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. We propose a drastically simplified quantum circuit called the minimal QSVT circuit, which uses only one ancilla qubit to simulate a class of $n$-qubit random Hamiltonians. 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. We demonstrate that QUES is directly related to the circuit fidelity, and the classical hardness of an associated quantum circuit sampling problem. Theoretical analysis suggests under suitable assumptions, 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 classical hardness.
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