Fast Quantum Amplitude Encoding of Typical Classical Data

• URL: http://arxiv.org/abs/2503.17113v2
• Date: Fri, 28 Mar 2025 19:50:05 GMT
• Title: Fast Quantum Amplitude Encoding of Typical Classical Data
• Authors: Vittorio Pagni, Sigurd Huber, Michael Epping, Michael Felderer,
• Abstract summary: We present an improved version of a quantum amplitude encoding scheme that encodes the $N$ entries of a unit classical vector.<n>We prove that the average runtime is $mathcalO(sqrtNlog N)$ for uniformly random inputs.<n>We present numerical evidence that this favourable runtime behaviour also holds for real-world data, such as radar satellite images.
• Score: 3.294877984684448
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We present an improved version of a quantum amplitude encoding scheme that encodes the $N$ entries of a unit classical vector $\vec{v}=(v_1,..,v_N)$ into the amplitudes of a quantum state. Our approach has a quadratic speed-up with respect to the original one. We also describe several generalizations, including to complex entries of the input vector and a parameter $M$ that determines the parallelization. The number of qubits required for the state preparation scales as $\mathcal{O}(M\log N)$. The runtime, which depends on the data density $\rho$ and on the parallelization paramater $M$, scales as $\mathcal{O}(\frac{1}{\sqrt{\rho}}\frac{N}{M}\log (M+1))$, which in the most parallel version ($M=N$) is always less than $\mathcal{O}(\sqrt{N}\log N)$. By analysing the data density, we prove that the average runtime is $\mathcal{O}(\log^{1.5} N)$ for uniformly random inputs. We present numerical evidence that this favourable runtime behaviour also holds for real-world data, such as radar satellite images. This is promising as it allows for an input-to-output advantage of the quantum Fourier transform.

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