Quantum Framework for Wavelet Shrinkage

TL;DR

This work addresses translating classical wavelet denoising into the quantum domain by reframing coefficient shrinkage as a physically realizable CPTP operation. It combines unitary quantum wavelet transforms with three shrinkage paradigms—ancilla-driven Kraus channels, controlled decoherence, and hybrid, feedback-assisted schemes—to achieve adaptive attenuation of wavelet coefficients while preserving quantum coherence where possible. Key contributions include explicit Kraus representations, a phase-damping surrogate for thresholding, and practical circuit designs plus NISQ-compatible hardware strategies, demonstrated through Qiskit-based discussions and notebooks. The approach creates a principled bridge between wavelet-based statistical inference and open quantum dynamics, turning decoherence into a programmable resource for multiscale denoising with potential integration into quantum sensing and information-processing pipelines.

Abstract

This paper develops a unified framework for quantum wavelet shrinkage, extending classical denoising ideas into the quantum domain. Shrinkage is interpreted as a completely positive trace-preserving process, so attenuation of coefficients is carried out through controlled decoherence rather than nonlinear thresholding. Phase damping and ancilla-driven constructions realize this behavior coherently and show that statistical adaptivity and quantum unitarity can be combined within a single circuit model. The same physical mechanisms that reduce quantum coherence, such as dephasing and amplitude damping, are repurposed as programmable resources for noise suppression. Practical demonstrations implemented with Qiskit illustrate how circuits and channels emulate coefficientwise attenuation, and all examples are provided as Jupyter notebooks in the companion GitHub repository. Encoding schemes for amplitude, phase, and hybrid representations are examined in relation to transform coherence and measurement feasibility, and realizations suited to current noisy intermediate-scale quantum devices are discussed. The work provides a conceptual and experimental link between wavelet-based statistical inference and quantum information processing, and shows how engineered decoherence can act as an operational surrogate for classical shrinkage.

Figures (9)

• Figure 1: Quantum implementation of the Daubechies DAUB2 wavelet transform as a single $8\times 8$ unitary operation. The input sequence is amplitude-encoded into three qubits and transformed through two levels of decomposition. The resulting approximation and detail coefficients coincide exactly with those obtained by the classical wavelet-matrix and Mallat algorithms as in the example on pages 116-117 of vidakovic1999, confirming the correctness and unitarity of the quantum realization.
• Figure 2: Phase-based thresholding map in the quantum wavelet framework. We start with wavelet coefficients $d=[2, 1, 9, 0, 3, -10, 2, 4]$ and rescale them to $[-1,1].$ This short sequence of classical values is used throughout this paper to zoom on the local action of wavelet shrinkage. The compact example makes each step of the transformation easy to follow, while the same procedures apply without difficulty to sequences of length in the thousands. Each rescaled wavelet coefficient undergoes a phase rotation determined by its magnitude, producing an effective thresholding rule in which small coefficients are strongly displaced toward destructive interference while large coefficients remain nearly unchanged. The resulting mapping provides a fully unitary realization of coefficient shrinkage.
• Figure 3: Soft-shrinkage behavior emerging from phase-controlled attenuation. The plot relates the input and output amplitudes under the phase-driven rule, showing a smooth contraction that parallels the classical soft-threshold function. This unitary realization anticipates the probabilistic shrinkage of later Kraus and CPTP channel formulations as in Subsection \ref{['subsec:phase_damping']}.
• Figure 4: Comparison of classical soft thresholding with CPTP-based shrinkage. The clean Doppler signal, its noisy version, the reconstruction via classical soft thresholding, and the reconstruction via CPTP attenuation of wavelet coefficients are shown. The CPTP method uses ancilla-0 diagonal scaling to impose coefficientwise attenuation that mimics soft shrinkage while remaining physically admissible as a quantum channel. The construction of the CPTP shrinkage channel is described in Section \ref{['subsec:ancilla_cptp']}.
• Figure 5: Ancilla-0 diagonal probabilities before and after CPTP attenuation. The plot displays the first 200 indices of the diagonal of the reduced density matrix associated with ancilla outcome zero. The post-attenuation curve shows the multiplicative action of the CPTP channel, confirming that the channel scales each coefficient level according to the prescribed attenuation factors.

Theorems & Definitions (7)

• Example 2.1
• Example 3.1
• Example 3.2
• Example 3.3
• Example 3.4
• Example 3.5
• Example 4.1