A Quantum Range-Doppler Algorithm for Synthetic Aperture Radar Image Formation

TL;DR

This paper introduces a quantum version of the SAR range-Doppler focusing algorithm, encoding the $N$-pixel image in a $n=\log_2 N$ qubit amplitude-encoded state and executing range and azimuth processing via QFT/IQFT stages and diagonal unitaries derived from the reference functions. The approach decomposes the classical pipeline into quantum subroutines: quantum range compression, quantum range cell migration correction, and quantum azimuth compression, employing multi-controlled multi-target gates and the principle of stationary phase to enforce unitarity. The authors find a leading quantum gate complexity of $O_\text{q}(N)$, with a one-time classical preprocessing cost of $O_\text{c}(\sqrt{N}\log N)$, and measurements contributing an $O(N^2)$ term that can be amortized, yielding an overall favorable scaling under ideal conditions. Simulations on a 128×128 scene demonstrate convergence toward the ground-truth image as measurement counts increase, highlighting the potential for quantum SAR processing to reduce computational overhead in large-scale remote sensing tasks, while noting that practical deployment will need to address noise and fault-tolerance.

Abstract

Synthetic aperture radar (SAR) is a well established technology in the field of Earth remote sensing. Over the years, the resolution of SAR images has been steadily improving and the pixel count increasing as a result of advances in the sensor technology, and so have the computational resources required to process the raw data to a focused image. Because they are a necessary step in the study of the retrieved data, new high-resolution and low-complexity focusing algorithms are constantly explored in the SAR literature. The theory of quantum computing proposes a new computational framework that might allow to process a vast amount of data in a more efficient way. Relevant to our case is the advantage proven for the quantum Fourier transform (QFT), the quantum counterpart of a fundamental element of many SAR focusing algorithms. Motivated by this, in this work we propose a quantum version of the range-Doppler algorithm. We show how in general reference functions, a key element in many SAR focusing algorithms, can be mapped to quantum gates; we present the quantum circuit performing the SAR raw data focusing and we discuss in detail its computational complexity. We find that the core of the quantum range-Doppler algorithm has a computational complexity, namely the number of single- and two-qubit gates, of $O(N)$, less than its classical counterpart with computational complexity $O(N \log N)$.

Figures (4)

• Figure 1: Representation of the amplitude encoding of the complex SAR raw data. $n/2$ qubits encode the range (r) dimension and the remaining $n/2$ qubits encode the azimuth (a) dimension.
• Figure 2: Complete circuit implementing the proposed quantum range-Doppler algorithm as described in algorithm \ref{['alg:complete-quantumrange-Doppler-algorithm']}. From the left, it consists of the state preparation gate $V$, and the three blocks of gates contained in the dashed boxes implementing the range compression, the range cell migration correction and the azimuth compression. Finally measurements on every qubit are added to retrieve the probability of every bit string, encoding the value of each pixel of the SAR image. The redundant gates, consisting in adjacent $X$-gates, can be omitted.
• Figure 3: Geometric setup of the simulation. The sensor starts at $\mathbf{x}_0 = (0, 0, 500km)$, observing a planar scene at $z=0$ over $x \in [0km, 10km]$ and $y \in [290km, 300km]$. The azimuth dimension, which corresponds to the x axis, and range (r) dimensions are depicted. Simulation parameters are summarized in Table \ref{['tab:sim-params']}.
• Figure 4: Results of the simulation for the quantum range-Doppler algorithm for a custom scene taken from geowgs84WhatImagery and resized to $N = 128 \times 128$ pixels, consisting of an urban area with a harbor. Upper row: original scene placed on the ground (left) and real part of the complex raw data signals collected by the simulated carrier flying over the scene (right). Central and bottom row: results obtained with the quantum range-Doppler algorithm with $\lceil N/100 \rceil$, $\lceil N/10 \rceil$, $N$ and $100N$ measurements.