The Structure and Interpretation of Quantum Programs I: Foundations

• URL: http://arxiv.org/abs/2509.04527v1
• Date: Wed, 03 Sep 2025 18:00:23 GMT
• Title: The Structure and Interpretation of Quantum Programs I: Foundations
• Authors: David Wakeham,
• Abstract summary: Qubits are a great way to build a quantum computer, but a limited way to program one.<n>We replace the usual "states and gates" formalism with a "props and ops" (propositions and operators) model.<n>We show how measurement modifies state, proving an operator-algebraic version of the Knill-Laflamme conditions.
• Score: 0.0
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: Qubits are a great way to build a quantum computer, but a limited way to program one. We replace the usual "states and gates" formalism with a "props and ops" (propositions and operators) model in which (a) the C*-algebra of observables supplies the syntax; (b) states, viewed as linear functionals, give the semantics; and (c) a novel diagrammatic calculus unifies the two. The first part develops the basic objects of the framework, encoding consistent patterns of operator correlation, recovering Hilbert space via the GNS construction, and re-deriving the Bloch sphere as the set of all consistent correlations of operators in the Pauli algebra. We then turn to intervention, showing how measurement modifies state, proving an operator-algebraic version of the Knill-Laflamme conditions, and expressing stabilizer codes with the same diagrammatic machinery. This provides a concise, representation-agnostic account of quantum error correction. The result is a self-contained foundation in which C*-algebras, and their dual Hilbert spaces, offer a rich and universal substrate for quantum programming; forthcoming papers will build a high-level language and quantum software applications on top of this substrate.

Related papers

• Denotational semantics for stabiliser quantum programs [0.0]
  We develop a sound, universal and complete denotational semantics for stabiliser operations.<n>We demonstrate the power of the resulting semantics by describing a small, proof-of-concept assembly language for stabiliser programs.
  arXiv  Detail & Related papers  (2025-11-27T19:51:43Z)
• Efficient Quantum State Preparation with Bucket Brigade QRAM [47.72095699729477]
  Preparation of data in quantum states is a critical component in the design of quantum algorithms.<n>One of the main approaches to achieve efficient state preparation is through the use of Quantum Random Access Memory (QRAM)<n>We present a framework that integrates the physical model of the Bucket Brigade QRAM (BBQRAM) with the classical data structure of the Segment Tree to achieve efficient state preparation.
  arXiv  Detail & Related papers  (2025-10-17T18:50:07Z)
• Graded Quantum Codes: From Weighted Algebraic Geometry to Homological Chain Complexes [0.0]
  We introduce graded quantum codes, unifying two classes of quantum error-correcting codes.<n> Applications include post-quantum cryptography, fault-tolerant quantum computing, and optimization via graded neural networks.
  arXiv  Detail & Related papers  (2025-08-11T01:44:51Z)
• Homology, Hopf Algebras and Quantum Code Surgery [55.2480439325792]
  We study quantum error-correction codes from an algebraic perspective.<n>We derive new methods of performing fault-tolerant quantum computation.<n>At its core, this thesis asks: what is lattice surgery?
  arXiv  Detail & Related papers  (2025-08-02T21:38:33Z)
• Quantum Homogenization as a Quantum Steady State Protocol on NISQ Hardware [42.52549987351643]
  Quantum homogenization is a reservoir-based quantum state approximation protocol.<n>We extend the standard quantum homogenization protocol to the dynamically-equivalent ($mathttSWAP$)$alpha$ formulation.<n>We show that our proposed protocol yields a completely positive, trace preserving (CPTP) map under which the code subspace is correctable.
  arXiv  Detail & Related papers  (2024-12-19T05:50:54Z)
• Fault-tolerant fermionic quantum computing [39.58317527488534]
  We introduce fermionic fault-tolerant quantum computing, a framework which removes this overhead altogether.<n>We show how our framework can be implemented in neutral atoms, overcoming the apparent inability of neutral atoms to implement non-number-conserving gates.
  arXiv  Detail & Related papers  (2024-11-13T19:00:02Z)
• Geometric structure and transversal logic of quantum Reed-Muller codes [51.11215560140181]
  In this paper, we aim to characterize the gates of quantum Reed-Muller (RM) codes by exploiting the well-studied properties of their classical counterparts. A set of stabilizer generators for a RM code can be described via $X$ and $Z$ operators acting on subcubes of particular dimensions.
  arXiv  Detail & Related papers  (2024-10-10T04:07:24Z)
• Reduction of unitary operators, quantum graphs and quantum channels [0.0]
  Given a unitary operator in a finite dimensional complex Hilbert space, its unitary reduction to a subspace is defined.<n>The application to quantum graphs is discussed.<n>The reduction of quantum channels is also defined.
  arXiv  Detail & Related papers  (2024-07-28T16:51:54Z)
• Quantum information recast via multiresolution in $L_2(0,1]$ [0.0]
  We present a multiresolution approach to the theory of quantum information. It arose from an effort to develop a systematic mathematical approach to the analysis of an infinite array of qubits.
  arXiv  Detail & Related papers  (2024-07-10T20:05:51Z)
• Hybrid Quantum-Classical Machine Learning with String Diagrams [49.1574468325115]
  This paper develops a formal framework for describing hybrid algorithms in terms of string diagrams. A notable feature of our string diagrams is the use of functor boxes, which correspond to a quantum-classical interfaces.
  arXiv  Detail & Related papers  (2024-07-04T06:37:16Z)
• Refinement calculus of quantum programs with projective assertions [5.151896714190243]
  This paper introduces a refinement calculus tailored for quantum programs. We first study the partial correctness of nondeterministic programs within a quantum. We also present their semantics in transforming a postcondition to the weakest liberal postconditions.
  arXiv  Detail & Related papers  (2023-11-23T22:12:57Z)
• The Schmidt rank for the commuting operator framework [58.720142291102135]
  The Schmidt rank is a measure for the entanglement dimension of a pure bipartite state. We generalize the Schmidt rank to the commuting operator framework. We analyze bipartite states and compute the Schmidt rank in several examples.
  arXiv  Detail & Related papers  (2023-07-21T14:37:33Z)
• On states of quantum theory [0.0]
  We study normal states, i.e. states which are represented by density operators, and singular states, i.e. states can not be represented by density operators. It is given an approach to the resolution of bounded linear functionals into quantum states by applying the GNS construction.
  arXiv  Detail & Related papers  (2021-10-02T12:42:01Z)
• PBS-Calculus: A Graphical Language for Coherent Control of Quantum Computations [77.34726150561087]
  We introduce the PBS-calculus to represent and reason on quantum computations involving coherent control of quantum operations. We equip the language with an equational theory, which is proved to be sound and complete. We consider applications like the implementation of controlled permutations and the unrolling of loops.
  arXiv  Detail & Related papers  (2020-02-21T16:15:58Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.