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The rational fragment of the ZX-calculus

Emmanuel Jeandel

arXiv:1810.05377v1quant-phcs.LO

TL;DR

The paper addresses how to axiomatize the rational fragment of the ZX-calculus without the earlier reliance on metarules. It uses generalized cyclotomic supplementarity and characterizes which arbitrary-angle equalities can be proved without rule (A), including the Euler equations in that class.

  • Problem

    The rational fragment had a complete axiomatisation, but it relied on an additional triangle generator or metarules.

    Source: Introduction · p. 2

  • Method

    The paper replaces those approaches with generalized cyclotomic supplementarity and uses results from diophantine geometry to characterize equalities provable without rule (A).

    Source: Introduction · p. 2, §3 · p. 18

  • Results

    Equalities provable without rule (A) are exactly limits of equalities involving rational angles, and the paper characterizes the Euler equations in this class.

    Source: Introduction · p. 2, §1.5 · pp. 8–9

  • Takeaways & Limitations

    The generalized supplementarity rule provides an easier-to-state-and-use alternative for the rational fragment, while the paper identifies the Euler equations provable without rule (A).

    Source: Introduction · p. 2, §1.5 · pp. 8–9

    The paper does not establish whether adding Euler equalities to the ZX rules suffices to prove rule (A) in full generality.

    Source: §5 · p. 28

Abstract

from arXiv · show

We introduce here a new axiomatisation of the rational fragment of the ZX-calculus, a diagrammatic language for quantum mechanics. Compared to the previous axiomatisation introduced in [8], our axiomatisation does not use any metarule , but relies instead on a more natural rule, called the cyclotomic supplementarity rule, that was introduced previously in the literature. Our axiomatisation is only complete for diagrams using rational angles , and is not complete in the general case. Using results on diophantine geometry, we characterize precisely which diagram equality involving arbitrary angles are provable in our framework without any new axioms, and we show that our axiomatisation is continuous, in the sense that a diagram equality involving arbitrary angles is provable iff it is a limit of diagram equalities involving rational angles. We use this result to give a complete characterization of all Euler equations that are provable in this axiomatisation.

Introduction

The introduction focuses on the rational fragment, where a complete axiomatisation already existed but relied on a triangle generator or metarules. The paper replaces those approaches with generalized cyclotomic supplementarity and characterizes equalities provable without rule (A) in the full fragment.

Introduction

  • The initial ZX-calculus rules were incomplete, prompting successive refinements and complete axiomatisations.The ZX-calculus represents quantum evolutions as diagrams and provides rules for transformations that preserve their interpretation.Source: Introduction · p. 1
  • The rational fragment includes angles that allow circuits such as the Quantum Fourier transform to be expressed exactly.The paper contrasts it with the π/4 fragment and the general fragment, where every matrix can be represented.Source: Introduction · pp. 1–2
  • The rational fragment already had a complete axiomatisation, but its rules used either a triangle generator or metarules.The generalized cyclotomic rule, introduced previously in [11], offers an easier-to-state-and-use replacement.Source: Introduction · p. 2
  • Without rule (A), provable equalities involving arbitrary angles are exactly limits of equalities involving rational angles.The paper uses this characterization, together with results on rotations in three-dimensional space, to list the Euler equations provable without rule (A).Source: Introduction · p. 2

2 Cyclotomic Supplementarity as a sufficient rule

The section proves that supplementarity rules suffice to derive the cyclotomic rules, yielding a complete axiomatisation of the rational fragment. Its proof extracts a polynomial coefficient diagrammatically, using a recursively defined diagram family.

