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Premonoidal Semantics and Scalable Diagrammatics of Fermionic Quantum Computing

Thomas Perez, Titouan Carette

arXiv:2610.02287v1quant-phcs.LO

TL;DR

Fermionic state spaces can be identified with qubit state spaces, yet fermionic parallel composition can fail the ordinary interchange law. The paper develops CAR-based categorical semantics and pronap diagrammatics, establishing normal forms and completeness for four fermionic ZW fragments, including a new FW matchgate normal form.

  • Problem

    Although Jordan–Wigner identifies LFM state spaces with qubit state spaces, fermionic parallel composition can fail the ordinary interchange law.

    Source: §1 · p. 1, Consequently there is a unique group… · p. 6

  • Method

    The paper derives LFM circuit semantics from CAR gate application and develops scalable diagrammatic rewriting in pronaps, which permit non-natural permutations.

    Source: §1 · p. 1, Conclusion · p. 20

  • Results

    The CAR semantics form a symmetric premonoidal category whose centre is the even subcategory, and the FW fragment has a new matchgate normal form distinct from the pW presentation’s form.

    Source: §1 · p. 1, Conclusion · p. 20

  • Takeaways & Limitations

    Matrix arrows and graph triangles provide uniform normal-form constructions for four fragments: EoW, FoW, EW, and FW.

    Source: Conclusion · p. 20

    The symmetric premonoidal CAR category contains only endomorphisms, so it cannot directly express preparations, effects, ancillae, or postselection.

    Source: §3.2 · p. 8

Abstract

from arXiv · show

Local fermionic mode (LFM) based quantum computation has pure state spaces that are isomorphic, via the Jordan-Wigner representation, to qubit state spaces, but its compositional structure is subtly different. Indeed, the algebraic formalism specifying how to embed fermionic systems underlies a notion of parallel composition, for which, in general, the usual interchange law of monoidal categories fails. We give a categorical account of this phenomenon. Starting from the CAR-algebraic semantics of fermionic gate application, we show that LFM processes form a symmetric premonoidal category whose centre is precisely the even, parity-preserving subcategory. The same processes can alternatively be organized with the ordinary tensor product and a fermionic presymmetry that is natural exactly on even maps. To reason diagrammatically in this latter presentation, we introduce pronaps, a relaxation of the notion of prop, in which permutations are not-necessarily-natural, and use them to organize a hierarchy of fragments of the fermionic ZW calculus relevant to the study of fermionic circuits on physically motivated gate-sets. We then extend scalable diagrammatic notation to pronaps. In this setting, syntactic sugars inspired by SZX and GSA are defined, with the specificity that matrix arrows encode minors and determinants, while graph triangles encode pfaffians of principal submatrices. These constructions yield elegant normal forms and completeness proofs for our presentations of the EoW, FoW, EW, and FW fragments. In particular, the FW presentation gives a new normal form for the matchgate fragment, distinct from the rWGS-X normal form of the planar-W (pW) presentation of the same category. The resulting framework connects fermionic circuit semantics, diagrammatic rewriting with scalable notations, and the algebra of determinants and pfaffians.

1 Introduction

Although Jordan–Wigner identifies LFM and qubit state spaces, fermionic exchange makes their compositional structures differ. The paper develops categorical semantics and diagrammatic tools for reasoning about fermionic circuits and their gate-set fragments.

Motivation

  • The LFM state-space identification with qubits does not make their compositional theories identical: parallel application can violate the ordinary interchange law.The failure arises from the algebraic embedding of fermionic systems into larger registers and matches the super-interchange law of monoidal supercategories.Source: §1 · p. 1

Contributions

  • The paper situates its account alongside prior complete FZW and EZW axiomatizations, aiming to connect CAR-induced semantics precisely to the monoidal presentation used for diagrammatic rewriting.The introduction outlines the progression from the LFM model and CAR algebra, through categorical semantics and pronap fragments, to scalable notation and completeness results.Source: §1 · pp. 1–2
  • CAR-algebraic gate application gives LFM circuits a symmetric premonoidal structure whose centre is exactly the even subcategory.The paper also identifies complementary presentations: ordinary tensor product with a fermionic permutation natural only on even maps, or a CAR-induced premonoidal product with natural fermionic permutation.Source: §1 · pp. 1–2
  • Pronaps relax props by allowing non-natural permutations, organizing fermionic ZW fragments for subsets of the Bravyi–Kitaev gate set.EoW and EW are props; adding relevant odd generators gives the pronap fragments FoW and FW, with EW and FoW identified as new fragments.Source: §1 · p. 2
  • Scalable notation extends to pronaps: matrix arrows encode minors, while graph triangles encode principal pfaffians.These constructions support normal forms and completeness proofs for EoW, FoW, EW, and FW; FW gives a matchgate normal form distinct from the pW presentation’s rWGS-X form.Source: §1 · p. 2

2 Local Fermionic Mode based quantum computing

This section introduces local fermionic mode computing through its mode-occupation encoding, CAR-algebraic semantics, Jordan–Wigner representation, and gate-application rules. It explains how effective odd transformations on disjoint modes violate the ordinary interchange law, motivating study of the resulting categorical structure.

