Quantum Path Finder: Navigating Constraint Space via Non-Abelian Gauge Dynamics

Abstract

Quantum algorithms are commonly designed within the circuit paradigm or through adiabatic evolution. Here we introduce \emph{Quantum Path Finder} (QPF), a distinct algorithm-design paradigm based on non-Abelian gauge dynamics in a constraint-protected degenerate manifold. Each Boolean variable is promoted to a qutrit whose additional state represents an unassigned variable. The problem Hamiltonian then defines a connected manifold of feasible partial assignments, and time-dependent local qutrit rotations generate coherent transport on this manifold. Unlike conventional adiabatic quantum computation, QPF does not require the state to follow an instantaneous eigenstate along a prescribed interpolation: adiabaticity with respect to the constraint gap protects the feasible manifold, while adiabatic following within that manifold is optional. The algorithmic task is therefore to design paths and interference in constraint space rather than only to engineer a sequence of gates or a ground-state interpolation. We illustrate the framework with three applications. For unstructured search, a two-level analysis and numerical evolution recover Grover-type scaling under explicitly stated assumptions. For complete $3$-XORSAT, the accessible gauge sector has a size-independent gap, in contrast to the exponentially closing gap of a conventional transverse-field adiabatic path. For satisfiable $3$-SAT instances, finite-size benchmarks demonstrate high satisfied-clause fractions at short classical comparison budgets, without claiming an equal-cost quantum advantage. We also outline a superconducting-qutrit implementation and discuss the conceptual relationship between QPF and nondeterministic Turing machine.