Rewriting Reality: A New Path to Quantum Logic

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Author: Denis Avetisyan

Researchers are demonstrating that the rules governing quantum operations can emerge from surprisingly simple, non-quantum systems of symbolic rewriting.

A multiway system, structured in three layers with words ordered according to the computational basis, offers a framework for representing diverse quantum circuits where the progression of time is visualized from left to right.
A multiway system, structured in three layers with words ordered according to the computational basis, offers a framework for representing diverse quantum circuits where the progression of time is visualized from left to right.

This work establishes a framework for deriving finite-dimensional quantum operators from Leibnizian strings and multiway systems based on principles of statistical mechanics and leveraging connections to ZX-calculus.

Despite the established success of quantum mechanics, its foundational relationship to computation and information remains an open question. This paper, ‘Quantum Gates from Wolfram Model Multiway Rewriting Systems’, explores a novel approach by demonstrating how finite-dimensional quantum operators-and thus quantum gates-can emerge from the dynamics of abstract, non-quantum rewriting systems based on Leibnizian strings and principles of statistical mechanics. Specifically, we show that these systems encode causal relations amenable to defining an $S$-matrix, ultimately yielding explicit representations of quantum circuits for qubits and qudits. Could this framework offer new insights into the fundamental connection between computation, information, and the laws governing the quantum realm?

Rewriting the Quantum Landscape: Beyond Circuit Constraints

Conventional quantum computation, while powerful, often represents quantum state transformations through the rigid structure of circuits. This approach, analogous to designing a specific series of electrical connections, can become cumbersome when dealing with complex or dynamically changing quantum processes. A fixed circuit struggles to naturally express certain state manipulations, requiring increasingly elaborate designs even for relatively simple operations. This limitation stems from the circuit model’s inherent focus on sequential, gate-based operations, which doesn’t always align with the fluid and interconnected nature of quantum mechanics. The difficulty in representing complex transformations efficiently within the circuit model motivates the search for alternative computational frameworks capable of expressing quantum dynamics with greater flexibility and conciseness, ultimately paving the way for more intuitive and scalable quantum algorithms.

A departure from traditional quantum computation’s reliance on circuits is proposed through a novel framework centered on nondeterministic rewriting systems. This approach allows for a significantly more flexible and expressive manipulation of quantum states, moving beyond the constraints of pre-defined pathways. Instead of rigidly structured circuits, quantum calculations are viewed as transformations governed by a set of rules applied to the system’s current state. This dynamic methodology promises greater adaptability in representing complex quantum algorithms and exploring state spaces, potentially unlocking computational possibilities beyond the reach of conventional circuit-based models. The system’s ability to evolve states based on probabilistic rules offers a powerful mechanism for encoding and processing quantum information, mirroring the inherent flexibility observed in natural quantum systems.

Quantum information, in this novel framework, is not represented by the conventional qubits or vectors, but by LeibnizianStrings – cyclic strings that embody a unique algebraic structure. These strings, derived from the work of Gottfried Wilhelm Leibniz, allow for the encoding of quantum states in a manner that naturally supports complex transformations. Calculations are then performed not through fixed quantum circuits, but by applying rewriting rules to these strings, effectively evolving the quantum state based on a set of defined operations. This approach moves beyond the limitations of circuit-based computation, offering a more flexible and expressive means of manipulating quantum information and potentially unlocking new computational possibilities through the dynamic interplay of string rewriting and quantum state evolution. The system’s foundation in cyclic strings allows for a fundamentally different paradigm where the state itself is the primary focus of computation, rather than a series of gates acting upon it.

Conventional quantum computation often relies on the rigid structure of circuits, limiting the natural expression of complex quantum processes. A new paradigm shifts this focus, proposing a dynamic, state-based computational framework that more closely reflects the inherent flexibility of quantum systems. Instead of pre-defined pathways, this approach centers on the manipulation of quantum states themselves, allowing calculations to evolve organically based on the current state rather than a fixed sequence of operations. This mirrors how quantum phenomena naturally unfold – not as a progression through a pre-determined circuit, but as a continuous transformation of possibilities. By embracing this dynamic nature, the framework aims to unlock computational approaches that are better suited to tackling problems where the inherent complexity and interconnectedness of quantum states are paramount, potentially offering a more intuitive and powerful means of harnessing quantum mechanics for computation.

The multiway system graph for the string “BBBAAACC” illustrates how applying the rules BA→AB and CB→BC generates a network of possibilities, with the physically realizable Leibnizian strings highlighted in red.
The multiway system graph for the string “BBBAAACC” illustrates how applying the rules BA→AB and CB→BC generates a network of possibilities, with the physically realizable Leibnizian strings highlighted in red.

The SS-Matrix: Encoding Quantum Operators Through Rewriting

The SS-Matrix functions as the primary data structure within the framework, explicitly representing finite-dimensional quantum operators that are generated through the process of rewriting. These operators, which act on quantum states, are not stored in a conventional matrix format but are instead encoded within the structure of the SS-Matrix itself. The matrix’s elements are determined by the specific rewriting rules applied and the resulting transformations of the quantum system. Consequently, the SS-Matrix provides a complete and computationally accessible representation of the quantum operator, enabling the simulation and analysis of quantum processes derived from the rewriting procedure. The structure allows for efficient computation of operator properties and the evolution of quantum states under these operators.