2 Cyclotomic Supplementarity as a sufficient rule

  • For each odd prime p, ZX with SUPp proves CY Cp, allowing supplementarity rules to replace the cyclotomic axioms in the rational-fragment completeness result.The section states that ZX with the cyclotomic axioms for prime numbers is complete for the rational fragment.Source: §2 · p. 9
  • The proof rewrites supplementarity as a polynomial equality, then aims to extract a specific coefficient diagrammatically to obtain the cyclotomic rule.The term of degree 1 in Q(e^iα, e^iβ) is identified as −1 − e^iβ − … − e^i(p−1)β.Source: §2 · pp. 10, 14, 15
  • W_n encodes a matrix whose first row has 1 in position 0 and whose second row has 2−n in positions 2i, providing a diagrammatic coefficient-extraction tool.The proof chooses n large enough that 2^n > 2p before applying the construction.Source: §2 · pp. 15–16
  • The construction yields a provable equality in ZX with SUPp, and the theorem completes the derivation of CY Cp for odd primes.The final step uses completeness for the π/4 fragment to establish equality of the remaining terms.Source: §2 · pp. 9, 16, 18

3 Equalities provable without rule (A)

This section characterizes diagram equalities provable without rule (A), showing that provability has structural consequences beyond the particular angles in an equality. The characterization uses algebraic and diophantine results and connects arbitrary-angle equalities to limits of rational-angle equalities.

3 Equalities provable without rule (A)

  • The section characterizes which arbitrary-angle diagram equalities can be proved without rule (A), using a framework of diagrams with angle variables.The diagrams are treated as functions of their angle variables, with rational affine transformations used to express the structural conditions.Source: §3 · pp. 18–19
  • The proof strategy applies rational-fragment completeness and algebraic geometry to the sets of angles where two diagrams agree.The coefficient functions are polynomials in exponentials of the angles; the argument uses results on algebraic groups and diophantine geometry, including Laurent’s result.Source: §3 · pp. 18, 22, 23
  • An equality provable without rule (A) is also provable when all its angles are multiplied by suitable constants, and is an instance of a more general diagram equality.These are two equivalent characterizations of the absence of coincidence at just the particular angles in the equality.Source: §3 · p. 19
  • An equality is provable without rule (A) exactly when it is a limit of provable equalities involving rational angles.The characterization is expressed through sequences of rational-angle instances converging to the angles in question.Source: §3 · p. 22

4 Applications to Euler transforms

This section uses the characterization of equalities provable without rule (A) to determine which Euler equalities are provable in the stated axiom system. It reduces the arbitrary-angle question to finding the Euler equalities satisfied by rational angles.

4 Applications to Euler transforms

  • Theorem 7 gives the Euler equalities provable in ZX ∪ {SUPp, p ≥ 3}, together with their color-swapped variants.The supplied passages do not reproduce the listed equalities themselves.Source: §4 · pp. 23–24
  • The method finds Euler equalities for rational angles and uses the preceding result to extend the characterization to arbitrary angles.The section invokes Theorem 8, a result originally formulated in the context of tilings.Source: §4 · p. 24
  • For the two diagrams of Proposition 6 to represent the same matrix up to a scalar, either α1 = β2 = nπ and β1 = (−1)^nα2, or α2 = β1 = nπ and β2 = (−1)^nα1.The proof handles cases by examining matrix coefficients and quotients, then uses red-green symmetry for corresponding cases.Source: §4 · p. 25
  • The rational-angle proof reduces an identity-up-to-scalar condition to cases with a coefficient in {0, π} or two consecutive coefficients equal to ±π/2.The remaining proof enumerates cases and applies Proposition 6 to recover conditions of the theorem.Source: §4 · pp. 25–27

5 Further work

The section identifies open work on replacing rule (A) in the full fragment. It also notes that the rational-fragment rule set can be made nicer using SUPp.

5 Further work

  • The condition for rule (A) can be rewritten as a sum of six exponential terms, a form studied by Mann and Conway and Jones [5].The most complicated rational-angle equality mentioned uses angles of the form π/3 and π/5.Source: §5 · p. 28
  • Replacing CY Cp or the metarule with SUPp gives a nicer rule set for the rational fragment, but the full fragment still needs a nicer replacement for rule (A).Euler equalities are a candidate, but the authors do not know whether adding them to ZX suffices to prove rule (A) in full generality.Source: §5 · p. 28
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