Representation and design of wavelets using unitary circuits

Here we offer a proof that scaling $\bm{h}$ and wavelet $\bm{g}$ sequences from unitary circuits always satisfy the relation of Eq. 12. We begin by defining the spatial reflection operator $\mathcal{R}$ which reverses the order of both rows and columns of a matrix, i.e if for a given $N\times N$ matrix $A$ we have that $\tilde{A}\equiv\mathcal{R}\left(A\right)$ then the matrix elements of $A$ and $\tilde{A}$ are related as

Notice that spatial reflection of a $2\times 2$ unitary $u$ from Eq. 8 changes the sign of the angle $\theta$,

It follows that, if $\mathcal{U}$ is a binary unitary circuit parameterized by angles $\{\theta_{1},\theta_{2},\ldots,\theta_{N}\}$, then its spatial reflection $\tilde{\mathcal{U}}\equiv\mathcal{R}\left(\mathcal{U}\right)$ is a binary unitary circuit with opposite sign angles $\{-\theta_{1},-\theta_{2},\ldots,-\theta_{N}\}$. Notice that spatial reflection of circuit $\mathcal{U}$ into $\tilde{\mathcal{U}}$ has exchanged the role of the scaling and wavelet sequences, such that if $\bm{g}=[g_{1},g_{2},\ldots,g_{2N}]$ is the wavelet sequence of $\mathcal{U}$ and $\bm{\tilde{h}}=[\tilde{h}_{1},\tilde{h}_{2},\ldots,\tilde{h}_{2N}]$ is the scaling sequence of $\tilde{\mathcal{U}}$, then

see also Fig. 3(a). Thus in order to demonstrate the relation between $\bm{h}$ and $\bm{g}$ of Eq. 12, it suffices to show that the elements of $\bm{h}$ and $\tilde{\bm{h}}$ are related with an alternating negative sign,

see also Fig. 3(b), which we now prove.

Consider the factorization of the circuit into its constituent layers,

where each $U_{k}$ is a function of some $\theta_{k}$. Let $\bm{s}=[s_{1},s_{2},\ldots s_{2M}]$ and $\tilde{\bm{s}}=[\tilde{s}_{1},\tilde{s}_{2},\ldots\tilde{s}_{2M}]$ be vectors of length $2M$ for integer $M$, and assume ${{\tilde{s}}_{r}}={(-1)^{r}}{s_{r}}$. Define $\bm{s}^{\prime}$ as given by the transform $\bm{s}$ by circuit layer $U(\theta)$,

and $\tilde{\bm{s}}^{\prime}$ as given by the transform of $\tilde{\bm{s}}$ by a layer $U(-\theta)$ of opposite sign,

If it can be shown that ${{\tilde{s}}^{\prime}_{r}}={(-1)^{r}}{s^{\prime}_{r}}$ then Eq. 50 follows from recursion over all layers $U_{k}$ of the binary circuit $\mathcal{U}$; in order to show this we evaluate elements of $\bm{s}^{\prime}$ from Eq. 52,

and similarly evaluate elements $\tilde{\bm{s}}^{\prime}$ from Eq. 53,

from which the desired result is observed. Thus we conclude that the scaling $\bm{h}$ and wavelet $\bm{g}$ sequences from a binary unitary circuit are always related as per Eq. 12.

Given a discrete, orthogonal WT described by the scaling sequence $\bm{h}=[h_{1},h_{2},\ldots,h_{2N}]$ of $2N$ coefficients, we now describe an algorithm to encode the sequence as a depth $N$ binary unitary circuit. More precisely, we describe how the angles $\{\theta_{1},\theta_{2},\ldots,\theta_{N}\}$ defining the circuit can be uniquely chosen such that the scaling sequence is mapped to a unit vector $\mathbbm{1}_{r}$ under transformation by the circuit,

in accordance with the definition of Eq. 45. Recall that the binary circuit decomposes into $N$ layers, $\mathcal{U}=U_{N}U_{N-1}\ldots U_{1}$, where each layer $U_{k}$ is the direct sum of a $2\times 2$ unitary matrices $u(\theta_{k})$. Let ${\bm{h}}^{\prime}$ be the scaling sequence after transforming by the bottom layer $U_{N}$ of the unitary circuit,

