Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Constructing $\mathrm{NP}^{\mathord{\#}\mathrm P}$-complete problems and ${\mathord{\#}\mathrm P}$-hardness of circuit extraction in phase-free ZH

Piotr Mitosek

TL;DR

The paper investigates the complexity of phase-free ZH diagrams, proving that StateEq and ContainsEntry_k are NP^{#P}-complete and that circuit extraction remains #P-hard for phase-free ZH. It achieves this by establishing NP^{#P} = NP^{C_=P[1]} and constructing an NP^{C_=P[1]}-complete problem SAT&Compare#SAT to drive reductions from counting-SAT to diagram problems via ZH encodings. The results extend known hardness from ZX calculus to phase-free ZH and illuminate the complexity of diagram equivalence, comparison, and extraction in graphical calculi. By connecting diagrammatic reasoning to counting complexity and oracle-based classes, the work provides foundational insights with potential implications for graphical languages, MBQC, and circuit optimization frameworks.

Abstract

The ZH calculus is a graphical language for quantum computation reasoning. The phase-free variant offers a simple set of generators that guarantee universality. ZH calculus is effective in MBQC and analysis of quantum circuits constructed with the universal gate set Toffoli+H. While circuits naturally translate to ZH diagrams, finding an ancilla-free circuit equivalent to a given diagram is hard. Here, we show that circuit extraction for phase-free ZH calculus is ${\mathord{\#}\mathrm P}$-hard, extending the existing result for ZX calculus. Another problem believed to be hard is comparing whether two diagrams represent the same process. We show that two closely related problems are $\mathrm{NP}^{\mathord{\#}\mathrm P}$-complete. The first problem is: given two processes represented as diagrams, determine the existence of a computational basis state on which they equalize. The second problem is checking whether the matrix representation of a given diagram contains an entry equal to a given number. Our proof adapts the proof of Cook-Levin theorem to a reduction from a non-deterministic Turing Machine with access to ${\mathord{\#}\mathrm P}$ oracle.

Constructing $\mathrm{NP}^{\mathord{\#}\mathrm P}$-complete problems and ${\mathord{\#}\mathrm P}$-hardness of circuit extraction in phase-free ZH

TL;DR

The paper investigates the complexity of phase-free ZH diagrams, proving that StateEq and ContainsEntry_k are NP^{#P}-complete and that circuit extraction remains #P-hard for phase-free ZH. It achieves this by establishing NP^{#P} = NP^{C_=P[1]} and constructing an NP^{C_=P[1]}-complete problem SAT&Compare#SAT to drive reductions from counting-SAT to diagram problems via ZH encodings. The results extend known hardness from ZX calculus to phase-free ZH and illuminate the complexity of diagram equivalence, comparison, and extraction in graphical calculi. By connecting diagrammatic reasoning to counting complexity and oracle-based classes, the work provides foundational insights with potential implications for graphical languages, MBQC, and circuit optimization frameworks.

Abstract

The ZH calculus is a graphical language for quantum computation reasoning. The phase-free variant offers a simple set of generators that guarantee universality. ZH calculus is effective in MBQC and analysis of quantum circuits constructed with the universal gate set Toffoli+H. While circuits naturally translate to ZH diagrams, finding an ancilla-free circuit equivalent to a given diagram is hard. Here, we show that circuit extraction for phase-free ZH calculus is -hard, extending the existing result for ZX calculus. Another problem believed to be hard is comparing whether two diagrams represent the same process. We show that two closely related problems are -complete. The first problem is: given two processes represented as diagrams, determine the existence of a computational basis state on which they equalize. The second problem is checking whether the matrix representation of a given diagram contains an entry equal to a given number. Our proof adapts the proof of Cook-Levin theorem to a reduction from a non-deterministic Turing Machine with access to oracle.
Paper Structure (18 sections, 18 theorems, 39 equations, 4 figures)

This paper contains 18 sections, 18 theorems, 39 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Structure
  3. Computational Complexity
  4. Phase-free ZH Calculus
  5. Boolean formulae in graphical calculi
  6. NP#P-completeness
  7. Containment
  8. Hardness -- oracle change
  9. Hardness -- crafting complete problem
  10. Hardness -- ZH encoding
  11. Circuit Extraction
  12. Conclusions and further work
  13. Comparing Diagrams
  14. Tensor networks and complexity classes
  15. Counting complexity
  16. ...and 3 more sections

Key Result

Theorem 1.1

$\mathbf{StateEq}$ and $\mathbf{ContainsEntry_k}$ are both $\mathrm{NP}^{\mathord{\#}\mathrm {P}}$-complete.

Figures (4)

  • Figure 1: Generators of phase-free ZH calculus. We refer to these as white spider, dark spider, white not, dark not, box and star respectively.
  • Figure 2: Basic logic values.
  • Figure 3: Basic logic gates.
  • Figure 4: $(x_1 \wedge x_2) \wedge (x_1 \wedge \neg x_3)$ in phase-free ZH

Theorems & Definitions (43)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 2.1
  • proof
  • Lemma 3.1
  • proof
  • Example 3.2
  • Theorem 4.1
  • proof
  • Theorem 4.2
  • ...and 33 more