Fast Quantum Amplitude Encoding of Typical Classical Data
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912299937169408 |
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| author | Pagni, Vittorio Huber, Sigurd Epping, Michael Felderer, Michael |
| author_facet | Pagni, Vittorio Huber, Sigurd Epping, Michael Felderer, Michael |
| contents | 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 $ρ$ and on the parallelization paramater $M$, scales as $\mathcal{O}(\frac{1}{\sqrtρ}\frac{N}{M}\log (M+1))$, 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_17113 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Fast Quantum Amplitude Encoding of Typical Classical Data Pagni, Vittorio Huber, Sigurd Epping, Michael Felderer, Michael Quantum Physics Data Structures and Algorithms 81P68 F.1.2; I.4.8; I.4.3; J.2 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 $ρ$ and on the parallelization paramater $M$, scales as $\mathcal{O}(\frac{1}{\sqrtρ}\frac{N}{M}\log (M+1))$, 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. |
| title | Fast Quantum Amplitude Encoding of Typical Classical Data |
| topic | Quantum Physics Data Structures and Algorithms 81P68 F.1.2; I.4.8; I.4.3; J.2 |
| url | https://arxiv.org/abs/2503.17113 |