The SS-Matrix is generated through a summation process, termed PathSums, which considers all valid rewriting paths applicable to a given quantum operation. Each path represents a specific sequence of rewriting steps, and the PathSums effectively aggregate the contributions from each of these pathways. This summation isn’t simply numerical; it’s a constructive process where each path yields a corresponding operator, and the matrix elements are determined by combining these operators according to the specific path. Consequently, the SS-Matrix directly embodies the quantum state’s evolution by encoding the cumulative effect of all possible rewriting sequences, thus providing a complete representation of the operator’s transformation.

The SS-Matrix construction intrinsically defines a causal structure within the quantum rewriting process by representing it as a partial order, or Poset. This Poset formalizes the dependencies between rewriting steps, where each element in the set corresponds to a specific rewriting operation and the order relation indicates which operations must precede others to maintain consistency. Specifically, the summation over all rewriting paths – PathSums – within the SS-Matrix inherently respects this ordering, preventing contributions from paths that violate the defined causal structure. This ensures that the resulting quantum evolution, represented by the matrix, adheres to the principles of consistent quantum mechanics by encoding the allowed sequence of operations.

This framework utilizes a symbolic representation to define and manipulate finite-dimensional quantum operators. Specifically, quantum operators acting on a $n$-dimensional Hilbert space are represented as symbolic expressions within the system. This allows for the application of symbolic computation techniques – such as rewriting rules and algebraic manipulation – to directly model quantum mechanical processes. The correspondence established enables the translation of quantum operations into equivalent symbolic transformations, and vice-versa, facilitating analysis and computation without explicit reliance on numerical methods for these finite-dimensional cases. This symbolic approach offers a pathway to explore quantum mechanics through the lens of computer science and formal systems.

Applying the rewriting rule BA→AB to the initial state AABAABBABAB generates a multiway system with a Leibnizian path (red) alongside a maximal variety path (green), illustrating diverging computational trajectories.
Applying the rewriting rule BA→AB to the initial state AABAABBABAB generates a multiway system with a Leibnizian path (red) alongside a maximal variety path (green), illustrating diverging computational trajectories.

Statistical Mechanics of Computation: An Ensemble Approach to Quantum Processes

The calculation of the Scattering-matrix, or $SS$-Matrix, utilizes methods derived from Statistical Mechanics by framing possible rewriting pathways as a statistical ensemble. Each pathway, representing a sequence of rewriting steps, is assigned a weight based on its probability of occurrence. This allows computation of ensemble averages, analogous to thermodynamic quantities, over these pathways. The $SS$-Matrix elements are then determined by summing over all possible rewriting paths, effectively treating the computational process as a statistical system in equilibrium. This approach enables analysis of complex computations through the lens of ensemble behavior, rather than tracking individual trajectories.

Applying principles from thermodynamics and information theory to the analysis of computational properties involves quantifying system behavior through established metrics. Specifically, concepts like entropy, free energy, and mutual information are used to characterize the complexity and efficiency of rewriting paths. The use of partition functions, borrowed from statistical mechanics, allows for the calculation of probabilities associated with different computational states and transitions. Information-theoretic measures, such as the Kullback-Leibler divergence, can assess the divergence between different rewriting sequences, providing insights into the system’s sensitivity to noise and the quality of approximations. These tools enable the determination of computational capacity and the identification of bottlenecks within the rewriting process, ultimately facilitating optimization and resource allocation.

PathSums, central to the SS-Matrix calculation, are weighted according to the probability of a specific rewriting sequence occurring during quantum computation. This weighting isn’t arbitrary; it directly correlates to the frequency with which a given path is traversed within the computational process. By assigning higher values to more probable sequences and lower values to less probable ones, the system effectively averages over a large ensemble of possible computational paths. This approach introduces robustness against noise and errors, as the contribution of any single, potentially flawed, path is diminished by the overall statistical distribution. Consequently, the resulting representation of quantum computation is less susceptible to minor perturbations and provides a more stable and reliable outcome, even in the presence of imperfections.

Maintenance of unitarity within the SS-Matrix, crucial for representing valid quantum state transitions, is directly dependent on the precise selection of parameters governing the rewriting process. Specifically, these parameters, which define the probabilities associated with each rewriting step, are constrained to satisfy the requirements of quantum mechanics – namely, conservation of probability. This is achieved by ensuring that the trace of the $SS^{\dagger}SS$ matrix equals one, effectively guaranteeing that the transformation preserves the norm of the quantum state vector. Deviations from these parameter constraints would introduce non-unitary transformations, leading to physically invalid results and the potential loss of quantum information.

Minimal multiway system representations effectively model key elements of ZX-calculus, including the Hadamard gate and both ZZ- and XX-spiders.
Minimal multiway system representations effectively model key elements of ZX-calculus, including the Hadamard gate and both ZZ- and XX-spiders.

