  Simulating quantum dynamics is expected to be performed more easily on a
quantum computer than on a classical computer. However, the currently available
quantum devices lack the capability to implement fault-tolerant quantum
algorithms for quantum simulation. Hybrid classical quantum algorithms such as
the variational quantum algorithms have been proposed to effectively use
current term quantum devices. One promising approach to quantum simulation in
the noisy intermediate-scale quantum (NISQ) era is the diagonalisation based
approach, with some of the promising examples being the subspace Variational
Quantum Simulator (SVQS), Variational Fast Forwarding (VFF), fixed-state
Variational Fast Forwarding (fs-VFF), and the Variational Hamiltonian
Diagonalisation (VHD) algorithms. However, these algorithms require a feedback
loop between the classical and quantum computers, which can be a crucial
bottleneck in practical application. Here, we present the Classical Quantum
Fast Forwarding (CQFF) as an alternative diagonalisation based algorithm for
quantum simulation. CQFF shares some similarities with SVQS, VFF, fs-VFF and
VHD but removes the need for a classical-quantum feedback loop and controlled
multi-qubit unitaries. The CQFF algorithm does not suffer from the barren
plateau problem and the accuracy can be systematically increased. Furthermore,
if the Hamiltonian to be simulated is expressed as a linear combination of
tensored-Pauli matrices, the CQFF algorithm reduces to the task of sampling
some many-body quantum state in a set of Pauli-rotated bases, which is easy to
do in the NISQ era. We run the CQFF algorithm on existing quantum processors
and demonstrate the promise of the CQFF algorithm for current-term quantum
hardware. Our work provides a $10^4$ improvement over the previous record.