We now propose that the free angle $\theta_{N}$ of unitary $u(\theta_{N})$ should be chosen as,

Under this choice the first element of the transformed scaling sequence ${\bm{h}}^{\prime}$ is zero,

for all three possibilities (i-iii) from Eq. 82. To understand this in the second instance (ii) notice that the orthogonality constraints of Eq. 11 impose,

which implies that $h_{1}=0$ given that $h_{2N-1}\neq 0$. Similarly, the trailing $(2N)^{\textrm{th}}$ element of ${\bm{h}}^{\prime}$ is also mapped to zero,

In instances (ii) and (iii) from Eq. 82 this follows trivially, whereas in the first instance (i) this can be understood by substituting the orthogonality constraint of Eq. 11 to give,

Thus we have demonstrated that under the choice of angle $\theta$ from Eq. 82 the scaling sequence $\bm{h}$ of $2N$ coefficients is mapped under $U_{N}$ to a new sequence $\bm{h}^{\prime}$ of $2N-2$ coefficients (where the first and last elements of $\bm{h}^{\prime}$, which were shown to be zero, have been dropped). It can also easily be seen that the orthogonality constraints on $\bm{h}$, see Eq. 11, map to an equivalent set of constraints for the new sequence $\bm{h}^{\prime}$. The set of angles $\{\theta_{N},\theta_{N-1},\ldots,\theta_{2},\theta_{1}\}$ is obtained by iterating this procedure a total of $N-1$ times, i.e. for unitary layers $\{U_{N},U_{N-1},\ldots,U_{2}\}$, which then maps the initial scaling sequence $\bm{h}$ of $2N$ coefficients into a sequence of length two denoted $\bm{h}^{\textrm{f}}=[h_{1}^{\textrm{f}},h_{2}^{\textrm{f}}]$. At this stage, the angle $\theta_{1}$ associated to the top level $U_{1}$ is fixed at,

such that $\bm{h}^{\textrm{f}}$ is mapped to a sequence with only a single non-zero element, indicating that the scaling sequence is encoded in the binary unitary circuit.

Employing this algorithm we generate the sets of angles that describe the Daubechies wavelets D2N for $N=\{1,2,\ldots 10\}$, see Tab. 1, the symlets of $2N$ coefficients for $N=\{1,2,\ldots 10\}$, see Tab. 2, and the coiflets of $2N$ coefficients for $N=\{3,6,9,12,15\}$, see Tab. 3. Notice that clear patterns are evident in the angles $\theta$ between different orders $N$ of wavelets in each of the families, see also Fig. 4.

Let $\mathcal{V}$ be a $3M\times 3M$ matrix for integer $M$; we say that $\mathcal{V}$ is a ternary unitary circuit of depth $N$ if it can be decomposed as,

see also the circuit diagram of Fig. 5. Here each of the $N$ sublayers $V_{k}$ is a direct sum of $M$ matrices $v(\theta_{k})$ from Eq. 90,

which are interspersed with sublayers $U_{\textrm{sw}}$ consisting of direct sums of swap gates $u_{\textrm{sw}}$ from Eq. 88,

with $\mathbb{I}$ the $1\times 1$ identity. The ternary circuit $\mathcal{V}$ includes a top sublayer $U_{\pm}$ composed of $2\times 2$ unitary gates $u(\pi/4)$ which are spaced by contributions of the identity,

Notice that the ternary unitary circuit of depth $N$ is fully parameterized by the set of angles $\{\theta_{1},\theta_{2},\ldots,\theta_{N}\}$. There are three coefficient sequences associated to a ternary circuit (which corresponds to a three-channel filter bank), which we label $\{\bm{s}^{+},\bm{b}^{+},\bm{b}^{-}\}$. Each of the three sequences is obtained by transforming a unit vector $\mathbbm{1}_{r}$ on different sites $r$ with the ternary circuit $\mathcal{V}$,

see also Fig. 5. The sequences possess the following properties:

• •

  $\bm{s}^{+}$ is a site-centered symmetric sequence of (odd) length $6N-3$ elements (i.e. that is symmetric w.r.t. reflections about its central element)

• •

  $\bm{b}^{+}$ is a edge-centered symmetric sequence of (even) length $6N$ elements (i.e. that is symmetric w.r.t. reflections centered between its two central elements)

• •

  $\bm{b}^{-}$ is a edge-centered antisymmetric sequence of (even) length $6N$ elements (i.e. that is antisymmetric w.r.t. reflections centered between its two central elements).

