  Hamiltonian simulation is one of the most important problems in quantum
computation, and quantum singular value transformation (QSVT) is an efficient
way to simulate a general class of Hamiltonians. However, the QSVT circuit
typically involves multiple ancilla qubits and multi-qubit control gates. In
order to simulate a certain class of $n$-qubit random Hamiltonians, we propose
a drastically simplified quantum circuit that we refer to as the minimal QSVT
circuit, which uses only one ancilla qubit and no multi-qubit controlled gates.
We formulate a simple metric called the quantum unitary evolution score (QUES),
which is a scalable quantum benchmark and can be verified without any need for
classical computation. Under the globally depolarized noise model, we
demonstrate that QUES is directly related to the circuit fidelity, and the
potential classical hardness of an associated quantum circuit sampling problem.
Under the same assumption, theoretical analysis suggests there exists an
`optimal' simulation time $t^{\text{opt}}\approx 4.81$, at which even a noisy
quantum device may be sufficient to demonstrate the potential classical
hardness.
