We study the problem of counting the number of homomorphisms from an input graph G to a fixed (quantum) graph ̄{H} in any finite field of prime order ℤ_p. The subproblem with graph H was introduced by Faben and Jerrum [ToC'15] and its complexity is still uncharacterised despite active research, e.g. the very recent work of Focke, Goldberg, Roth, and Zivný [SODA'21]. Our contribution is threefold. First, we introduce the study of quantum graphs to the study of modular counting homomorphisms. We show that the complexity for a quantum graph ̄{H} collapses to the complexity criteria found at dimension 1: graphs. Second, in order to prove cases of intractability we establish a further reduction to the study of bipartite graphs. Lastly, we establish a dichotomy for all bipartite (K_{3,3}$1{e}, {domino})-free graphs by a thorough structural study incorporating both local and global arguments. This result subsumes all results on bipartite graphs known for all prime moduli and extends them significantly. Even for the subproblem with p = 2 this establishes new results.
We study the problem of counting the number of homomorphisms from an input graph G to a fixed (quantum) graph ̄{H} in any finite field of prime order ℤ_p. The subproblem with graph H was introduced by Faben and Jerrum [ToC'15] and its complexity is still uncharacterised despite active research, e.g. the very recent work of Focke, Goldberg, Roth, and Zivný [SODA'21]. Our contribution is threefold. First, we introduce the study of quantum graphs to the study of modular counting homomorphisms. We show that the complexity for a quantum graph ̄{H} collapses to the complexity criteria found at dimension 1: graphs. Second, in order to prove cases of intractability we establish a further reduction to the study of bipartite graphs. Lastly, we establish a dichotomy for all bipartite (K_{3,3}$1{e}, {domino})-free graphs by a thorough structural study incorporating both local and global arguments. This result subsumes all results on bipartite graphs known for all prime moduli and extends them significantly. Even for the subproblem with p = 2 this establishes new results.
We study the complexity of counting (weighted) planar graph homomorphism problem Pl-GH(M) parametrized by an arbitrary symmetric non-negative real valued matrix M. For matrices with pairwise distinct diagonal values, we prove a complete dichotomy theorem: Pl-GH(M) is either polynomial-time tractable, or #P-hard, according to a simple criterion. More generally, we obtain a dichotomy whenever every vertex pair of the graph represented by M can be separated using some planar edge gadget. A key question in proving complexity dichotomies in the planar setting is the expressive power of planar edge gadgets. We build on the framework of Mančinska and Roberson to establish links between \textit{planar} edge gadgets and the theory of the \textit{quantum automorphism group} Qut(M). We show that planar edge gadgets that can separate vertex pairs of M exist precisely when Qut(M) is \emph{trivial}, and prove that the problem of whether Qut(M) is trivial is undecidable. These results delineate the frontier for planar homomorphism counting problems and uncover intrinsic barriers to extending nonplanar reduction techniques to the planar setting.
Quantum graph homomorphisms, introduced by Mančinska and Roberson, form a natural quantum relaxation of classical graph homomorphisms. Since this relaxation may create new comparabilities, it could in principle collapse antichains and other order-theoretic configurations. We prove that this does not happen: the quantum homomorphism quasi-order of finite directed graphs is countably universal, and the quantum homomorphism quasi-order of finite planar graphs of maximum degree at most 7 is also countably universal. Consequently, the same universality holds for finite undirected graphs and for the corresponding quotient partial orders. The result is constructive for finite patterns. Given any finite poset P, we explicitly construct finite planar graphs Gp, p∈P, with Δ(Gp)≤7, such that \[ p\le_P q \quad\Longleftrightarrow\quad G_p\toq G_q. \] For directed graphs, the proof uses disjoint unions of clockwise directed cycles, where quantum and classical homomorphisms coincide. For undirected graphs, the main ingredient is a finite ordered indicator whose terminals are quantum endpoint-forcing. This gives a quantum analogue of the classical ordered-indicator method: the classical endpoint-image condition is replaced by the projection-level vanishing condition Fa,pFb,q=0 for every illegal ordered terminal pair. The fixed indicator encodes directed-cycle constructions inside planar bounded-degree graphs without creating extra quantum homomorphisms.
We introduce a category qGph of quantum graphs, whose definition is motivated entirely from noncommutative geometry. For all quantum graphs G and H in qGph, we then construct a quantum graph [G,H] of homomorphisms from G to H, making qGph a closed symmetric monoidal category. We prove that for all finite graphs G and H, the quantum graph [G,H] is nonempty iff the (G,H)-homomorphism game has a winning quantum strategy, directly generalizing the classical case.
