The spies are taking so many photos that they catch the room before the DJ can mix it up. They have enough clues to figure out exactly what the starting state was. You can guess correctly almost every time.

The paper argues that this ability to "learn" from the spy's photos is the same as the physical change in entanglement.

• Phase 1 (Low Measurement): You can't learn anything. The system is "scrambled."
• Phase 2 (High Measurement): You can learn everything. The system is "unscrambled" and localized.

Part 3: The "Post-Selection" Problem (Why it's hard to see)

Here is a tricky part: If you look at the average of all possible outcomes (ignoring which specific photos the spy took), the transition disappears. It looks like a boring, messy soup.

• The Analogy: Imagine a magician shuffling a deck of cards. If you look at the average position of the cards over a million shuffles, it looks random. But if you look at one specific shuffle where the magician accidentally dropped a card, you see a pattern.
• The Problem: To see the transition, you have to look at specific "trajectories" (specific sequences of photos). But in real life, getting the exact same sequence of photos over and over is nearly impossible because the outcomes are random. This is called the "post-selection problem."
• The Solution: The author argues this isn't a bug; it's a feature. It's like a code. The transition isn't about what you can see with your eyes, but about whether the information is recoverable if you have a supercomputer to decode the spy's photos.

Part 4: The Magic Trick (Statistical Mechanics Mapping)

This is the most technical part of the paper, but here is the simple version.

Calculating the average of these quantum systems is mathematically impossible to do directly because of the "non-linear" nature of the math (it's like trying to average the square of a number by squaring the average—it doesn't work).

The "Replica Trick":

• The Analogy: Imagine you want to know the average height of a crowd, but the formula is too hard. So, you create 100 identical copies of the crowd (replicas). You calculate the average height of the group of 100 crowds. Then, you use a mathematical magic trick to pretend you only have 1 crowd again.
• The Result: By doing this, the author transforms the messy quantum problem into a Classical Physics problem.
  • The quantum circuit becomes a grid of magnets (an Ising model).
  • The "spins" of the magnets represent whether the information is scrambled or not.
  • The "measurements" act like heating the magnets.
    • Cold (Low Measurement): The magnets line up (Ordered Phase) = High Entanglement.
    • Hot (High Measurement): The magnets spin randomly (Disordered Phase) = Low Entanglement.

Part 5: The "Minimal Cut" (The Rope Analogy)

Finally, the paper uses a visual way to understand the transition.

• The Analogy: Imagine the quantum circuit is a 3D block of rope. To separate a piece of the rope (to measure its entanglement), you have to cut the ropes holding it together.
• The Rule: The "cost" of entanglement is the number of cuts you need to make.
  • In the "Messy" Phase: The ropes are woven so tightly that to separate a piece, you have to cut a huge number of ropes (proportional to the size of the piece). This is Volume Law.
  • In the "Clean" Phase: The measurements have already cut so many ropes that the piece is only held by a few. You only need to cut a small number of ropes (proportional to the surface area). This is Area Law.

Summary

This paper teaches us that:

1. Quantum systems naturally want to scramble information (entanglement).
2. Measurements try to stop the scrambling by revealing information.
3. There is a tipping point where the system switches from being a "scrambled mess" to a "clean, readable state."
4. We can understand this complex quantum behavior by turning it into a simpler problem about magnets or percolation (like water flowing through a sponge).
5. Even though we can't easily see this in a lab (because of the "post-selection" issue), it is a fundamental truth about how information flows in our universe, relevant for building future quantum computers that can correct their own errors.

Technical Summary

1. Problem Statement

The central problem addressed is understanding the universal features of non-equilibrium quantum dynamics in many-body systems, specifically how quantum information is encoded, scrambled, and potentially "leaked" to an observer.

• Context: Traditional many-body physics focuses on equilibrium phases. However, modern quantum simulators allow for the study of isolated systems far from equilibrium.
• The Conflict: There is a competition between unitary dynamics (random quantum gates), which scramble local information into non-local degrees of freedom (leading to volume-law entanglement), and projective measurements, which extract information and tend to collapse the wavefunction (leading to area-law entanglement).
• The Phenomenon: This competition gives rise to Measurement-Induced Phase Transitions (MIPTs). Unlike standard phase transitions driven by temperature or Hamiltonian parameters, MIPTs occur in the space of quantum trajectories (conditioned on measurement outcomes) and are characterized by a change in the scaling of entanglement entropy (from volume-law to area-law) or the observer's ability to learn about the initial state.
• The Challenge: Calculating properties of individual quantum trajectories is generically intractable due to the nonlinearity introduced by the Born rule (normalization of the state after measurement) and the randomness of the circuit.

2. Methodology

The lecture notes employ a combination of random matrix theory, tensor network methods, and statistical mechanics mappings using the replica trick.

A. Random Quantum Circuits (Unitary Case)

• Model: A 1D brick-work circuit of qudits with local random unitary gates drawn from the Haar measure.
• Haar Averaging: To compute average quantities (like purity), the author utilizes the Weingarten calculus. The average of products of unitary matrices over the Haar measure is expanded into a sum over permutations of replicas.
• Tensor Networks: The circuit is viewed as a tensor network. Entanglement is bounded by the "minimal cut" (the minimum number of links to sever to isolate a region). In the unitary case, this leads to linear entanglement growth.

