Robustness of Quantum Algorithms Against Approximate Data Representations
Vladyslav Los, Mykola Maksymenko, Maciej Koch-Janusz, Yuriy Pryyma, Richard Givhan · 2023
Quantum algorithms face practical difficulties and overheads in execution on realistic devices, potentially thwarting their theoretical advantages. A key practical challenge is efficiently encoding classical data, since data embedding layers, often exponential in the number of qubits, can consume a significant portion of the circuit depth. One possible strategy to address this is to compress the standard data embeddings using variational circuits. The approximate amplitude encoding (AAE) algorithm successfully applies this approach to financial calculations and image pattern matching, achieving only polynomially deep embedding layers. While prior works focus on optimizing the approximate embedding in isolation, the ultimate goal is the compound algorithm's performance. This study systematically examines the impact of embedding compression on the fidelity of the subsequent quantum algorithm in a noiseless setting, specifically investigating amplitude encoding and its dependence on input data set size and complexity. Trainability of the different variational ansaetze, their output state fidelity, and the degree of the embedding layer depth compression are investigated. Performance benchmarks of standard quantum algorithms, e.g. Grover search, amplitude estimation, and quantum Fourier transform (QFT), are conducted based on the degree of compression and quality of the compressed input state. Results show that the amplitude encoding circuit can be compressed substantially in depth, for generic data and a small number of qubits, with only marginal loss in state fidelity. Certain algorithms, like QFT, demonstrate robustness to such approximations.