  We propose an approach to approximately implement the variational coupled
cluster (VCC) theory on quantum computers, which struggles with exponential
scaling of computational costs on classical computers. To this end, we employ
expanding the exponential cluster operator using Chebyshev polynomials and
introduce two methods: the Chebyshev approximated VCC (C$^d$-VCC) and the
Hermitian-part Chebyshev approximated VCC (HC$^d$-VCC), where $d$ indicates the
maximum degree of the Chebyshev polynomials. The latter method decomposes the
cluster operator into anti-Hermitian and Hermitian parts, with the
anti-Hermitian part represented by the disentangled unitary coupled cluster
ansatz and the Hermitian part approximated using Chebyshev expansion. We
illustrate the implementation of the HC$^d$-VCC in a quantum circuit using the
quantum singular value transformation technique. Numerical simulations show
that the C$^d$-VCC rapidly converges to the exact VCC with increasing
truncation degree $d$, and the HC$^d$-VCC effectively reduces the Chebyshev
expansion error compared to the C$^d$-VCC. The HC$^d$-VCC method for realizing
non-unitary coupled cluster wave functions on quantum computers is expected to
be useful for initial state preparation on quantum computers and efficient
tomography of the quantum state for post-processing on classical computers
after quantum computations.
