Quantum Wave Atom Transforms

TL;DR

This work tackles assigning quantum speedups to wavelet-based transforms by developing quantum circuits for both Shannon wavelet transforms and wave atom transforms with parabolic scaling. The approach relies on encoding wave packet indices into compact quantum registers, using retaining decoding and tree-aware helper functions to navigate multilevel structures, and introducing blending to manage frequency-domain overlaps in wave atoms. The main contributions are: (i) a quantum Shannon wavelet transform with $O(L^2)$ gate complexity for monotonic wave-packet trees, (ii) a quantum wave atom transform that extends to parabolic-scaling wave packets, leveraging a decomposition $C^A=F^A R^agger G^A R$ and achieving $O(L^2)$ gate complexity under suitable $G^A$ constructions, and (iii) a detailed implementation blueprint that includes frequency encoding, retaining decoding, and permutation strategies to realize these transforms on quantum hardware. The results suggest potential exponential speedups for solving wave equations and related PDEs by enabling efficient wavelet-based preconditioning and sparse representations, with natural extensions to higher dimensions and more general tree structures.

Abstract

This paper constructs the first quantum algorithm for wavelet packet transforms with a "parabolic scaling" tree structure, sometimes called wave atom transforms. Classically, wave atoms are used to construct sparse representations of differential operators, which enable fast numerical algorithms for partial differential equations. Compared to previous work, our quantum algorithm can implement a larger class of wavelet and wave atom transforms, by using an efficient representation for a larger class of possible tree structures. Our quantum implementation has O(poly(n)) gate complexity for applying a transform of dimension 2^n, while classical implementations use O(n*2^n) floating point operations. The result can be used to improve existing quantum algorithms for solving hyperbolic partial differential equations, such as wave equations.

Figures (14)

• Figure 1: Circuit representations of standard quantum gates. Left column (from top to bottom): phase shift $P_k$-, $X$-, $Z$- and $H$-gates. Center-top: $\mathrm{SWAP}$ gate. Center-bottom: controlled-$U$ gate, which is equivalent to $\ket{1}\bra{1}\otimes U + \ket{0}\bra{0}\otimes I_2$ Right: example of controlled-$U$ gate, when applied to qubits $3$ and $4$ if qubit $1$ is $\ket{0}$ and qubit $2$ is $\ket{1}$. In this case controlled-$U$ is equivalent to $\ket{01}\bra{01}\otimes U + (I_4 - \ket{01}\bra{01}) \otimes I_4$.
• Figure 2: The circuit of the QFT on $L$ qubits, consisting of QFT rotations $Q_i^R$, followed by a QFT swap $Q_L^S$ (see Eq. (\ref{['eq:fourier']})).
• Figure 3: Shannon wavelet $\hat{\varphi}^j_{m}$: one bump per each side of the axis in the frequency domain.
• Figure 4: The function $g$, satisfying equations (\ref{['prop:sum_of_squares']}) and (\ref{['prop:change_of_sign']})
• Figure 5: Wave atoms. $\hat{\psi}^0_0$ is unique, $\hat{\psi}^0_1$ is an example of odd $m$'s, $\hat{\psi}^0_2$ represents even $m$'s.

Theorems & Definitions (22)

• Definition 2.1: Admissible wavelet packet binary tree, adapted WaveletTourofSignalProcessing
• Proposition 2.1: enumeration properties of admissible wave packet tree
• Definition 4.1
• Theorem 4.1
• Proposition 5.1: Integer support of wave atoms
• Definition 5.1
• Proposition 5.2
• Definition 5.2
• Definition 5.3
• Definition 5.4