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AEU - International Journal of Electronics and Communications

QOLE: A browser-native quantum circuit simulator based on decision diagrams

Asimakis Kydros, Konstantinos P. Prousalis, Vasilis Skarlatos, Nikos Konofaos

Published Oct 14, 2026• 0 citations
Publisher

Abstract

The classical simulation of quantum circuits is constrained by the exponential growth of the quantum state. Decision-diagram methods, such as the Quantum Multiple-valued Decision Diagram (QMDD), mitigate this by exploiting regularities in the tensor descriptions of quantum operators and states. This paper presents QOLE, a browser-native quantum circuit simulator built on QMDDs. It pairs a simple, high-level programming interface with an internal compiler that maps gates and state vectors into compact graphs for efficient simulation. Extending the conference version of this work, we add an empirical evaluation of the open implementation across four families of circuits (GHZ states, linear cluster states, the Bernstein–Vazirani algorithm and random Clifford + T circuits) of up to 29 qubits. We measure diagram size, the wall-clock cost of construction and sampling, and the memory behaviour of the lazy output interface, contrasting diagram sizes against a dense state-vector baseline. For structured circuits, QOLE represents the full state in a number of nodes that grows linearly with the qubit count, whereas a dense representation grows exponentially, and the lazy interface keeps the working set bounded independently of state size. Diagram-based simulation is thus both accessible and effective within modern web execution environments.

