A Quantum Framework for K Coloring of Graphs

Lord Sen; Shyamapada Mukherjee

Home/Archive/arXiv

arXiv · 2609.32348

A Quantum Framework for K Coloring of Graphs

Lord Sen·Shyamapada Mukherjee

View original source ↗Read source document ↗Export metadata ↓

Abstract

Graph coloring is a well-known NP-complete problem with applications in scheduling, register allocation, frequency assignment, and network optimization. Quantum computing offers the potential for polynomial or even super-polynomial speedups in certain problem instances, yet practical quantum graph coloring methods must carefully balance qubit resources, circuit depth, and constraint enforcement. We proposed a solver agnostic quantum framework for $K$-coloring. Along with a novel encoding and efficient constraint implementation for its exact coloring exploiting Grover search and almost optimal coloring by the Quantum Approximate Optimization Algorithm \emph{(QAOA)}, and \emph{Quantum Annealing}. Unlike using $O(NK)$ qubits as in \emph{SOTA}, we reduced the qubit requirements to $O(N \log_2 K)$, along with optimized comparator circuits enforcing adjacency constraints. Further symmetry-based graph reduction is incorporated as an optional preprocessing step to further reduce the instance size before quantum execution.

Explore related subjects

quant-ph

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Lord Sen, Shyamapada Mukherjee. 2026-09-26. A Quantum Framework for K Coloring of Graphs. https://arxiv.org/abs/2609.32348

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Self-duality and Jordan structure of quantum theory follow from homogeneity and pure transitivity

Quantum theory satisfies the mathematically powerful property of self-duality, meaning that its state and effect spaces are made isomorphic by an inner product. This is why both quantum states and effects can be represented by positive Hermitian operators. The seminal Koecher-Vinberg theorem shows that self-duality is a rather special property: any system that is self-dual and also homogeneous, meaning that the group of symmetries of its cone of unnormalised states acts transitively on the strictly positive states, must be isomorphic to a Euclidean Jordan algebra. EJAs have been classified and are known to be just a slight generalisation of quantum systems. This is why the Koecher-Vinberg theorem, and hence self-duality and homogeneity, have been used in several reconstructions of quantum theory from first principles. While homogeneity has an operational derivation from the ability to steer states, self-duality currently lacks such a clear operational motivation. In this paper we prove an alternative to the Koecher-Vinberg theorem, substituting for self-duality the more operational property of pure transitivity: symmetries of the normalised state space act transitively on the pure states. We show that any system that is homogeneous and satisfies pure transitivity must be self-dual, and hence isomorphic to an EJA. Together with various ways of singling out the complex matrix algebras from among EJAs, and known ways of ruling out classicality in these, this yields several new and concise reconstructions of quantum theory. For example, quantum systems, classical systems, and composites of these are the only ones that are homogeneous, satisfy pure transitivity, and allow locally tomographic composites; fully quantum systems are the only ones that are homogeneous, have continuous pure transitivity, and allow a correspondence between observables and generators of reversible transformations.

quant-ph↗

arXiv2026

Quantum algorithm for the gradient of a logarithm-determinant

The logarithm-determinant is a widely-present operation in many areas of physics and computer science. Derivatives of the logarithm-determinant compute physically relevant quantities in statistical physics models, quantum field theories, as well as the inverses of matrices. A multi-variable version of the quantum gradient algorithm is developed here to evaluate the derivative of the logarithm-determinant. From this, the pseudo-inverse of a sparse-rank input operator may be determined efficiently. Measuring an expectation value of the quantum state--instead of all $N^2$ elements of the input operator--can be accomplished in $O(k/\varepsilon^2)$ time in the idealized case for $k$ relevant eigenvectors of the input matrix with precision $\varepsilon$. A practical implementation of the required operator will likely need $\log_2N$ overhead, giving an overall complexity of $O((k\log_2 N)/\varepsilon^2)$. The method applies widely and converges super-linearly in $k$ when the condition number is high. The best classical method we are aware of scales as $N$. Given the same resource assumptions as other algorithms, such that an equal superposition of eigenvectors is available efficiently, the algorithm is evaluated in the practical case as $O(\log_2 N/\varepsilon^2)$. The output is given in $O(1)$ queries of an oracle, which is given explicitly here and only relies on time-evolution operators that can be implemented with arbitrarily small error. The algorithm is envisioned for fully error-corrected quantum computers but may be implementable on near-term machines. We discuss how this algorithm can be used for kernel-based quantum machine-learning.

quant-ph↗

arXiv2026

Spinor Bose-Einstein condensate as an analog simulator of molecular bending vibrations

We demonstrate that spinor Bose-Einstein condensates (BECs) can be operated as an analog simulator of the two-dimensional vibron model. This algebraic model describes bending vibrations of molecules and, in the case of triatomic molecules, exhibits two phases where linear and bent configurations are stabilised. Spinor BECs can be engineered to simulate states that correspond to linear or bent triatomic molecules, with the Wigner function of the BEC encoding information about the molecular configuration. We show how quantum simulations of the bending dynamics of linear molecules can be realised, and how preparing a linear configuration in the bent phase leads to a dynamical instability. In the dynamics triggered by the corresponding instability, a significant amount of entanglement is generated, and we characterise the dynamics with the squeezing parameter and the quantum Fisher information (QFI). The scaling of the non-Gaussian sensitivity, described by the difference between squeezing and QFI, grows with the system size once the spinor system crosses from the linear to the bent phase, thus serving as a dynamical witness for the quantum phase transition.

quant-ph↗