Characterizing a non-equilibrium phase transition on a quantum computer

Using the Quantinuum H1-1 quantum computer, researchers successfully simulated a quantum extension of a classical disease spreading model to quantitatively characterize its non-equilibrium phase transition, demonstrating how mid-circuit resets and conditional logic enable the study of complex open quantum system dynamics.

The Gist

Imagine you are watching a game of "Zombie Tag" played on a long line of people.

In this game, there are two states: Alive (standing up) and Zombie (sitting down).

• The Rules: If you are Alive and standing next to a Zombie, there's a chance the Zombie will "infect" you, turning you into a Zombie.
• The Twist: Sometimes, just by bad luck, an Alive person might spontaneously sit down (die) without being infected.

Now, imagine you can tune the game.

• If the "spontaneous death" rate is very high, the zombies can't spread fast enough. The whole line eventually sits down. This is the Dead State.
• If the "spontaneous death" rate is low, the infection spreads like wildfire. The line stays mostly standing. This is the Alive State.

But what happens right in the middle? That's the Phase Transition. It's a delicate balance where the infection spreads just enough to keep a few people standing, but not enough to take over the whole line. In physics, this specific balance point is called the Directed Percolation threshold. It's a universal rule that applies to everything from forest fires to disease outbreaks.

The Problem: The "Too Hard" Simulation

Scientists have long wanted to study what happens if you add Quantum Mechanics to this game. In the quantum version, people can be in a "superposition" (both standing and sitting at the same time) and can get "entangled" (their fates become linked in spooky ways).

The problem? Simulating this quantum game on a regular computer is like trying to count every possible outcome of a coin flip for a billion people simultaneously. The math gets so huge, so fast, that even the world's fastest supercomputers give up. They simply run out of memory and time.

The Solution: The Quantum Computer as a "Magic Blackboard"

This paper describes how a team of scientists used a Quantum Computer (specifically the Quantinuum H1-1) to play this game for real, instead of just calculating it.

Think of the quantum computer not as a calculator, but as a magic blackboard that can actually be the game. Instead of simulating the rules, the computer is the game.

However, quantum computers are notoriously fragile. They are like a house of cards in a hurricane; a tiny breeze (error) knocks them over. Usually, to run a long game like this, you would need thousands of cards (qubits), but the computer only had 20.

The Clever Tricks: "Recycling" and "Smart Logic"

To make this work with only 20 cards, the team used two brilliant tricks:

1. Qubit Reuse (The "Hot Potato" Strategy):
   Imagine you are playing a game that requires 73 players, but you only have 20 actors. Instead of giving up, you let the actors play a scene, then tell them to sit down, wipe their makeup, and immediately stand up to play a different character in the next scene.
   The team used "mid-circuit resets" to wipe the memory of a qubit (a quantum bit) and reuse it for a new part of the simulation. This allowed them to simulate a 73-site chain using only 20 physical qubits.

2. Error Avoidance (The "Don't Touch the Sleeping Giant" Rule):
   In their quantum game, if a player is already "dead" (in the |0⟩ state), the infection rules shouldn't apply to them. But, if the computer tries to apply a rule to a dead player, it might accidentally wake them up due to a glitch (an error).
   The team programmed the computer to check a "log" in real-time. If it saw a player was already dead, it would skip the rule entirely. It's like a referee seeing a player is already out of the game and saying, "No need to blow the whistle, just move on." This prevented errors from ruining the simulation.

The Big Discovery

They ran the game at three different settings:

• Too much death: The line went quiet (Dead State).
• Too little death: The line stayed active (Alive State).
• Just right (The Critical Point): They watched the "edge of chaos."

The Result: Even with the spooky quantum rules (superposition and entanglement), the game behaved exactly like the classical version. The "universal rules" of the phase transition didn't change. The quantum fluctuations didn't break the pattern; they just played along with the same old dance steps.

Why This Matters

This is a huge step forward for two reasons:

1. Proof of Concept: It shows that quantum computers can now tackle "open systems" (systems that lose energy or information to the environment), which are much harder to simulate than closed systems.
2. The Future: It proves that with clever tricks like recycling qubits and avoiding errors in real-time, we can use today's imperfect quantum computers to solve problems that are impossible for classical computers.

In a nutshell: The scientists built a tiny, magical version of a disease-spreading game on a quantum computer. They used clever shortcuts to keep the game running without crashing, and they discovered that even in the quantum world, the rules of how things spread and die remain surprisingly familiar.

Technical Summary

1. Problem Statement

The paper addresses the challenge of simulating non-equilibrium phase transitions in open quantum systems, specifically those involving dissipative dynamics and absorbing states.

• Context: While equilibrium phase transitions are well-understood via universality classes (e.g., Ising model), non-equilibrium transitions (like the Directed Percolation or DP class) are less understood, especially when quantum effects (entanglement, superposition) are introduced.
• The Challenge: Classically simulating open quantum systems is computationally expensive, often requiring memory or time that scales exponentially with system size. Furthermore, determining whether quantum fluctuations alter the universality class of a transition (e.g., does a quantum contact process still follow DP scaling?) has remained elusive due to the difficulty of reaching large system sizes and long evolution times in classical simulations.
• Goal: To experimentally realize and characterize a quantum generalization of the classical Contact Process (a model for disease spreading) on a quantum computer to determine if it exhibits Directed Percolation (DP) critical scaling.

2. Methodology

The authors utilized the Quantinuum H1-1 trapped-ion quantum computer to simulate the 1D Floquet Quantum Contact Process (1D FQCP).

