  Quantum reservoir computing (QRC) is a highly promising computational
paradigm that leverages quantum systems as a computational resource for
nonlinear information processing. While its application to time-series analysis
is eagerly anticipated, prevailing approaches suffer from the collapse of the
quantum state upon measurement, resulting in the erasure of temporal input
memories. Neither repeated initializations nor weak measurements offer a
fundamental solution, as the former escalates time complexity while the latter
restricts the information extraction from the Hilbert space. To address this
issue, we propose a feedback-driven QRC framework. This methodology employs
projective measurements on all qubits for unrestricted access to the quantum
state, with the measurement outcomes subsequently fed back into the reservoir
to restore the memory of prior inputs. We demonstrate that our QRC successfully
acquires the fading-memory property through the feedback, a critical element in
time-series processing. Notably, analysis of measurement trajectories reveal
three distinct phases depending on the feedback strength, with the memory
performance maximized at the edge of chaos. We also evaluate the predictive
capabilities of our QRC, demonstrating its suitability for forecasting signals
originating from quantum spin systems.