Implementing Core Quantum Functionality: A Foundation for Quantum Algorithms

The framework’s capacity to represent fundamental quantum gates, such as the HadamardGate and CNOTGate, serves as a critical demonstration of its expressive power. These gates, essential building blocks for any quantum algorithm, are faithfully encoded within the rewriting system, allowing for complex quantum operations to be constructed from simpler components. The HadamardGate, which creates superposition, and the CNOTGate, responsible for entanglement, are both accurately modeled, proving the system isn’t merely theoretical but can handle core quantum mechanical transformations. This successful representation indicates a capacity to move beyond basic operations and potentially implement more elaborate quantum circuits, establishing a foundation for practical quantum algorithm development. The ability to express these gates within the rewriting system confirms its potential as a versatile and powerful tool for exploring and manipulating quantum information.

The system exhibits a noteworthy compatibility with established quantum computational methods through its natural representation of ZXCalculus. This diagrammatic language allows quantum circuits to be visualized and manipulated as equations, offering an alternative to traditional circuit diagrams. By seamlessly integrating with ZXCalculus, the framework enables researchers to leverage existing tools and techniques developed within this established field, fostering interoperability and accelerating the development of new quantum algorithms. This compatibility isn’t merely representational; it facilitates a translation between the system’s rewriting rules and the established principles of ZXCalculus, opening avenues for verification and optimization of quantum processes through well-understood diagrammatic reasoning.

The successful representation of fundamental quantum gates – specifically the Hadamard, π/8, and Controlled-NOT (CNOT) – within this framework demonstrates a crucial ability to capture core quantum operations. The Hadamard gate, essential for creating superposition, and the CNOT gate, vital for entanglement, serve as building blocks for more complex quantum circuits. Achieving a formal representation of these, alongside the phase gate represented by π/8, validates the system’s capacity to not only express quantum mechanics but also to potentially manipulate quantum states. This accomplishment signifies a key step towards constructing a complete system for quantum computation, offering a robust foundation for implementing and exploring a wider range of quantum algorithms and protocols.

The successful representation of fundamental quantum gates – including Hadamard and CNOT – within this framework establishes a compelling pathway toward practical quantum computation. This isn’t merely a theoretical exercise; by accurately modeling these core elements, the system facilitates the construction of more complex quantum algorithms using a novel approach. Unlike traditional circuit-based models, this framework offers a different means of manipulating quantum information, potentially unlocking new optimization strategies and algorithmic designs. The ability to express these foundational operations demonstrates the system’s capacity to move beyond simple gate representation and toward the realization of fully functional quantum programs, offering a unique contribution to the field and paving the way for further exploration of quantum computational possibilities.

ZZ- and XX-spider diagrams, alongside the Hadamard gate, demonstrate distinct entanglement structures for quantum information processing.
ZZ- and XX-spider diagrams, alongside the Hadamard gate, demonstrate distinct entanglement structures for quantum information processing.

The pursuit of representing quantum operators through classical systems, as detailed in this work, echoes a desire for fundamental unity. It seeks to demonstrate that what appears intrinsically quantum may, in fact, emerge from more basic, non-quantum principles. This resonates with the sentiment expressed by Paul Dirac: “I have not the slightest idea of what I am doing.” Though seemingly paradoxical, this quote captures the humility required when probing the deepest levels of reality – acknowledging the limitations of current understanding while striving for elegant, underlying principles. The framework presented, utilizing rewriting systems and statistical mechanics, exemplifies this pursuit of elegance, seeking a consistent and empathetic structure for understanding quantum phenomena-a structure where the ‘invisible architecture’ of quantum mechanics is laid bare.

The Road Ahead

The correspondence established between rewriting systems and quantum operators, while elegant, does not erase the lingering questions at the heart of quantum foundations. It merely shifts the burden of explanation. One now asks not simply ‘what is quantum mechanics?’ but ‘why should these particular rewriting rules govern reality?’ The framework’s current reliance on statistical mechanics as a bridge, while effective, feels provisional – a pragmatic step rather than a final articulation. A truly satisfying account will likely reveal an intrinsic connection, a deep reason why these systems must evolve in this manner, not just that they can.

Furthermore, the practical implications of this approach remain largely unexplored. While the derivation of quantum gates from abstract rules is conceptually powerful, scaling this to represent complex quantum algorithms presents a significant challenge. The computational cost of simulating these multiway systems, and the potential for discovering genuinely novel quantum circuits through this method, warrants careful investigation. Beauty in code emerges through simplicity and clarity; a truly useful theory will be both computationally tractable and conceptually illuminating.

The connection to ZX-calculus is a promising avenue, hinting at a visual and compositional language for rewriting rules that mirrors the structure of quantum circuits. Perhaps, through a deeper understanding of this interplay, every interface element is part of a symphony, and the abstract elegance of rewriting systems will not only illuminate the foundations of quantum theory but also guide the design of future quantum technologies.

Original article: https://arxiv.org/pdf/2512.20587.pdf

Contact the author: https://www.linkedin.com/in/avetisyan/

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2025-12-25 03:24