Similar to the construction applied to the binary unitary circuit in Sect. II, layers of the ternary circuit can be composed to form a multi-scale circuit,

which encodes the dilation factor 3 discrete WT. Notice that there are two distinct ways of implementing this composition dependent on whether $\bm{s}^{+}$ or $\bm{b}^{+}$ is considered as the scaling sequence, which we refer to as the site-centered or edge-centered multi-scale circuits respectively, see also Fig. 6.

A depth $N$ ternary unitary circuit offers a parameterization of exactly reflection symmetric and antisymmetric wavelets with $N$ free parameters $\{\theta_{1},\theta_{2},\ldots,\theta_{N}\}$. In this section we consider the family of wavelets that result from choosing these parameters to maximize the number of vanishing moments that the wavelets possess.

We use the site-centered form of the multi-scale circuit as depicted in Fig. 6(a), where sequence $\bm{s}^{+}$ is treated as the scaling sequence and $\bm{b}^{+}$, $\bm{b}^{-}$ are the wavelet sequences, and we optimize the angles $\theta_{k}$ such the wavelets have vanishing moments,

for $\alpha=[0,1,2,\ldots]$. Notice that $\bm{b}^{+}$ is automatically orthogonal to odd $\alpha$ polynomials due to its symmetry, and similarly $\bm{b}^{-}$ is automatically orthogonal to even $\alpha$ polynomials. Thus satisfying Eq. 97 for orders $\alpha=[0,1,2,\ldots,N-1]$, i.e. such that the wavelets have $N$ vanishing moments, requires $N$ non-trivial constraints to be met.

The sets of angles $\{\theta_{1},\theta_{2},\ldots,\theta_{N}\}$ that satisfy the vanishing moment criteria for circuit depths $N=2,4,6$ are given in Tab. 4. These were obtained numerically using the standard Nelder-Mead algorithm to minimize the cost function given by the sum of the first $N$ moments of the wavelet sequences. The resulting wavelets are depicted in Fig. 7. Interestingly, while the scaling functions associated to these WTs appear similar to those obtained previously in Figs. 3 and 5 of Ref. Sym1, , the wavelets themselves are seen to have much smaller effective support (i.e. the region over which the wavelets have non-vanishingly small amplitude is reduced), which suggests they may perform better in practical applications.

Choosing the angles $\{\theta_{1},\theta_{2},\ldots,\theta_{N}\}$ that parameterize the ternary circuit $\mathcal{V}$ to maximize the vanishing moments of the wavelets, as per Sect. IV.1.2, has several notable deficiencies. One of these deficiencies if that the symmetric and anti-symmetric wavelets seen in Fig. 7 are not well separated when resolved in frequency space; ideally we want the three-channel filter bank represented by circuit $\mathcal{V}$ to possess well resolved low, mid, and high frequency components. A second deficiency is that the scaling functions in Fig. 7 still contain a significant high frequency component, which results in the corresponding wavelets not having a high degree of smoothness. In this Section we explore an alternative optimization criteria that resolves these deficiencies.

The alternative criteria we propose is that the circuit should be optimized such that some sequences $\bm{h}=[h_{1},h_{2},\ldots]$ have vanishing moments at high frequencies,

A three-channel wavelet filter bank with well resolved low/mid/high frequencies can be achieved by requiring that the scaling function and one of the wavelets satisfy the (high frequency) orthogonality constraint of Eq. 98 for some $\alpha=[0,1,\ldots]$ (while also requiring that both wavelets satisfy the standard vanishing moment criteria of Eq. 14). Recall that the ternary unitary circuit has three associated sequences $\{\bm{s}^{+},\bm{b}^{+},\bm{b}^{-}\}$, as such there freedom as to how the sequences are associated to the low/mid/high frequency components (which also informs whether the bond-centered or edge-centered form of the ternary circuit is used, see Fig. 6). Specifically, there are three choices (which we call Types I-III) that are compatible with the symmetries of the sequences, see Tab. 5.