  This study considers a virtual multiuser multiple-input multiple-output
system with PSK modulation realized via the reconfigurable intelligent
surface-based passive transmitter setup. Under this framework, the study
derives the formulation for the union-bound symbol-error probability, which is
an upper bound on the actual symbol-error probability. Based on this, a
symbol-level precoding power minimization problem under the condition that the
union-bound symbol-error probability is below a given requirement is proposed.
The problem is formulated as a constrained optimization on an oblique manifold,
and solved via a bisection method. The method consists of successively
optimizing transmit power while evaluating the feasibility of the union-bound
symbol-error probability requisite by solving, via the Riemannian conjugate
gradient algorithm, an auxiliary problem dependent only on the reflection
coefficients of the reconfigurable intelligent surface elements. Numerical
results demonstrate the effectiveness of the proposed approach in minimizing
the transmit power for different symbol-error probability requirements.
