Abstract Quantum reservoir computing has emerged as a promising machine learning paradigm for processing temporal data on near-term quantum devices, as it exploits the large computational capacity of qubits without suffering from typical issues arising when training variational quantum circuits. In particular, quantum gate-based echo state networks have proven effective when the evolution of the reservoir circuit is non-unital. Nonetheless, a method for ensuring a tunable and stable non-unital circuit evolution was lacking. We propose an algorithm that induces damping by applying a controlled rotation to each qubit in the reservoir. It enables tunable, circuit-level amplitude amplification of the zero state, maintaining the system away from the maximally mixed state and preventing information loss caused by repeated mid-circuit measurements. The algorithm is inherently stable over time, as it can process arbitrarily long input sequences, beyond the coherence time of individual qubits, by inducing arbitrary damping on each qubit. Moreover, we show that quantum correlations between qubits improve memory retention, underscoring the potential utility of a quantum system as a computational reservoir. We demonstrate, through standard reservoir computing benchmarks, that this algorithm enables robust and scalable quantum random computing on fault-tolerant quantum hardware. Similar content being viewed by others Data availability The datasets generated and/or analyzed during the current study are not publicly available due to technical limitations and their use in ongoing related research, but are available from the corresponding author on reasonable request. References Deutsch, I. H. Harnessing the power of the second quantum revolution. PRX Quantum 1, 020101 (2020). Bharti, K. et al. Noisy intermediate-scale quantum algorithms. Rev. Mod. Phys. 015004 (2022). Prati, E. Quantum neuromorphic hardware for quantum artificial intelligence. In Journal of Physics: Conference Series, vol. 880 (IOP Publishing, 2017). Schuld, M. & Petruccione, F. Chapter in: Machine Learning With Quantum Computers, vol. 676 (Springer, 2021).