[Paper Review] A universal crossover in quantum circuits governed by a proximate classical error correction transition

Anasuya Lyons, Soonwon Choi+1 more|arXiv (Cornell University)|Aug 3, 2022

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper introduces a semi-classical quantum circuit model to study the interplay between quantum entanglement and classical error correction, showing that even a small density of quantum gates (via Hadamard operations) destroys the classical absorbing state at a directed percolation critical point, turning a sharp phase transition into a universal crossover. The crossover is governed by directed percolation critical exponents, demonstrating that quantum effects act as a relevant perturbation that suppresses the error threshold and alters information decay dynamics.

ABSTRACT

We formulate a semi-classical circuit model to clarify the role of quantum entanglement in the recently discovered encoding phase transitions in quantum circuits with measurements. As a starting point we define a random circuit model with nearest neighbor classical gates interrupted by erasure errors. In analogy with the quantum setting, this system undergoes a purification transition at a critical error rate above which the classical information entropy in the output state vanishes. We show that this phase transition is in the directed percolation universality class, consistent with the fact that having zero entropy is an absorbing state of the dynamics; this classical circuit cannot generate entropy. Adding an arbitrarily small density of quantum gates in the presence of errors eliminates the transition by destroying the absorbing state: the quantum gates generate internal entanglement, which can be effectively converted to classical entropy by the errors. We describe the universal properties of this instability in an effective model of the semi-classical circuit. Our model highlights the crucial differences between information dynamics in classical and quantum circuits.

Motivation & Objective

• To understand the role of quantum entanglement in information dynamics near a classical error correction transition in quantum circuits.
• To model a semi-classical circuit with classical gates and erasure errors to isolate the classical encoding transition.
• To investigate how the addition of quantum gates—specifically Hadamard operations—alters the critical behavior of the system.
• To determine whether quantum effects can stabilize long-lived quantum information or instead induce a universal crossover in information decay.
• To establish a connection between quantum circuit dynamics and statistical mechanics models like directed percolation.

Proposed method

• Formulate a random circuit model with nearest-neighbor classical gates and probabilistic erasure errors (reset to 0) on each site.
• Map the classical circuit dynamics to a directed percolation problem, where encoded bits propagate and are destroyed by errors.
• Identify a classical encoding transition at a critical error rate $ p_c = 0.081 $, belonging to the directed percolation universality class with critical exponents $ z=1.58 $, $ \gamma=0.75 $, $ \eta=2.34 $.
• Introduce a small density of quantum gates (Hadamard operations) to generate entanglement, which acts as an additive noise perturbation on the classical critical point.
• Use stabilizer circuit simulations to verify the scaling collapse of entropy decay at criticality, confirming directed percolation behavior.
• Compare with a purely classical noise process (indeterminate bit replacement) that also destroys the absorbing state and yields matching critical exponents.

Experimental results

Research questions

• RQ1 Does a classical circuit with erasure errors exhibit a phase transition in information encoding, and if so, what universality class does it belong to?
• RQ2 How does the addition of quantum gates—specifically Hadamard operations—modify the critical behavior of the classical encoding transition?
• RQ3 Can quantum entanglement be understood as a relevant perturbation that destroys the absorbing state at the directed percolation critical point?
• RQ4 What is the nature of the information decay dynamics when quantum effects are introduced, and does it lead to a sharp transition or a crossover?
• RQ5 Is the critical behavior of the system with quantum gates equivalent to that of a classical noise process, suggesting a universal scaling behavior?

Key findings

• The classical circuit with erasure errors undergoes a phase transition at $ p_c = 0.081 $, belonging to the directed percolation universality class with critical exponents $ z=1.58 $, $ \gamma=0.75 $, $ \eta=2.34 $.
• Adding an arbitrarily small density of Hadamard gates destroys the absorbing state, eliminating the sharp phase transition and replacing it with a universal crossover.
• The crossover is governed by the same critical exponents as directed percolation, confirming that quantum effects act as a relevant additive noise perturbation at the critical point.
• System entropy decays to zero only in the classical limit ($ q=0 $); for $ q>0 $, entropy reaches a finite equilibrium value, indicating loss of the absorbing state.
• A purely classical noise process (bit indeterminacy) produces identical critical scaling, confirming that the key effect is the introduction of internal entropy generation.
• Despite the loss of the absorbing state in entropy, the initial-to-final mutual information still decays to zero for any $ p>0 $, but the timescale of decay is governed by the critical exponents, indicating a crossover in information loss dynamics.