2 Local Fermionic Mode based quantum computing

  • Lloyd proposed fermions as carriers of quantum information in 1996, and Bravyi and Kitaev later gave LFM computing a precise gate-based formulation using creation and annihilation operators.The section develops operational intuition, the operator algebra, and then the model’s denotational semantics.Source: §2 · p. 2

2.1 A Fermionic shell game

  • LFM registers compute with individually addressable modes and binary occupation patterns, while indistinguishable fermions produce exchange signs that can become observable through interference.Optical tweezers provide a concrete realization, but typically conserve total atom number, a stronger constraint than the LFM model’s parity superselection rule.Source: §2.1 · pp. 2–3

2.2 Fermionic creation and annihilation operators

  • The CAR algebra models registers through creation and annihilation operators; its parity grading captures the superselection constraint, and Jordan–Wigner identifies its Hilbert-space representation with the state space of qubits.The CAR formalism remains important because its operator semantics exposes compositional subtleties that a Hilbert-space-only description could miss.Source: §2.2 · pp. 3–4

2.3 Bravyi–Kitaev’s computational model

  • Fermionic gates are homogeneous unitaries, and applying one to selected modes relabels its creation and annihilation operators within the register.The model includes the odd mode-flip, but its physical implementation depends on platform conservation laws and may require additional degrees of freedom or resources.Source: §2.3 · pp. 4–5

Consequently there is a unique group homomorphism

  • Fermionic swaps represent mode permutations and induce a symmetric-group action on the CAR algebra; even operators on disjoint modes commute, whereas odd operators need not.The swaps are even Hermitian unitaries, and conjugation by them implements ordinary permutations of mode labels.Source: §2.3 · p. 5, Consequently there is a unique group… · pp. 5–6
  • Including effective odd transformations makes application on disjoint modes violate the ordinary monoidal interchange law.The following section studies the categorical structure induced by this obstruction.Source: Consequently there is a unique group… · p. 6

3 Categorical denotational semantics

The section builds categorical semantics from CAR gate application, then extends them via Jordan–Wigner to processes between registers of different sizes. It identifies how fermionic signs shape parallel composition and the relationship between monoidal and premonoidal presentations.

3.1 The CAR category

  • The CAR category captures square maps on each register but lacks processes between register sizes, including state preparations and effects.Jordan–Wigner extends the semantics to rectangular processes, including ancilla preparation and postselection.Source: §3.1 · p. 6, §3.3 · p. 8

3.2 Premonoidality and Non-natural symmetry

  • CAR gate application yields a strict symmetric premonoidal category, whose centre is exactly the even subcategory.For homogeneous maps, graded interchange contributes the fermionic sign; odd operations on disjoint modes therefore cannot be freely exchanged.Source: §3.2 · pp. 7–8

3.3 Categorical aspects of the Hilbert space semantics

  • Under Jordan–Wigner, homogeneous CAR operators become homogeneous linear maps, and the resulting LFM category includes maps between registers of different sizes.The extensions include homogeneous state vectors and effects.Source: §3.3 · p. 8
  • The same LFM processes admit two presentations: ordinary tensor composition is monoidal with fermionic symmetry natural only on even maps, while the CAR-induced product makes that symmetry natural for homogeneous processes.With the premonoidal product, two odd processes on disjoint systems fail interchange by a sign.Source: §3.3 · pp. 8–9

4 Diagrammatic syntax of LFM-based quantum computing

The section develops diagrammatic theories for fermionic circuits, using props for even gates and pronaps where fermionic presymmetry is not natural, and organizes fragments for different gate sets and circuit capabilities. It establishes completeness for the EoW, FoW, EW, and FW presentations using scalable notations for pronaps.

