  We describe a general procedure for mapping arbitrary $n$-qubit states to
two-dimensional (2D) vector fields. The mappings use complex rational function
representations of individual qubits, producing classical vector field
configurations that can be interpreted in terms of 2D inviscid fluid flows or
electric fields. Elementary qubits are identified with localized defects in 2D
harmonic vector fields, and multi-qubit states find natural field
representations via complex superpositions of vector field products. In
particular, separable states appear as highly symmetric flow configurations,
making them both dynamically and visually distinct from entangled states. The
resulting real-space representations of entangled qubit states enable an
intuitive visualization of their transformations under quantum logic
operations. We demonstrate this for the quantum Fourier transform and the
period finding process underlying Shor's algorithm, along with other quantum
algorithms. Due to its generic construction, the mapping procedure suggests the
possibility of extending concepts such as entanglement or entanglement entropy
to classical continuum systems, and thus may help guide new experimental
approaches to information storage and non-standard computation.
