Fast quantum amplitude encoding of typical classical data

Abstract

We present an improved version of a quantum amplitude encoding scheme that encodes the N entries of a unit classical vector \(\boldsymbol{v}=(v_{1},..,v_{N})\) 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 \(\mathcal{O}(\frac{1}{\sqrt{\rho}}\frac{N}{M}\log (M+1))\), which in the most parallel version (\(M=N\)) is always less or equal 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 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.