4.1 Props and their theories

  • Even-gate Bravyi–Kitaev circuits form a prop, so they can be manipulated like ordinary quantum circuits.The prop’s arrows are central and its symmetry is natural; the paper uses the complete EZW theory to derive sound circuit rewrites.Source: §4.1 · pp. 9–10

EoW EW EZW

  • EZW, EW, and EoW organize complete diagrammatic theories into fragments for progressively smaller even-gate sets.EW omits 𝒙, while EoW also removes cups and caps; EoW suffices for circuits using {G1, G2}.Source: §4.1 · pp. 10–11, EoW EW EZW · p. 12

4.2 Pronaps and their theories

  • A pronap relaxes the prop structure to presymmetric monoidal categories, accommodating LFM circuits where presymmetry is not natural.The paper identifies (LFM, ⊗, Π) as a pronap because its presymmetry is only presymmetric, not natural in general.Source: §4.2 · p. 12, EoW EW EZW · p. 12
  • FZW translates circuits with odd F gates, while its FW fragment covers {G1, G2, G3, F} and presents the same category as pW.Unlike pW’s essentially open planar graphs, FW has a paradigmatic presymmetry and only one non-natural generator.Source: §4.2 · pp. 12–13
  • FoW extends EoW with non-natural generators for preparing and postselecting filled fermionic modes in circuits using {G1, G2}.These operations correspond to the states and effects |1⟩ and ⟨1|.Source: §4.2 · p. 13

EoW EW EZW

  • Building on completeness for FZW, EZW, and pW (which presents the same category as FW), Section 5 proves completeness for EoW, FoW, EW, and FW.The FW result is a new completeness proof using scalable notations for pronaps.Source: EoW EW EZW · p. 13

5 Scalable notations and completeness results

This section extends scalable notation to pronaps, using matrix arrows and graph triangles to develop compact normal forms and completeness results for fermionic fragments. It establishes completeness for the specified scaled state diagrams and, via map–state duality, for arbitrary-arity maps and the unscaled EoW, FoW, EW, and FW fragments.

5.1 Scalable notations

  • Scaled pronap notation gives compact syntax for large diagrams, and completeness of a scaled theory is equivalent to completeness of its original pronap theory.Equations are scalable when they involve only natural generators or no monoidal product; width-dependent phases can make an equation non-scalable.Source: §5.1 · pp. 14–15

5.2 Matrix arrows

  • Matrix arrows encode all minors of a complex matrix, making diagrammatic rewrites express determinant identities such as Cauchy–Binet, cofactor, and Sylvester formulas.Fermionic swaps contribute crossing signs, so the resulting weighted matching sum is a determinant rather than a permanent.Source: §5.2 · p. 16

5.3 Graph triangles

  • Graph triangles encode Pfaffians of principal submatrices of antisymmetric matrices and support diagrammatic Pfaffian identities, including inverse and Schur-complement formulas.Their crossings are fermionic swaps, linking the graph-triangle interpretation to Pfaffians rather than determinants.Source: §5.3 · pp. 17–18

5.4 Completeness

  • Completeness follows by rewriting diagrams to normal forms whose interpretations uniquely determine their syntax; in SEoW, each diagram has a unique matrix normal form, and equal semantics coincide with equal normal forms.For SEoW, the normal form is specified by a unique matrix in C^m×n.Source: §5.4 · p. 18
  • SFoW state diagrams have a unique normal form parameterized by a scalar, rank, and full-rank RCEF matrix; matching normal forms are equivalent to matching semantics.The rank ranges from 0 to n, and the matrix has size n×r.Source: §5.4 · p. 19
  • Compact map–state duality extends state normal forms to arbitrary-arity FW and EW morphisms, and scaled completeness yields completeness of the unscaled EoW, FoW, EW, and FW fragments.The semantic interpretation functor is faithful for each of these four unscaled fragments.Source: §5.4 · pp. 19–20
  • SFW state diagrams have a unique normal form adding an antisymmetric matrix whose pivot rows and columns are zero, with normal-form equality equivalent to semantic equality.The parameters are rank, scalar, antisymmetric matrix, and a full-rank RCEF matrix.Source: §5.4 · p. 19
  • SEW state diagrams have the SFW-style unique normal form, but the matrix rank parameter is even; matching normal forms are equivalent to matching semantics.The rank is 2k, with 0 ≤ 2k ≤ n.Source: §5.4 · pp. 19–20

Conclusion

The conclusion presents fermionic circuit semantics as premonoidal in general, with even maps forming the centre, and develops complete diagrammatic theories for four fermionic ZW fragments. Scalable notation yields uniform normal forms, including a new FW presentation of the matchgate fragment.

