  Quantum circuit synthesis is the process in which an arbitrary unitary
operation is decomposed into a sequence of gates from a universal set,
typically one which a quantum computer can implement both efficiently and
fault-tolerantly. As physical implementations of quantum computers improve, the
need is growing for tools which can effectively synthesize components of the
circuits and algorithms they will run. Existing algorithms for exact,
multi-qubit circuit synthesis scale exponentially in the number of qubits and
circuit depth, leaving synthesis intractable for circuits on more than a
handful of qubits. Even modest improvements in circuit synthesis procedures may
lead to significant advances, pushing forward the boundaries of not only the
size of solvable circuit synthesis problems, but also in what can be realized
physically as a result of having more efficient circuits.
  We present a method for quantum circuit synthesis using deterministic walks.
Also termed pseudorandom walks, these are walks in which once a starting point
is chosen, its path is completely determined. We apply our method to construct
a parallel framework for circuit synthesis, and implement one such version
performing optimal $T$-count synthesis over the Clifford+$T$ gate set. We use
our software to present examples where parallelization offers a significant
speedup on the runtime, as well as directly confirm that the 4-qubit 1-bit full
adder has optimal $T$-count 7 and $T$-depth 3.
