  In chaotic dynamical systems, extreme events manifest in time series as
unpredictable large-amplitude peaks. Although deterministic, extreme events
appear seemingly randomly, which makes their forecasting difficult. By learning
the dynamics from observables (data), reservoir computers can time-accurately
predict extreme events and chaotic dynamics, but they may require many degrees
of freedom (large reservoirs). In this paper, by exploiting quantum-computer
ans\"atze and entanglement, we design reservoir computers with compact
reservoirs and accurate prediction capabilities. First, we propose the
recurrence-free quantum reservoir computer (RF-QRC) architecture. By developing
ad-hoc quantum feature maps and removing recurrent connections, the RF-QRC has
quantum circuits with small depths. This allows the RF-QRC to scale well with
higher-dimensional chaotic systems, which makes it suitable for hardware
implementation. Second, we forecast the temporal chaotic dynamics and their
long-term statistics of low- and higher-dimensional dynamical systems. We find
that RF-QRC requires smaller reservoirs than classical reservoir computers.
Third, we apply the RF-QRC to the time prediction of extreme events in a model
of a turbulent shear flow with turbulent bursts. We find that the RF-QRC has a
longer predictability than the classical reservoir computer. The results and
analyses indicate that quantum-computer ans\"atze offer nonlinear expressivity
and computational scalability, which are useful for forecasting chaotic
dynamics and extreme events. This work opens new opportunities for using
quantum machine learning on near-term quantum computers.
