We consider the encoding of graph problems as Quadratic Unconstrained Binary Optimization (QUBO) problems, solvable by quantum or classical annealers. However, nowhere-zero flows have not previously been included among graph problems encoded as QUBO problems. Nowhere-zero flows are related to Tutte’s \(5\)-flow conjecture and occur in many contexts in graph theory. We provide a QUBO Hamiltonian encoding of nowhere-zero flows and prove the correctness of the construction. The resulting Hamiltonian \(H_{\mathrm{mod},k}\) has zero ground-state energy if and only if the graph \(G\) has a nowhere-zero \(\mathbb{Z}_k\)-flow. By Tutte’s equivalence theorem, zero ground energy is equivalent to \(\varphi(G)\le k\), and the zero-energy degeneracy is given by the flow polynomial \(F(G;k)\). The construction uses one-hot variables for edge flow residues modulo \(k\) and auxiliary variables for the per-vertex modular quotient. We prove that correctness is independent of the choice of orientation, root vertex, and positive penalty weights. We verify the construction on \(59\) graph and \(k\) examples, including both yes-instances and no-instances. We also sweep orientations and root choices on selected robustness instances and test a finite suite of positive penalty weights. The Hamiltonian is implemented using the dimod.BinaryQuadraticModel class, compatible with the D-Wave Ocean SDK. Quantum-hardware runs and claims about potential speedup are left to future work.