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Proceedings/Proceedings of the ... International Conference on Software Engineering and Knowledge Engineering

Quantum Software Models: Software Density Matrix is a Perfect Direct Sum of Module Matrices

Iaakov Exman, Alexey V. Nechaev

Published Jul 1, 2022• 3 citations
Publisher

Abstract

Quantum Software Models is a theoretical framework to systematically design and analyze any software system -be it quantum, classical or hybrid -representing it by a design Density Matrix.Recently, we have demonstrated a top-down approach, to decompose a whole software system Density Matrix into modules, using basis vector projectors of the Matrix.However, it would be even more natural to have a systematic bottom-up procedure, to compose a whole software system Density Matrix, given a set of well-designed software module matrices, taken as sub-systems.This is exactly the paper's purpose.The result obtained: the whole software system Density Matrix is a perfect Direct Sum of module density matrices.This result yields clear software design benefits: it is bidirectional, one can traverse the software system hierarchy top-down or bottom-up, in particular, gradually building up the whole system from verified correct modules, assured by spectral decoupling techniques.The claim is formally validated and is illustrated by software system studies.

DOI
10.18293/seke2022-158

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Quantum Software Models is a theoretical framework to systematically design and analyze any software system -be it quantum, classical or hybrid -representing it by a design Density Matrix.Recently, we have demonstrated a top-down approach, to decompose a whole software system Density Matrix into modules, using basis vector projectors of the Matrix.However, it would be even more natural to have a systematic bottom-up procedure, to compose a whole software system Density Matrix, given a set of well-designed software module matrices, taken as sub-systems.This is exactly the paper's purpose.The result obtained: the whole software system Density Matrix is a perfect Direct Sum of module density matrices.This result yields clear software design benefits: it is bidirectional, one can traverse the software system hierarchy top-down or bottom-up, in particular, gradually building up the whole system from verified correct modules, assured by spectral decoupling techniques.The claim is formally validated and is illustrated by software system studies.
It is widely understood that quantum computing — quantum gates upon qubits — is the general case, encompassing computing by classical means, viz. Boolean logic upon classical bits. It also seems reasonable that Quantum Software should encompass Classical Software. However, to accept such a statement regarding software, the feeling that it seems reasonable is not enough. One needs clear-cut definitions and formal conclusions. This is exactly the purpose of this paper. Previously, we have represented Classical Software by the Laplacian Matrix. More recently, we have shown that Quantum Software is faithfully represented by Density Matrices. It turns out that a Laplacian Matrix normalized by the Laplacian Trace easily obtains a Density Matrix. This opens the horizons for Quantum Software operations — such as unitary and reversible evolution — not naturally available with the classical Laplacian. This paper provides the necessary definitions and conclusions, illustrating the more general Quantum operations with a relevant case study, playing the double role of both classical and quantum software.
Quantum Tomography partially measures and then recovers the remaining density matrix quantum state, in order to verify that a certain device -processor or detector -indeed outputs the intended quantum state.However, single matrix element measurements rapidly increase in number with the density matrix dimension and are error-prone.This work proposes a novel quantum software viewpoint on quantum tomography.Instead of individual matrix elements, measurements and density matrix recovery are performed on higher-abstraction modules, i.e. semantically meaningful groups of matrix elements.Quantum Modules Tomography potentially reduce the necessary number of explicit measurements, while still allowing recovery of the whole density matrix.A Recovery Theorem is formulated and proved: density matrix diagonal measurements suffice to recover the whole system density matrix, in terms of modules.The recovery procedure is illustrated by a few case studies.
Abstract Richard P. Feynman’s 1982 paper “Simulating Physics with Computers” is often recognized as a pioneer of quantum computing. However, careful reading between the lines finds further meaningful content. This work reinterprets the pioneering paper, as a precursor of Quantum Software. Feynman’s proposal to represent the simulated system by a Density Matrix opens the way toward a mathematical Quantum Software systems theory. Density Matrix modularization leads to software modules as high-level abstractions unifying conceptual software units and matrix basis kets, stimulating new software-related questions and novel quantum solutions. Software modules are building blocks for any imaginable Quantum Software computations in practice, such as software system evolution, measurement, compositionality, and future potential applications.
The rapid advancements in quantum computing necessitate a scientific and rigorous approach to the construction of a corresponding software ecosystem, a topic underexplored and primed for systematic investigation. This chapter takes an important step in this direction: It presents scientific considerations essential for building a quantum software ecosystem that makes quantum computing available for scientific and industrial problem solving. Central to this discourse is the concept of hardware-software co-design, which fosters a bidirectional feedback loop from the application layer at the top of the software stack down to the hardware. This approach begins with compilers and low-level software that are specifically designed to align with the unique specifications and constraints of the quantum processor, proceeds with algorithms developed with a clear understanding of underlying hardware and computational model features, and extends to applications that effectively leverage the capabilities to achieve a quantum advantage. We analyze the ecosystem from two critical perspectives: the conceptual view, focusing on theoretical foundations, and the technical infrastructure, addressing practical implementations around real quantum devices necessary for a functional ecosystem. This approach ensures that the focus is towards promising applications with optimized algorithm-circuit synergy, while ensuring a user-friendly design, an effective data management and an overall orchestration. Our chapter thus offers a guide to the essential concepts and practical strategies necessary for developing a scientifically grounded quantum software ecosystem.
Abstract The rapid advancements in quantum computing necessitate a scientific and rigorous approach to the construction of a corresponding software ecosystem, a topic underexplored and primed for systematic investigation. This chapter takes an important step in this direction. It presents scientific considerations essential for building a quantum software ecosystem that makes quantum computing available for scientific and industrial problem-solving. Central to this discourse is the concept of hardware–software co-design, which fosters a bidirectional feedback loop from the application layer at the top of the software stack down to the hardware. This approach begins with compilers and low-level software that are specifically designed to align with the unique specifications and constraints of the quantum processor, proceeds with algorithms developed with a clear understanding of underlying hardware and computational model features, and extends to applications that effectively leverage the capabilities to achieve a quantum advantage. We analyze the ecosystem from two critical perspectives: the conceptual view, focusing on theoretical foundations, and the technical infrastructure, addressing practical implementations around real quantum devices necessary for a functional ecosystem. This approach ensures that the focus is toward promising applications with optimized algorithm–circuit synergy, while ensuring a user-friendly design, an effective data management, and an overall orchestration. This chapter thus offers a guide to the essential concepts and practical strategies necessary for developing a scientifically grounded quantum software ecosystem.
Introduction: This article is the result of doctoral research conducted in 2025 on quantum software architecture, addressing the transformative potential of quantum computing and the challenges associated with integrating quantum capabilities into software systems. Problem: The lack of standardized architectural models for hybrid classical–quantum systems limits interoperability, reuse, and systematic software design. Objective: This study proposes a reference architecture to support the design and evolution of hybrid classical–quantum software systems. Methodology: The architecture was derived through a harmonization process. Results: The proposed framework defines ten architectural elements organized across three dimensions: Conceptual, Organizational, and Quality. Conclusion: The architecture provides a structured foundation for designing hybrid classical–quantum solutions. Originality: The work consolidates dispersed knowledge into a unified framework for hybrid classical–quantum systems. Limitations: The model is based on the currently limited body of literature and industrial experience, requiring future validation.