DOI
10.1016/j.aeue.2026.156673

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The classical simulation of quantum circuits is constrained by the exponential growth of the quantum state. Decision-diagram methods, such as the Quantum Multiple-valued Decision Diagram (QMDD), mitigate this by exploiting regularities in the tensor descriptions of quantum operators and states. This paper presents QOLE, a browser-native quantum circuit simulator built on QMDDs. It pairs a simple, high-level programming interface with an internal compiler that maps gates and state vectors into compact graphs for efficient simulation. Extending the conference version of this work, we add an empirical evaluation of the open implementation across four families of circuits (GHZ states, linear cluster states, the Bernstein–Vazirani algorithm and random Clifford + T circuits) of up to 29 qubits. We measure diagram size, the wall-clock cost of construction and sampling, and the memory behaviour of the lazy output interface, contrasting diagram sizes against a dense state-vector baseline. For structured circuits, QOLE represents the full state in a number of nodes that grows linearly with the qubit count, whereas a dense representation grows exponentially, and the lazy interface keeps the working set bounded independently of state size. Diagram-based simulation is thus both accessible and effective within modern web execution environments.
The classical simulation of quantum circuits is inherently constrained by the exponential growth of the quantum state. Strategies based on decision diagrams, such as the Quantum Multiple-valued Decision Diagram structure, exploit regularities in the tensor descriptions of quantum operators and states to mitigate this limitation. This paper presents QOLE, a browser-compatible simulator for quantum circuits, that utilizes these diagrams in its implementation. QOLE offers a simple and intuitive high-level programming interface for building circuits, while its internal diagram-based compiler maps quantum gates and state vectors into compact graphs that enable more efficient simulation. The open implementation of QOLE demonstrates how diagram-based simulation can be made accessible within the constraints of modern web execution environments.
Quantum-mechanical phenomena are playing an increasing role in information processing as transistor sizes approach the nanometer level, while the securest forms of communication rely on quantum data encoding. When they involve a finite number of basis states, these phenomena can be modeled as quantum circuits, the quantum analogue of conventional or classical logic circuits. Simulation of quantum circuits can therefore be used as a tool to evaluate issues in the design of quantum information processors. Unfortunately, simulating such phenomena efficiently is exceedingly difficult. The matrices representing quantum operators (gates) and vectors modeling quantum states grow exponentially with the number of quantum bits. The information represented by quantum states and operators often exhibits structure that can be exploited when simulating certain classes of quantum circuits. We study the development of simulation methods that run on classical computers and take advantage of such repetitions and redundancies. In particular, we define a new data structure for simulating quantum circuits called the quantum information decision diagram (QuIDD). A QuIDD is a compressed graph representation of a vector or matrix and permits computations to be performed directly on the compressed data. We develop a comprehensive set of algorithms for operating on QuIDDs in both the state-vector and density-matrix formats, and evaluate their complexity. These algorithms have been implemented in a general-purpose simulator program for quantum-mechanical applications called QuIDDPro. Through extensive experiments conducted on representative quantum simulation applications, including Grover's search algorithm, error characterization, and reversible circuits, we demonstrate that QuIDDPro is faster than other existing quantum-mechanical simulators such as the National Institute of Standards and Technology's QCSim program, and is far more memory-efficient. Using QuIDDPro, we explore the advantages of quantum computation over classical computation, simulate quantum errors and error correction, and study the impact of numerical precision on the fidelity of simulations. We also develop several novel algorithms for testing quantum circuit equivalence and compare them empirically. The QuIDDPro software is equipped with a user-friendly interface and is distributed with numerous example scripts. It has been used as a laboratory supplement for quantum computing courses at several universities.
The decision diagram (DD) data structure enables fast linear-algebra calculations by bringing vectors into a normal form and subsequently merging equivalent ones, yielding a minimally-sized DD modulo the equivalence relation. A fruitful application area is quantum-circuit simulation, where the vectors represent quantum states. The Local Invertible Map Decision Diagram (LIMDD) type, merges LIM-equivalent (typically Pauli-gate equivalent) vectors, can efficiently simulate Clifford circuits as well as some high-T-count circuits, and has theoretically been proven exponentially faster for simulation than other well-developed data structures, including other common DD variants. However, these exponential advantages have not fully materialized yet in existing implementations, for which the normal-form procedure, which is a highly complex algorithm, is either absent or only partially implemented. We here present a novel normal-form algorithm for Pauli-LIMDDs, achieving a worst-case speedup from O(n3)O(n^3)O(n3) to O(n2)O(n^2)O(n2) for an nnn-qubit DD node with a single child node while keeping the O(n3)O(n^3)O(n3) run time in case of two distinct children nodes. We implement the algorithm as part of QolDDer, our Pauli-LIMDD simulator for quantum circuits, written from scratch in C/C++. The implementation realizes the theoretically-proven advantages of Pauli-LIMDDs on Clifford circuits, is significantly faster than the existing LIMDD simulators on such circuits, and on a public quantum-circuit data set often outperforms them by an order of magnitude. In the future, we envision that our work will enable further application and development of LIMDD variants, not only for quantum design tasks, but also for analysis of linear-algebra-based systems in general.
Quantum computers promise to solve important problems faster than conventional computers ever could. Underneath is a fundamentally different computational primitive that introduces new challenges for the development of software tools that aid designers of corresponding quantum algorithms. The different computational primitives render classical simulation of quantum circuits particularly challenging. While the logic simulation of conventional circuits is comparatively simple with linear complexity with respect to the number of gates, quantum circuit simulation has to deal with the exponential memory requirements to represent quantum states on non-quantum hardware with respect to the number of qubits. Decision Diagrams (DDs) address this challenge through exploitation of redundancies in matrices and vectors to provide significantly more compact representations in many cases. Moreover, the probabilistic nature of quantum computations enables another angle to tackle the complexity: Quantum algorithms are resistant to some degree against small inaccuracies in the quantum state as these only lead to small changes in the outcome probabilities. We propose to exploit this resistance against (small) errors to gain even more compact decision diagrams. In this work, we investigate the potential of approximation in quantum circuit simulation in detail. To this end, we first present four dedicated schemes that exploit the error resistance and efficiently approximate quantum states represented by decision diagrams. Subsequently, we propose two simulation strategies that utilize those approximations schemes in order to improve the efficiency of DD-based quantum circuit simulation, while, at the same time, allowing the user to control the resulting degradation in accuracy. We empirically show that the proposed approximation schemes reduce the size of decision diagrams substantially and also analytically prove the effect of multiple approximations on the attained accuracy. Eventually, this enables speed-ups of the resulting approximate quantum circuit simulation of up to several orders of magnitudes—again, while controlling the fidelity of the result.
Simulating quantum computation on a classical computer is a difficult problem. The matrices representing quantum gates, and vectors modeling qubit states grow exponentially with an increase in the number of qubits. However, by using a new data structure called the Quantum Information Decision Diagram (QuIDD) that exploits the structure of quantum operators, many of these matrices and vectors can be represented in a form that grows polynomially. Using QuIDDs, we implemented a general-purpose quantum computing simulator in C++ called QuIDDPro and tested it on Grover's algorithm. Our QuIDD technique asymptotically outperforms other known simulation techniques.
As quantum circuits grow in complexity, traditional simulation methods face significant challenges, particularly regarding memory usage and computational time. This paper investigates the performance of decision diagram-based quantum circuit simulations when scaled across high-performance computing (HPC) environments. Specifically, we benchmark a multi-core implementation of the Quantum Multi-Valued Decision Diagram (QMDD) circuit simulator using the COARE HPC facility at the Department of Science and Technology-Advanced Science and Technology Institute. Our results demonstrate that leveraging decision diagrams in a multi-core setup enhances simulation scalability and efficiency in several circuits, although not all, compared to single-core implementations and other classical simulation techniques, such as Qiskit Aer and Munich Quantum Toolkit’s DDSim. This work provides valuable insights into the practical scalability of quantum circuit simulators and suggests potential directions for future advancements in quantum computing simulation technologies.