  Given two quantum states of N q-bits we are interested to find the shortest
quantum circuit consisting of only one- and two- q-bit gates that would
transfer one state into another. We call it the quantum maze problem for the
reasons described in the paper. We argue that in a large N limit the quantum
maze problem is equivalent to the problem of finding a semiclassical trajectory
of some lattice field theory (the dual theory) on an N+1 dimensional space-time
with geometrically flat, but topologically compact spatial slices. The spatial
fundamental domain is an N dimensional hyper-rhombohedron, and the temporal
direction describes transitions from an arbitrary initial state to an arbitrary
target state. We first consider a complex Klein-Gordon field theory and argue
that it can only be used to study the shortest quantum circuits which do not
involve generators composed of tensor products of multiple Pauli Z matrices.
Since such situation is not generic we call it the Z-problem. On the dual field
theory side the Z-problem corresponds to massless excitations of the phase
(Goldstone modes) that we attempt to fix using Higgs mechanism. The simplest
dual theory which does not suffer from the massless excitation (or from the
Z-problem) is the Abelian-Higgs model which we argue can be used for finding
the shortest quantum circuits. Since every trajectory of the field theory is
mapped directly to a quantum circuit, the shortest quantum circuits are
identified with semiclassical trajectories. We also discuss the complexity of
an actual algorithm that uses a dual theory prospective for solving the quantum
maze problem and compare it with a geometric approach. We argue that it might
be possible to solve the problem in sub-exponential time in 2^N, but for that
we must consider the Klein-Gordon theory on curved spatial geometry and/or more
complicated (than N-torus) topology.
