Fast quantum amplitude encoding of typical classical data
Vittorio Pagni, Sigurd Huber, Michael Epping, Michael Felderer · EPJ Quantum Technology · 2026
Abstract We present an improved version of a quantum amplitude encoding scheme that encodes the N entries of a unit classical vector $\boldsymbol{v}=(v_{1},..,v_{N})$ 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 $\mathcal{O}(M\log N)$ O ( M log N ) . The runtime, which depends on the data density ρ and on the parallelization paramater M , scales as $\mathcal{O}(\frac{1}{\sqrt{\rho}}\frac{N}{M}\log (M+1))$ O ( 1 ρ N M log ( M + 1 ) ) , which in the most parallel version ( $M=N$ M = N ) is always less or equal than $\mathcal{O}(\sqrt{N}\log N)$ O ( N log N ) . By analysing the data density, we prove that the average runtime is $\mathcal{O}(\log ^{1.5} N)$ O ( log 1.5 N ) for input vectors that are uniformly sampled on the N -sphere. 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.