  Clifford group lies at the core of quantum computation -- it underlies
quantum error correction, its elements can be used to perform magic state
distillation and they form randomized benchmarking protocols, Clifford group is
used to study quantum entanglement, and more. The ability to utilize Clifford
group elements in practice relies heavily on the efficiency of their
circuit-level implementation. Finding short circuits is a hard problem; despite
Clifford group being finite, its size grows quickly with the number of qubits
$n$, limiting known optimal implementations to $n{=}4$ qubits. For $n{=}6$, the
number of Clifford group elements is about $2.1{\cdot}10^{23}$. In this paper,
we report a set of algorithms, along with their C/C++ implementation, that
implicitly synthesize optimal circuits for all 6-qubit Clifford group elements
by storing a subset of the latter in a database of size 2.1TB (1KB=1024B). We
demonstrate how to extract arbitrary optimal 6-qubit Clifford circuit in
$0.0009358$ and $0.0006274$ seconds using consumer- and enterprise-grade
computers (hardware) respectively, while relying on this database.