B. The Replica Trick for Monitored Circuits

To handle the nonlinearity of the normalization factor ($\text{tr}(\rho_m)$) in monitored circuits, the author uses the replica trick:

• Identity: $\ln x = \lim_{k \to 0} \frac{d}{dk} x^k$.
• Formulation: The averaged Rényi entropy $S_A^{(n)}$ is expressed as a limit involving $Q = nk + 1$ replicas. The term $k$ corresponds to the Rényi index, and the extra $+1$ replica accounts for the Born probability weight of the trajectory.
• Mapping to Statistical Mechanics:
  • After averaging over Haar-random unitaries and measurement outcomes, the problem maps to a classical statistical mechanics model on a honeycomb lattice.
  • Degrees of Freedom: Permutation group elements $g_i \in S_Q$ (where $Q$ is the number of replicas) live on the lattice vertices.
  • Boltzmann Weights:
    • Unitary Links: Weighted by Weingarten functions $Wg_D(g_i^{-1}g_j)$, favoring alignment of permutations (ferromagnetic interaction).
    • Measurement Links: With probability $p$, measurements project all replicas to the same state, effectively forcing the permutation to the identity. This acts as a "disordering" field or increased temperature.
  • Partition Function: The entanglement entropy is identified with the free energy cost of creating a domain wall between different boundary conditions (trivial vs. swapped permutations) in this classical spin model.

C. Large $d$ Limit and Percolation

• In the limit of infinite local Hilbert space dimension ($d \to \infty$), the statistical mechanics model simplifies to a Potts model with $Q!$ colors.
• Using the Fortuin-Kasteleyn cluster representation, this maps exactly to classical bond percolation.
• The entanglement transition corresponds to the percolation threshold: volume-law entanglement exists when unmeasured links percolate (forming a connected path), while area-law occurs when measurements break all paths.

3. Key Contributions

1. Unified Statistical Mechanics Framework: The notes provide a rigorous derivation mapping the dynamics of monitored random quantum circuits to a classical statistical mechanics model of interacting permutations. This transforms a complex quantum many-body problem into a tractable classical problem.
2. "Learnability" Perspective: The author emphasizes an information-theoretic interpretation of MIPTs. The transition is framed not just as an entanglement change, but as a transition in the observer's ability to learn (distinguish) between two orthogonal initial states based on the measurement record.
   • Volume-law phase: The observer cannot distinguish states (random guessing, $P_{success} = 1/2$).
   • Area-law phase: The observer can distinguish states ($P_{success} > 1/2$).
3. Resolution of the "Post-Selection Problem": The notes clarify that while MIPTs are technically defined by post-selecting specific measurement trajectories (which is exponentially hard experimentally), they represent fundamental information-theoretic phase transitions. The transition is about the recoverability of information, analogous to decoding thresholds in quantum error correction, rather than a direct observable in the averaged density matrix.
4. Conformal Field Theory (CFT) Analysis: The notes discuss the critical point of the transition, predicting it is described by a non-unitary CFT with central charge $c=0$ (in the replica limit). The entanglement entropy at criticality scales logarithmically, $S \sim \alpha_n \ln L$, where the prefactor $\alpha_n$ is a boundary scaling dimension, distinct from the bulk central charge.
5. Universality and Crossover: The notes explain the discrepancy between numerical simulations (often showing percolation exponents for small $d$) and theoretical predictions. They identify a crossover scale $\xi(d) \sim d^{4/3}$: for small systems, the system behaves like the infinite-$d$ percolation limit; for large systems, it crosses over to the true finite-$d$ universality class.

4. Key Results

• Entanglement Growth: In pure unitary circuits, the 2nd Rényi entropy grows linearly with time ($S \sim t$) until saturation at volume-law ($S \sim L$). This is derived via a mapping to an Ising model where domain walls have finite line tension.
• Phase Transition: There exists a critical measurement rate $p_c$.
  • $p < p_c$ (Volume-Law Phase): The system retains extensive entanglement. The statistical mechanics model is in an ordered (ferromagnetic) phase. The observer cannot learn the initial state.
  • $p > p_c$ (Area-Law Phase): Frequent measurements collapse the state. The statistical mechanics model is disordered (paramagnetic). Domain walls have zero tension, and entanglement scales with the boundary size (constant in 1D). The observer can learn the initial state.
• Critical Exponents:
  • In the $d \to \infty$ limit, the transition maps to bond percolation with $p_c = 1/2$ and correlation length exponent $\nu = 4/3$.
  • For finite $d$, the transition belongs to a different universality class, though percolation exponents are observed at intermediate scales.
• Purification Dynamics: Starting from a mixed state, the system purifies (entropy goes to 0) rapidly in the area-law phase, while it remains mixed (high entropy) in the volume-law phase.

5. Significance

• Theoretical Bridge: The work bridges quantum information theory, many-body physics, and statistical mechanics. It demonstrates that complex quantum dynamics can be understood through the lens of classical phase transitions (symmetry breaking of replica permutations).
• Experimental Relevance: While direct observation of MIPTs is challenging due to the post-selection problem, the "learnability" and "error correction" perspectives provide a roadmap for experimental verification using quantum simulators and classical post-processing (decoding).
• Foundational Insight: The notes clarify that MIPTs are not merely artifacts of averaging but represent a fundamental shift in how quantum information is stored and protected in open systems. The volume-law phase acts as a self-correcting quantum memory, while the area-law phase represents a regime where information is lost to the environment.
• Pedagogical Value: By focusing on exact methods (Haar averaging, replica trick) and avoiding dual-unitary circuits (which are solvable but less generic), the notes provide a clear, self-contained derivation of the statistical mechanics mapping, serving as a foundational resource for researchers entering the field.