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.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.

quant-pharXiv:2610.02287Oct 2, 20260 citations

Contraction of matchgate tensor networks on non-planar graphs

Sergey B. Bravyi

A tensor network is a product of tensors associated with vertices of some graph $G$ such that every edge of $G$ represents a summation (contraction) over a matching pair of indexes. It was shown recently by Valiant, Cai, and Choudhary that tensor networks can be efficiently contracted on planar graphs if components of every tensor obey a system of quadratic equations known as matchgate identities. Such tensors are referred to as matchgate tensors. The present paper provides an alternative approach to contraction of matchgate tensor networks that easily extends to non-planar graphs. Specifically, it is shown that a matchgate tensor network on a graph $G$ of genus $g$ with $n$ vertices can be contracted in time $T=poly(n) + 2^{2g} O(m^3)$ where $m$ is the minimum number of edges one has to remove from $G$ in order to make it planar. Our approach makes use of anticommuting (Grassmann) variables and Gaussian integrals.A tensor network is a product of tensors associated with vertices of some graph $G$ such that every edge of $G$ represents a summation (contraction) over a matching pair of indexes. It was shown recently by Valiant, Cai, and Choudhary that tensor networks can be efficiently contracted on planar graphs if components of every tensor obey a system of quadratic equations known as matchgate identities. Such tensors are referred to as matchgate tensors. The present paper provides an alternative approach to contraction of matchgate tensor networks that easily extends to non-planar graphs. Specifically, it is shown that a matchgate tensor network on a graph $G$ of genus $g$ with $n$ vertices can be contracted in time $T=poly(n) + 2^{2g} O(m^3)$ where $m$ is the minimum number of edges one has to remove from $G$ in order to make it planar. Our approach makes use of anticommuting (Grassmann) variables and Gaussian integrals.

arXiv:0801.2989Jan 18, 20081 citation

Foundations of tensor network theory

Harvey H. Happ

Journal of the Franklin InstituteDec 1, 19684 citations

Gaussianity and simulability of Cliffords and matchgates

Andrew M. Projansky, Jason Necaise, James Daniel Whitfield

Abstract Though Cliffords and matchgates are both examples of classically simulable circuits, they are considered simulable for different reasons. The celebrated Gottesman–Knill explains the simulability Cliffords, and the efficient simulability of matchgates is understood via Pfaffians of antisymmetric matrices. We take the perspective that by studying Clifford-matchgate hybrid circuits, we expand the set of known simulable circuits and reach a better understanding of what unifies these two circuit families. While the simulability of Clifford conjugated matchgate circuits for single qubit outputs has been briefly considered, the simulability of Clifford and matchgate hybrid circuits has not been generalized up to this point. In this paper we extend that work, studying simulability of marginals as well as Pauli expectation values of Clifford and matchgate hybrid circuits. We describe a hierarchy of Clifford circuits, and find that as we consider more general Cliffords, we lose some amount of simulability of bitstring outputs. We then show that the known simulability of Pauli expectation values of Clifford circuits acting on product states can be generalized to Clifford circuits acting after any matchgate circuit. We conclude with general discussion about the relationship between Cliffords and matchgates, and show that both circuit families can be understood as being Gaussian.Abstract Though Cliffords and matchgates are both examples of classically simulable circuits, they are considered simulable for different reasons. The celebrated Gottesman–Knill explains the simulability Cliffords, and the efficient simulability of matchgates is understood via Pfaffians of antisymmetric matrices. We take the perspective that by studying Clifford-matchgate hybrid circuits, we expand the set of known simulable circuits and reach a better understanding of what unifies these two circuit families. While the simulability of Clifford conjugated matchgate circuits for single qubit outputs has been briefly considered, the simulability of Clifford and matchgate hybrid circuits has not been generalized up to this point. In this paper we extend that work, studying simulability of marginals as well as Pauli expectation values of Clifford and matchgate hybrid circuits. We describe a hierarchy of Clifford circuits, and find that as we consider more general Cliffords, we lose some amount of simulability of bitstring outputs. We then show that the known simulability of Pauli expectation values of Clifford circuits acting on product states can be generalized to Clifford circuits acting after any matchgate circuit. We conclude with general discussion about the relationship between Cliffords and matchgates, and show that both circuit families can be understood as being Gaussian.

arXiv:2410.10068Journal of Physics A Mathematical and TheoreticalApr 15, 20255 citations

A tensor network is a product of tensors associated with vertices of some graph

G

such that every edge of

G

represents a summation (contraction) over a matching pair of indexes. It was shown recently by Valiant, Cai, and Choudhary that tensor networks can be efficiently contracted on planar graphs if components of every tensor obey a system of quadratic equations known as matchgate identities. Such tensors are referred to as matchgate tensors. The present paper provides an alternative approach to contraction of matchgate tensor networks that easily extends to non-planar graphs. Specifically, it is shown that a matchgate tensor network on a graph

G

of genus

g

with

n

vertices can be contracted in time

T=poly(n) + 2^{2g} O(m^3)

where

m

is the minimum number of edges one has to remove from

G

in order to make it planar. Our approach makes use of anticommuting (Grassmann) variables and Gaussian integrals.