[Paper Review] Entangled quantum cellular automata, physical complexity, and Goldilocks rules

Logan E. Hillberry, Matthew Timothy Jones+7 more|arXiv (Cornell University)|May 4, 2020

Quantum-Dot Cellular AutomataComputer Science135 references30 citations

TL;DR

This paper introduces Goldilocks rules for quantum cellular automata (QCA) that balance activity and stasis to generate physical complexity in simple, unitary, one-dimensional QCA. Using mutual information networks and persistent entropy fluctuations, the authors demonstrate emergent complexity—such as entangled breathers and scale-free network structures—quantifiable via complexity science tools, with implementations feasible on Rydberg arrays, trapped ions, and superconducting qubits.

ABSTRACT

Cellular automata are interacting classical bits that display diverse emergent behaviors, from fractals to random-number generators to Turing-complete computation. We discover that quantum cellular automata (QCA) can exhibit complexity in the sense of the complexity science that describes biology, sociology, and economics. QCA exhibit complexity when evolving under 'Goldilocks rules' that we define by balancing activity and stasis. Our Goldilocks rules generate robust dynamical features (entangled breathers), network structure and dynamics consistent with complexity, and persistent entropy fluctuations. Present-day experimental platforms- Rydberg arrays, trapped ions, and superconducting qubits- can implement our Goldilocks protocols, making testable the link between complexity science and quantum computation exposed by our QCA.

Motivation & Objective

• To establish a bridge between quantum computation and complexity science by demonstrating that simple QCA can exhibit physical complexity.
• To define and implement 'Goldilocks rules'—evolution rules that balance qubit activity and stasis to avoid trivial or chaotic dynamics.
• To quantify complexity in QCA using tools from complexity science, including mutual information networks and persistent entropy fluctuations.
• To demonstrate that present-day quantum platforms can realize these QCA protocols, making the theoretical findings experimentally testable.
• To shift focus from single-particle or classically simulatable QCA to emergent many-body phenomena driven by entanglement and non-trivial dynamics.

Proposed method

• Define Goldilocks rules: a qubit evolves only if exactly half its neighbors are in the |1⟩ state (ˆσz = +1), otherwise it remains static, balancing activity and stasis.
• Implement unitary, one-dimensional, nearest-neighbor QCA circuits using controlled-controlled-Hadamard gates, with alternating layers for even and odd sites.
• Calculate pairwise mutual information Mjk between qubits to construct weighted complex networks, treating qubits as nodes and Mjk as edge weights.
• Apply complex network measures (e.g., short path length, high clustering) to detect structural complexity in the evolved QCA states.
• Analyze persistent entropy fluctuations in the von Neumann entropy of individual sites to detect dynamical complexity.
• Use numerical simulations and analytical tools (e.g., matrix product states) to study entanglement and dynamics in finite-size chains (L=19).

Experimental results

Research questions

• RQ1 Can simple, unitary, one-dimensional QCA exhibit physical complexity akin to that seen in biological, social, and economic systems?
• RQ2 What dynamical rules—specifically, what balance between activity and stasis—generate robust, complex behavior in QCA?
• RQ3 Can complexity metrics from neuroscience and statistical physics (e.g., mutual information networks, persistent entropy) be applied to quantify complexity in quantum many-body systems?
• RQ4 Are the emergent complex behaviors in QCA—such as entangled breathers and scale-free network structures—experimentally realizable with current quantum hardware?
• RQ5 How does the interplay between entanglement, non-equilibrium dynamics, and network topology give rise to complexity in quantum systems?

Key findings

• Goldilocks rules—where a qubit evolves only if exactly half its neighbors are in the |1⟩ state—produce robust, complex dynamics in 1D QCA, avoiding both complete stasis and uncontrolled chaos.
• The QCA evolution generates entangled breathers: localized, oscillating, many-body entangled states that are the quantum analogs of classical blinkers.
• Mutual information networks formed from QCA dynamics exhibit short path lengths and high clustering, indicating complex network topology consistent with complexity science.
• Persistent fluctuations in site-specific von Neumann entropy are observed, signaling non-equilibrium, complex dynamical behavior across the system.
• The complexity metrics used—mutual information networks and entropy fluctuations—demonstrate that the system exhibits multiple hallmarks of complexity, including robustness-fragility trade-offs and power-law-like statistics.
• The QCA protocols based on Goldilocks rules are realizable on current quantum platforms, including Rydberg atom arrays, trapped ions, and superconducting qubits, enabling experimental validation of quantum complexity.