  The paradigm of variational quantum classifiers (VQCs) encodes
\textit{classical information} as quantum states, followed by quantum
processing and then measurements to generate classical predictions. VQCs are
promising candidates for efficient utilization of a near-term quantum device:
classifiers involving $M$-dimensional datasets can be implemented with only
$\lceil \log_2 M \rceil$ qubits by using an amplitude encoding. A general
framework for designing and training VQCs, however, has not been proposed, and
a fundamental understanding of its power and analytical relationships with
classical classifiers are not well understood. An encouraging specific
embodiment of VQCs, quantum circuit learning (QCL), utilizes an ansatz: it
expresses the quantum evolution operator as a circuit with a predetermined
topology and parametrized gates; training involves learning the gate parameters
through optimization. In this letter, we first address the open questions about
VQCs and then show that they, including QCL, fit inside the well-known kernel
method. Based on such correspondence, we devise a design framework of efficient
ansatz-independent VQCs, which we call the unitary kernel method (UKM): it
directly optimizes the unitary evolution operator in a VQC. Thus, we show that
the performance of QCL is bounded from above by the UKM. Next, we propose a
variational circuit realization (VCR) for designing efficient quantum circuits
for a given unitary operator. By combining the UKM with the VCR, we establish
an efficient framework for constructing high-performing circuits. We finally
benchmark the relatively superior performance of the UKM and the VCR via
extensive numerical simulations on multiple datasets.