Conclusion

  • In full generality, LFM circuit composition is premonoidal rather than monoidal, and its centre is exactly the even subcategory.The paper notes that an operational interpretation involving computational effects would be coherent with this failure of monoidality.Source: Conclusion · p. 20
  • EW represents all Gaussian gates in the Bravyi–Kitaev gate set, while FoW is the smallest theory where the paradigmatic swap is non-natural.FoW is therefore formulated as a pronap; EoW plays a structural role in developing the other fragments.Source: Conclusion · p. 20
  • Scalable notation supports uniform normal forms and completeness proofs for EoW, FoW, EW, and FW.The constructions also diagrammatically represent operations governing determinants and Pfaffians and give short proofs of classical identities involving them.Source: Conclusion · p. 20
  • FW gives a new canonical normal form for the matchgate fragment, distinct from the reduced WGS-X normal form for planar-W calculus.The FW presentation captures the previously studied pW calculus in a different presentation formulated with pronaps.Source: Conclusion · p. 20
  • The framework organizes increasingly expressive fermionic circuit fragments as complete graphical languages and connects circuit semantics, scalable diagrammatics, and determinant and Pfaffian algebra.The authors identify compilation, optimisation, and classical simulation as directions for further investigation.Source: Conclusion · p. 20

A Proofs of section 2

The proofs establish that mode embeddings preserve the CAR structure, are injective and parity-preserving, and that fermionic swaps satisfy the relations needed for permutations.

A Proofs of section 2

  • Injective mode embeddings extend to unital *-algebra morphisms and preserve monomials' parity.The normally ordered monomials form a basis, and their images under an injective mode map are linearly independent.Source: A Proofs of section 2 · p. 22
  • A fermionic swap is a Hermitian, even unitary, and its action on creation and annihilation operators follows from the CAR.Its square is the identity; the proof uses its even degree and support on the swapped modes.Source: A Proofs of section 2 · p. 22
  • Adjacent fermionic swaps commute on disjoint supports and satisfy the braid relation, yielding the symmetric-group representation.The braid proof identifies the two induced conjugation actions, then uses the vacuum to fix their scalar factor to one.Source: A Proofs of section 2 · p. 22
  • The induced algebra morphisms agree on generators, so they agree on the full CAR algebra and give the stated result when applied to G.The proof establishes equality of the morphisms by checking their generator actions.Source: A Proofs of section 2 · p. 22

B Proofs of section 3

The proofs establish the functorial mode embeddings and the graded interchange behavior, then characterize centrality and naturality through parity.

B Proofs of section 3

  • Mode embeddings preserve identities and composition, and their composition gives the associativity equations for the induced functors.The unit equations follow from the identity embedding.Source: B Proofs of section 3 · p. 23
  • Exchanging homogeneous operators on disjoint mode blocks contributes a sign determined by their parities.The proof derives the sign by counting CAR anticommutations between generators in the two blocks.Source: B Proofs of section 3 · p. 23
  • Even operators are central, whereas a nonzero odd operator fails to be central when compared with an odd mode-flip.The mode-flip is unitary, and injectivity of the application embedding ensures the relevant interchange composites are nonzero.Source: B Proofs of section 3 · p. 23
  • In the CAR-algebra presentation, the block fermionic permutations are natural with respect to every morphism.Their coherence follows from block permutations, and naturality follows from the equivariant action on the modes.Source: B Proofs of section 3 · p. 23
  • For the fermionic presymmetry on LFM maps, naturality holds exactly for even maps.The converse follows by applying naturality with a one-mode input and comparing computational-basis coefficients.Source: B Proofs of section 3 · p. 24
  • The presymmetry and tensor product yield the strict symmetric premonoidal structure, while the centre of its induced product consists exactly of even maps.The proof shows that maps natural for the tensor-product presymmetry are central, and that nonzero odd maps are not central.Source: B Proofs of section 3 · pp. 24–25

C Proofs of section 5

The section 5 proofs establish scaled-diagram distribution under naturality conditions and transfer equations to scaled diagrams when their generators are natural.

C Proofs of section 5

  • The scaled-diagram induction uses naturality of dividers and gatherers for permutations, while the composition step requires all generators in the diagram to be natural.The other inductive steps remain valid for arbitrary diagrams.Source: C Proofs of section 5 · p. 26
  • The corresponding induction for the second distribution result omits the tensor-product step.This is the stated distinction from the preceding proof.Source: C Proofs of section 5 · p. 26
  • Equations between diagrams containing only natural generators transfer to their scaled diagrams using the two distribution results.The proof applies the respective lemmas to obtain both points of the theorem.Source: C Proofs of section 5 · p. 26

C.1 Proofs of the derived equations

The appendix derives equations for the SEoW, SEW, SFoW, and SFW fragments using induction and auxiliary linear-algebraic decompositions.

