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Double categories for adaptive quantum computation

Cihan Okay, Walker Stern, Redi Haderi, Selman Ipek

TL;DR

A unified categorical framework that encompasses prominent adaptive quantum models and their interrelations using the language of double categories is developed, leading to a categorical formulation of the result that non-contextual resources can compute only affine Boolean functions.

Abstract

Quantum computation can be formulated through various models, each highlighting distinct structural and resource-theoretic aspects of quantum computational power. This paper develops a unified categorical framework that encompasses these models and their interrelations using the language of double categories. We introduce double port graphs, a bidirectional generalization of port graphs, to represent the quantum (horizontal) and classical (vertical) flows of information within computational architectures. Quantum operations are described as adaptive instruments, organized into a one-object double category whose horizontal and vertical directions correspond to quantum channels and stochastic maps, respectively. Within this setting, we capture prominent adaptive quantum computation models, including measurement-based and magic-state models. To analyze computational power, we extend the theory of contextuality to an adaptive setting through the notion of simplicial instruments, which generalize simplicial distributions to double categorical form. This construction yields a quantitative characterization of computational power in terms of contextual fraction, leading to a categorical formulation of the result that non-contextual resources can compute only affine Boolean functions. The framework thus offers a new perspective on the interplay between adaptivity, contextuality, and computational power in quantum computational models.

Double categories for adaptive quantum computation

TL;DR

A unified categorical framework that encompasses prominent adaptive quantum models and their interrelations using the language of double categories is developed, leading to a categorical formulation of the result that non-contextual resources can compute only affine Boolean functions.

Abstract

Quantum computation can be formulated through various models, each highlighting distinct structural and resource-theoretic aspects of quantum computational power. This paper develops a unified categorical framework that encompasses these models and their interrelations using the language of double categories. We introduce double port graphs, a bidirectional generalization of port graphs, to represent the quantum (horizontal) and classical (vertical) flows of information within computational architectures. Quantum operations are described as adaptive instruments, organized into a one-object double category whose horizontal and vertical directions correspond to quantum channels and stochastic maps, respectively. Within this setting, we capture prominent adaptive quantum computation models, including measurement-based and magic-state models. To analyze computational power, we extend the theory of contextuality to an adaptive setting through the notion of simplicial instruments, which generalize simplicial distributions to double categorical form. This construction yields a quantitative characterization of computational power in terms of contextual fraction, leading to a categorical formulation of the result that non-contextual resources can compute only affine Boolean functions. The framework thus offers a new perspective on the interplay between adaptivity, contextuality, and computational power in quantum computational models.

Paper Structure

This paper contains 28 sections, 22 theorems, 74 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Acknowledgments.
  3. Double port graphs
  4. Port graphs
  5. Double port graphs
  6. Port labels
  7. Pasting double port graphs
  8. Functoriality in port labels
  9. Adaptive instruments
  10. Instruments
  11. The double category
  12. The horizontal and vertical categories
  13. Quantum computation with qubits
  14. Qubit instruments
  15. Boolean circuits
  16. ...and 13 more sections

Key Result

Theorem 1

Let $f : \mathbb{Z}_2^m \to \mathbb{Z}_2^\ell$ be a Boolean function, and let $(\rho, \Phi, h)$ be a triple consisting of a quantum state $\rho$, an $m$-Bell instrument $\Phi$, and an affine Boolean function $h : \mathbb{Z}_2^m \to \mathbb{Z}_2^\ell$ that together compute $f$. If the simplicial dist

Figures (1)

  • Figure 1: (a) A double port graph consisting of a single node. (b) An adaptive instrument $\Phi$ with input ${X}$ output ${Y}$ operating on the input Hilbert space $V$ with output Hilbert space $W$. The horizontal direction indicated the quantum computation, whereas the vertical direction is the classical computation.

Theorems & Definitions (95)

  • Theorem
  • Definition 1
  • Definition 2
  • Definition 3
  • Example 4
  • Example 5
  • Proposition 6
  • Definition 7
  • Definition 8
  • Definition 9
  • ...and 85 more