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arXiv:2510.07540 · Research in the Mathematical Sciences

Polyhedral classical simulators for quantum computation

Cihan Okay

Published Jun 1, 2026• 0 citations
arXivHTMLPDF

Abstract

Abstract Identifying the precise boundary between quantum and classical computational power is a central challenge in quantum computing. We introduce polyhedral classical simulators , a framework for classical simulation grounded in polyhedral geometry. This framework encompasses well-known methods such as the Gottesman–Knill algorithm, while extending naturally to more recent models including quantum computation with magic states and measurement-based quantum computation. The framework is compositional: The correctness of a simulation reduces to verifying a preservation property for the basic instruments of the model, from which the full adaptive simulation is assembled. Beyond unifying existing simulation methods, this provides a geometric roadmap for pushing the boundary of efficient classical simulation further.

DOI
10.1007/s40687-026-00626-x

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Abstract Identifying the precise boundary between quantum and classical computational power is a central challenge in quantum computing. We introduce polyhedral classical simulators , a framework for classical simulation grounded in polyhedral geometry. This framework encompasses well-known methods such as the Gottesman–Knill algorithm, while extending naturally to more recent models including quantum computation with magic states and measurement-based quantum computation. The framework is compositional: The correctness of a simulation reduces to verifying a preservation property for the basic instruments of the model, from which the full adaptive simulation is assembled. Beyond unifying existing simulation methods, this provides a geometric roadmap for pushing the boundary of efficient classical simulation further.
Efficiently simulating quantum circuits on classical computers is a fundamental challenge in quantum computing. This paper presents a novel theoretical approach that achieves substantial speedups over existing simulators for a wide class of quantum circuits. The technique leverages advanced group theory and symmetry considerations to map quantum circuits to equivalent forms amenable to efficient classical simulation. Several fundamental theorems are proven that establish the mathematical foundations of this approach, including a generalized Gottesman-Knill theorem. The potential of this method is demonstrated through theoretical analysis and preliminary benchmarks. This work contributes to the understanding of the boundary between classical and quantum computation, provides new tools for quantum circuit analysis and optimization, and opens up avenues for further research at the intersection of group theory and quantum computation. The findings may have implications for quantum algorithm design, error correction, and the development of more efficient quantum simulators.
This paper introduces a novel approach to geometry simulation leveraging the unique properties of quantum computing. Traditional geometric modeling often relies on explicit algorithms, which can be computationally expensive and limited in their ability to accurately represent complex geometric structures. Quantum computing offers the potential for exponentially greater computational power, enabling the design of a quantum-based geometric simulation framework capable of simulating intricate geometric objects with unprecedented precision. This work explores the fundamental principles of this framework, detailing its architecture, key algorithms, and potential applications. We propose a method predicated on quantum entanglement and superposition to enhance the representation and manipulation of geometric states, ultimately facilitating more accurate and efficient simulations.
We study classical simulation of quantum computation, taking the Gottesman-Knill theorem as a starting point. We show how each Clifford circuit can be reduced to an equivalent, manifestly simulatable circuit (normal form). This provides a simple proof of the Gottesman-Knill theorem without resorting to stabilizer techniques. The normal form highlights why Clifford circuits have such limited computational power in spite of their high entangling power. At the same time, the normal form shows how the classical simulation of Clifford circuits fits into the standard way of embedding classical computation into the quantum circuit model. This leads to simple extensions of Clifford circuits which are classically simulatable. These circuits can be efficiently simulated by classical sampling (``weak simulation'') even though the problem of exactly computing the outcomes of measurements for these circuits (``strong simulation'') is proved to be #P\#\mathbf{P}#P-complete---thus showing that there is a separation between weak and strong classical simulation of quantum computation.
We study classical simulation of quantum computation, taking the Gottesman-Knill theorem as a starting point. We show how each Clifford circuit can be reduced to an equivalent, manifestly simulatable circuit (normal form). This provides a simple proof of the Gottesman-Knill theorem without resorting to stabilizer techniques. The normal form highlights why Clifford circuits have such limited computational power in spite of their high entangling power. At the same time, the normal form shows how the classical simulation of Clifford circuits fits into the standard way of embedding classical computation into the quantum circuit model. This leads to simple extensions of Clifford circuits which are classically simulatable. These circuits can be efficiently simulated by classical sampling ('weak simulation') even though the problem of exactly computing the outcomes of measurements for these circuits ('strong simulation') is proved to be #P-complete--thus showing that there is a separation between weak and strong classical simulation of quantum computation.
Quantum entanglement, a cornerstone of quantum mechanics, presents a unique challenge to conventional classical geometry. This paper explores the potential of utilizing entangled states as fundamental building blocks for computation, proposing a novel geometric paradigm based on entangled states. We introduce a non-classical geometric model that leverages the properties of entanglement to simulate complex geometric structures and topological relationships. This model emphasizes the application of geometric transformations driven by entangled states to achieve precise control over these structures. The core mechanism involves the design of an entanglement-based geometric transformation algorithm, offering a fundamentally different approach to computational geometry compared to traditional methods. We demonstrate the feasibility of this approach through illustrative examples and present a conceptual framework for future research. This work aims to push the boundaries of quantum computation by integrating the inherently non-classical nature of entanglement into the core of geometric processing.
Quantum computation provides new avenues for the simulation of complex systems, which are beyond the capability of classical computation. In this paper, we describe an efficient simulator for quantum computing models, particularly designed for simulating complex systems. Through hybrid quantum–classical methods, noise-tolerant algorithms, and adaptive parameter updating, the method addresses several limitations of NISQ-era methods. The framework proves to be more scalable, accurate, and domain-transparent in applications to molecular dynamics, material science, and quantum chemistry. Experimental results verify the benefit compared to the state of the art in terms of both convergence rate and fidelity, demonstrating its applicability for future simulation tasks.