Quantum frequency resampling

TL;DR

This work presents a toolset of quantum algorithms to resample data encoded in the probabilities of a quantum register, using the quantum Fourier transform to adjust the number of high-frequency encoding qubits.

Abstract

In signal processing, resampling algorithms can modify the number of resources encoding a collection of data points. Downsampling reduces the cost of storage and communication, while upsampling interpolates new data from limited one, e.g. when resizing a digital image. We present a toolset of quantum algorithms to resample data encoded in the probabilities of a quantum register, using the quantum Fourier transform to adjust the number of high-frequency encoding qubits. We discuss advantage over classical resampling algorithms.

Figures (5)

• Figure 1: Little-endian ordering when encoding a 3D signal $\mathcal{S}$. Classically, $\mathcal{S}$ is represented by a $4\times4\times4$ array, with a total of $64$ samples arranged along axes $e_{0}, e_{1}, e_{2}$. According to \ref{['eq:Input']}, these values are encoded in the probabilities of a $6$ qubits register $E$, which can be decomposed into $3$ subregisters, $E_0$, $E_1$, $E_2$, arranged, top to bottom, in order of increasing significance.
• Figure 2: Quantum downsampling of a $d$-dimensional signal encoded in the probabilities of a register $E$, divided in $d$ subregisters $E_i$, of $n_0$ qubits each. (a) Circuit scheme of the algorithm, which discards $\tilde{n} < n_0$ qubit for each axis, compressing the input into a state of register $D$, with $n_D = n_E - d\,\tilde{n}$ qubits (consistently divided into $d$ subregisters). (b) Effect on the classical signal. The discarding parameter $\tilde{n}$ determines the output resolution, from $N_0^d$ to $N_0^d/2^{d\tilde{n}}$.The algorithm is equivalent to averaging the original signal on a set of hyper-cubic blocks of side $\Tilde{N}$.
• Figure 3: Quantum upsampling of a $d$-dimensional signal encoded in the probabilities of register $E$, composed of $d$ subregisters $E_i$, each containing $n_0$ qubits. (a) Circuit implementation of the protocol: for the $i$-th axes, $\tilde{n}$ additional qubits (the subregister $P_i$) are appended to $E_i$. The upsampled state $\ket{\Omega}_U$, encoded in $d$ subregisters $U_i$, consists of $n_0 + \tilde{n}$ qubits. In contrast to downsampling, this protocol is purity-preserving. (b) Effect on the underlying classical signal, expanded from $N_0$ to $N_0\, 2^{\tilde{n}}$ samples per axis. The protocol acts as a nearest neighborhood interpolation: if one neglects the normalization, the input values are duplicated along all axes, for a number of times determined by the padding parameter.
• Figure 4: Simulation of quantum resampling. A truncated sinc function (shifted by a unit value along the $y$-axis), is initially sampled over $256$ possible values at rate $\mathsf{f}_E=256Hz$ in $[0\,s,2\,s]$ and encoded in a register of $9$ qubits (left), whose computational basis indexes are obtained by discretizing and normalizing the $x$-axis (the sampling interval). The signal is first downsampled to a rate $\mathsf{f}_D=32Hz$ (6 qubits) via \ref{['alg:Downsampling']} (center) and then upsampled to $\mathsf{f}_U=512Hz$ (10 qubits) beyond the original resolution, employing \ref{['alg:Upsampling']} (right). The insets at $[0.56\, s, \,0.59\,s]$ show the effects of block-averaging and nearest neighbour interpolation. For the latter, upsampled values vary due to artefacts and statistical fluctuations. The simulation is conducted with Qiskit Aercode:qiskit2024 and $256^2 \times 2^{n}$ shots, with $n$ being the number of output qubits.
• Figure 5: Quantum advantage bounds for the downsampling algorithms, as a function of the number of encoding and discarded qubits per signal axis. The dotted red line and the solid black one represent the lower and upper bounds of \ref{['eq:BoundsForAdvantage']}, respectively. The colormap expresses the ratio between classical and quantum cost $(\mathscr{D}_{\text{c}} /\mathscr{D}_{\text{q}})$, taken as figure of merit for quantifying advantage. (a) One-dimensional binary signal. (b) Two-dimensional 8-bit signal (e.g. a traditional grey-scale digital image). In both cases, the averaged MSE is set to $\delta^2 = 1/L^2$, i.e. by requiring fluctuations to be no larger than the bit-resolution.