C.1.2 Derived equations in SEW

  • The SEW proof of L12 identifies A15 as its base case.The passage states the base case after unfolding the scalar case of the matrix-arrow definition.Source: §C.1.2 · p. 28

C.1.1 Derived equations in SEoW

  • The L17 proof proceeds by induction on n, with base case exactly equation A21; symmetric horizontal rewrites and repeating an operation m times move from case (n + m, 0) to (n, m).For L18 and L24, the cited passage says it is enough to establish the displayed small-matrix cases and check stability under direct sums and products.Source: §C.1.2 · pp. 30–31

Auxiliary linear algebra

  • Elementary matrices generate GL_n(C), allowing invertible-matrix arguments to be reduced to transvections, dilations, and their products.Gaussian elimination supplies the generation argument, and the appendix describes decompositions of elementary matrices using products, direct sums, and permutation conjugations.Source: §C.1.2 · p. 31

Auxiliary decompositions

C.1.3 Derived equations in SFoW

  • SFoW proofs proceed by induction for L25 and L26, and use A29 as the base case for L25.The proof of L24 separately records a base case.Source: §C.1.3 · p. 35

C.1.4 Derived equations in SFW

  • SFW proofs use induction for L2, while L9 uses a rank-r factorization completed to invertible matrices.The factorization writes an m × n rank-r matrix as P′Q′ with both factors of rank r.Source: §C.1.1 · p. 28

D Normal-form rewrites

This section develops normal-form rewrites by reducing diagrams to matrix-arrow forms and using reduced column echelon form (RCEF) to organize normal forms and their uniqueness arguments. For EW, the argument covers parity-dependent zero forms and full-rank forms rewritten to RCEF, comparing RCEF normal forms and relating EW forms to FW forms.

D Normal-form rewrites

  • RCEF places zero columns last and gives each nonzero column a leading 1 whose row contains no other nonzero entries.The pivot rows increase strictly from one nonzero column to the next.Source: D Normal-form rewrites · p. 38

D.1 Completeness of EoW

  • In EoW, rewrites express the generators as matrix arrows, and L6 and L3 then let any diagram be written as a matrix arrow.The uniqueness argument compares normal-form arrows labelled by matrices M and M′.Source: §D.1 · p. 39

D.2 Completeness of FoW

  • FoW state diagrams reduce to a normal form using a full-rank RCEF matrix, with elementary-matrix determinants absorbed into the scalar.When the matrix rank is deficient, the rewrites return the diagram to the zero normal form.Source: §D.2 · p. 40
  • FoW normal forms have zero interpretation exactly when their global scalar vanishes, and two normal forms of the same interpretation reduce to identical forms.The uniqueness proof compares their normal-form data.Source: §D.2 · pp. 41, 43

D.3 Completeness of FW

  • FW state diagrams reduce to normal forms with a full-rank RCEF matrix and an antisymmetric component zero on the matrix’s pivot rows and columns.The uniqueness proof establishes equality of the normal-form data.Source: §D.3 · pp. 44, 48
  • FW normal forms have zero interpretation exactly when their global scalar vanishes, and equal interpretations yield identical normal forms.The concluding equality covers the rank, scalar, antisymmetric matrix, and RCEF matrix data.Source: §D.3 · pp. 45, 48

D.4 Completeness of EW

  • EW state diagrams are rewritten by moving cups to the top-left and caps to the top-right, then expressing the remaining EoW diagram as a matrix arrow.Source: §D.4 · p. 48
  • When rank(M8) ≠ 2r3, rewrites L9, L18, SA14, and SA13 reduce the diagram to a parity-dependent zero form, up to L30.The passage says this is essentially the same zero-form result as for SFW.Source: §D.4 · p. 49
  • When rank(M8) = 2r3 ≤ n, rewrites put M8 in RCEF by right multiplication with elementary matrices, while adjusting x3 by their determinant.This produces full-rank M9 in RCEF and x4 for the resulting diagram.Source: §D.4 · p. 49
  • Because M9 is in RCEF, Lemma D.0.1 supplies a permutation matrix P3.The passage introduces the matrix before stating the ensuing expression.Source: §D.4 · p. 49
  • The uniqueness argument compares two SEW normal forms of type 0 → n whose matrices are in RCEF and have ranks 2r and 2r′.Their respective pivot rows are denoted j1,…,j2r and j′1,…,j′2r′.Source: §D.4 · p. 49
  • Applying L30 recovers an SFW normal form, whose uniqueness is already established.Source: §D.4 · p. 49
  • An EW normal form with parameters r′, x′, Z′, and M′ is converted to an FW normal form with parameters 2r′, (−1)^(r′(r′−1)/2)(x′ + 1), Z′, and M′.The passage then says a further conclusion can be directly deduced but does not include it.Source: §D.4 · p. 49
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