A. The Model: 1D Floquet Quantum Contact Process

The model is a discrete-time quantum circuit generalization of the classical contact process:

• Lattice: A 1D chain of qubits where $|1\rangle$ represents an "active" site and $|0\rangle$ an "inactive" (absorbing) site.
• Dynamics: Each time step consists of:
  1. Probabilistic Reset (Dissipation): Each qubit is reset to $|0\rangle$ with probability $p$. This drives the system toward an absorbing state ($|0\dots0\rangle$).
  2. Unitary Driving: Four layers of controlled-rotation gates ($CR_x$ and $CR_y$) with angle $\theta$. These gates allow active sites to "spread" to neighbors (branching) or coalesce.
• Phase Transition: The competition between spreading (unitary) and decay (dissipation) creates a phase transition.
  • Inactive Phase ($p > p_c$): Active sites decay exponentially.
  • Active Phase ($p < p_c$): A finite density of active sites survives indefinitely.
  • Critical Point ($p = p_c$): The system exhibits scale-invariant, power-law growth of observables.

B. Experimental Implementation & Optimization

To overcome hardware limitations (noise and qubit count), the team employed several advanced techniques:

1. Qubit Reuse & Mid-Circuit Measurement/Reset:
   • Instead of using $4t+1$ qubits for $t$ time steps, they used qubit reuse. By measuring qubits and resetting them mid-circuit, they simulated up to 72 circuit layers (time steps) on a 73-site lattice using only 20 physical qubits.
   • They exploited reflection symmetry ($r \to -r$) of the initial single-seed state to reduce the required qubits further (simulating only $r \geq 0$).
2. Error Avoidance via Conditional Logic:
   • The H1-1 allows real-time conditional logic. The team maintained a real-time log of qubits known to be in the $|0\rangle$ state (due to prior resets).
   • They conditionally skipped applying two-qubit gates if the control qubit was known to be $|0\rangle$. Since a $|0\rangle$ control does not change the target in this specific gate set, skipping the gate avoids introducing gate errors without altering the physics. This reduced the number of two-qubit gates by ~30%.
3. Error Mitigation:
   • Zero-Noise Extrapolation (ZNE): For the critical point ($p=0.3$), they ran circuits with original gate counts and "stretched" circuits (3x gate count) to extrapolate results to the zero-noise limit.

3. Key Contributions

• First Experimental Characterization: This is one of the first demonstrations of a non-equilibrium phase transition on a quantum computer, moving beyond equilibrium ground-state simulations.
• Scalability via Qubit Reuse: Demonstrated that mid-circuit measurement and reset capabilities allow for the simulation of deep circuits (72 layers) on a 20-qubit device, effectively simulating a 73-site system.
• Validation of Universality: Provided strong experimental and numerical evidence that the quantum contact process belongs to the Directed Percolation (DP) universality class, despite the introduction of quantum entanglement.
• Technique Development: Showcased "error avoidance" via real-time conditional logic as a powerful primitive for simulating open quantum systems, distinct from traditional error correction.

4. Results

The team simulated the model for three regimes: Active ($p=0.1$), Critical ($p=0.3$), and Inactive ($p=0.5$), with $\theta = 3\pi/4$.

• Critical Scaling:
  • Near the critical point, observables followed the expected power-law scaling of the DP class:
    • Total active sites: $\langle N(t) \rangle \sim t^\Theta$
    • Mean-squared extent: $\langle R^2(t) \rangle \sim t^{2/z}$
  • The experimentally extracted exponents ($\Theta \approx 0.31$, $z \approx 1.58$) matched the known 1D DP values within experimental error margins.
• Scaling Collapse:
  • The active-site density profile $\langle n(r,t) \rangle$ collapsed onto a universal curve when rescaled by $t^{1/z}$ and $t^{\Theta-1/z}$, confirming the self-similar fractal nature of the critical cluster.
• Error Impact:
  • Deviations from DP scaling were observed at late times ($t > 12$) due to two-qubit gate errors creating "fake" active sites.
  • However, the error avoidance technique and ZNE allowed the data to remain quantitatively accurate up to $t \approx 12$, sufficient to verify the critical scaling regime.
• Numerical Validation:
  • Classical Matrix Product Operator (MPO) simulations confirmed that the quantum model ($\theta = 3\pi/4$) exhibits a DP transition with a critical point $p_c \approx 0.30$, distinct from the classical point ($\theta = \pi, p_c \approx 0.39$).

5. Significance

• Quantum Advantage for Open Systems: The work demonstrates that quantum computers are uniquely suited for simulating open quantum systems where classical methods struggle due to the exponential cost of tracking mixed states and non-unitary dynamics.
• Universality Robustness: It suggests that the Directed Percolation universality class is robust against the introduction of quantum fluctuations (at least for the specific initial states and parameters studied), bridging the gap between classical statistical mechanics and quantum many-body physics.
• Future Directions: The techniques developed (qubit reuse, conditional error avoidance) are presented as essential primitives for future studies of more complex non-equilibrium phenomena, such as measurement-induced phase transitions (MIPT) and higher-dimensional open system dynamics.

In summary, this paper successfully leverages the unique capabilities of a trapped-ion quantum computer (mid-circuit reset, high fidelity, conditional logic) to solve a problem in non-equilibrium statistical physics that is classically intractable at the required scale, confirming that quantum generalizations of classical absorbing state transitions retain their classical universality.