The finite quantum graphs in qGph are tracial, real, and self-adjoint, and the morphisms between them are CP morphisms that are adjoint to a unital ∗-homomorphism. We prove that Weaver's two notions of a CP morphism coincide in this context. We also include a short proof that every finite reflexive quantum graph is the confusability quantum graph of a quantum channel.
We study the approximation of the number of solutions of Laurent polynomials over finite fields. For a Laurent polynomial f(x)=∑j=1sajxuj∈Fq[x1±1,…,xn±1], let U be its augmented support matrix whose columns are (1,uj) with rank ρ and N(f):=#{x∈(Fq×)n∣f(x)=0} be its torus point count. Our first main result is a quantum algorithm that outputs N(f) satisfying ∣N(f)−N(f)∣≤εqn+s/2−ρ with success probability 1−δ. Provided that ρ and ∥U∥∞ are bounded, the algorithm runs in both classical bit and quantum gate complexity poly(n,s,logq,1/ε,log(1/δ)). It provides finer resolution than relative-error approximations in general settings. To the best of our knowledge, in the explicit finite-field input model considered here, no previous algorithm achieves this additive accuracy with running time polynomial in logq. Van Dam (arXiv:quant-ph/0405081) conjectured the existence of such an algorithm under the assumption of an oracle reflecting the algebraic properties of the polynomial. In contrast, by exploiting a point-counting formula derived from character sums over finite fields, we develop an alternative approach that efficiently approximates the number of points without assuming the existence of such an oracle. As a second main result, we prove that the same approximation problem becomes #P-hard under randomized polynomial-time Turing reductions when the support matrix U varies freely as part of the input. Thus, taken together, our results clarify how the effectiveness of the quantum approach depends on the tradeoff between the accuracy scale and the support parameters of the input polynomial.
We explore the frontier beyond the recently discovered barrier represented by the \emph{quantum automorphism group} qut(M) in the classification theory of planar graph homomorphisms PlGH(M). We show that analyzing the spectral relations of M can prove \#P-hardness when traditional vertex separation and domain-reduction methods with planar edge gadgets provably fail due to the $\qut(M)$ barrier. We prove two criteria of \#P-hardness for PlGH(M): a spectral criterion and a determinant criterion. It is known that the core problem for the classification of PlGH(M) for nonnegative matrices M is for positive definite entry-wise positive matrices. We use the spectral criterion to show that PlGH(M) is \#P-hard for all circulant matrices of prime order q≥3, while for q=2 it is precisely the matchgate case and is P-time computable by the FKT algorithm (for planar perfect matching). We also prove a complexity dichotomy for $\PlGH$ problems defined by tensor products of 2 by 2 matrices. This gives a complete complexity classification for this class of matrices, and the FKT algorithm together with a holographic transformation is \emph{universal}---every PlGH(M) is either (1) P-time computable over all graphs, or (2) \#P-hard in general but P-time computable over planar graphs, or (3) \#P-hard over planar graphs; furthermore, PlGH(M) in (2) consists of precisely those computable by FKT with a holographic transformation.
Let H be a (non-empty) graph on n vertices, possibly containing isolated vertices. Let f_H(G) = 1 iff the input graph G on n vertices contains H as a (not necessarily induced) subgraph. Let alpha_H denote the cardinality of a maximum independent set of H. In this paper we show: Q(f_H) = Omega( sqrt{alpha_H * n}), where Q(f_H) denotes the quantum query complexity of f_H. As a consequence we obtain lower bounds for Q(f_H) in terms of several other parameters of H such as the average degree, minimum vertex cover, chromatic number, and the critical probability. We also use the above bound to show that Q(f_H) = Omega(n^{3/4}) for any H, improving on the previously best known bound of Omega(n^{2/3}) [M. Santha/A. Chi-Chih Yao, unpublished manuscript]. Until very recently, it was believed that the quantum query complexity is at least square root of the randomized one. Our Omega(n^{3/4}) bound for Q(f_H) matches the square root of the current best known bound for the randomized query complexity of f_H, which is Omega(n^{3/2}) due to Groger. Interestingly, the randomized bound of Omega(alpha_H * n) for f_H still remains open. We also study the Subgraph Homomorphism Problem, denoted by f_{[H]}, and show that Q(f_{[H]}) = Omega(n). Finally we extend our results to the 3-uniform hypergraphs. In particular, we show an Omega(n^{4/5}) bound for quantum query complexity of the Subgraph Isomorphism, improving on the previously known Omega(n^{3/4}) bound. For the Subgraph Homomorphism, we obtain an Omega(n^{3/2}) bound for the same.