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Algebraic Structure of Quantum Controlled States and Operators

Edwin Agnew, Lia Yeh, Richie Yeung

Abstract

Quantum control is an important logical primitive of quantum computing programs, and an important concept for equational reasoning in quantum graphical calculi. We show that controlled diagrams in the ZXW-calculus admit rich algebraic structure. The perspective of the higher-order map Ctrl recovers the standard notion of quantum controlled gates, while respecting sequential and parallel composition and multiple-control. In this work, we prove that controlled square matrices form a ring and therefore satisfy powerful rewrite rules. We also show that controlled states form a ring isomorphic to multilinear polynomials. Putting these together, we have completeness for polynomials over same-size square matrices. These properties supply new rewrite rules that make factorisation of arbitrary qubit Hamiltonians achievable inside a single graphical calculus.

Algebraic Structure of Quantum Controlled States and Operators

Abstract

Quantum control is an important logical primitive of quantum computing programs, and an important concept for equational reasoning in quantum graphical calculi. We show that controlled diagrams in the ZXW-calculus admit rich algebraic structure. The perspective of the higher-order map Ctrl recovers the standard notion of quantum controlled gates, while respecting sequential and parallel composition and multiple-control. In this work, we prove that controlled square matrices form a ring and therefore satisfy powerful rewrite rules. We also show that controlled states form a ring isomorphic to multilinear polynomials. Putting these together, we have completeness for polynomials over same-size square matrices. These properties supply new rewrite rules that make factorisation of arbitrary qubit Hamiltonians achievable inside a single graphical calculus.
Paper Structure (16 sections, 33 theorems, 89 equations, 1 figure)

This paper contains 16 sections, 33 theorems, 89 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The ZXW-Calculus
  4. Controlled Diagrams
  5. Quantum Control as a Higher-Order Map
  6. Ring Axioms for Controlled Diagrams
  7. Isomorphism between the Ring of Controlled States and Multilinear Polynomials
  8. Isomorphism
  9. Completeness for Factoring Controlled Operators
  10. Factorising Hamiltonians
  11. Conclusion
  12. Acknowledgements
  13. Proofs for Section \ref{['sec:ctrlmap']}
  14. Basic Lemmas
  15. Proofs for Section \ref{['sec:ring']}
  16. ...and 1 more sections

Key Result

Proposition 1

Given controlled matrices $\tilde{M_1}, ..., \tilde{M_k}$ and $c_1, ..., c_k \in \mathbb{C}$, the controlled square matrices $\widetilde{\Pi_i M_i}$ and $\widetilde{\Sigma_i c_i M_i}$ are respectively given by

Figures (1)

  • Figure 1: These ZX, ZW, and ZXW Rules are altogether complete for qubit linear maps poor2023completeness, where $k \in \{0, 1\}$ and $a \in \mathbb{C}$. The white background ZX, ZW, and ZXW Rules here suffice for completeness of arithmetic diagrams (Definition \ref{['def:arithmetic']}), where \ref{['rule:TA']} was used only to prove Lemma \ref{['lem:kill_quad']} and we did not use the X spider case of \ref{['rule:S2']}. Equivalently, these complete rules for arithmetic diagrams can all be derived from the minimal qubit ZW-calculus of deVisme2025minzw. Adding to these the \ref{['rule:CScpy']}, \ref{['rule:CMcpy']}, and \ref{['rule:CMcom']} rules, we can show that controlled states and controlled operators form rings, culminating in Theorem \ref{['thm:ctrl_pnf']} achieving completeness and minimality for all operations over these rings. We did not use the gray background rules in this work.

Theorems & Definitions (72)

  • Definition 2.1
  • Definition 2.2
  • Proposition 1: Propositions 3.3 and 3.4 of shaikh2022sum
  • Proposition 2
  • Remark 1
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Lemma 4.1
  • Lemma 4.2
  • ...and 62 more