Representation and design of wavelets using unitary circuits

The representation of discrete, compact wavelet transformations (WTs) as circuits of local unitary gates is discussed. We employ a similar formalism as used in the multi-scale representation of quantum many-body wavefunctions using unitary circuits, further cementing the relation established in [Phys. Rev. Lett. 116, 140403 (2016)] between classical and quantum multi-scale methods. An algorithm for constructing the circuit representation of known orthogonal, dyadic, discrete WTs is presented, and the explicit representation for Daubechies wavelets, coiflets, and symlets is provided. Furthermore, we demonstrate the usefulness of the circuit formalism in designing novel WTs, including various classes of symmetric wavelets and multi-wavelets, boundary wavelets and biorthogonal wavelets.