Designing energy-efficient digital FIR filters remains a challenge due to the trade-off between computational accuracy and resource utilization. This study addresses this issue, a novel approach to scheming a highly effective digital Finite Impulse Response (FIR) filter for noise signal and image noise removal applications by integrating an optimized Red Piranha Based Approximate Logarithm Posit Multiplier (RP-ALPM) with a Toffoli-Depth Quantum Adder (TDQA) (DF-TDQA-RP-ALPM) is proposed. The RP-ALPM leverages the accuracy and dynamic range advantages of posit arithmetic while utilizing approximate computing to reduce power and area overheads. The integration of a TDQA ensures efficient quantum operations with minimal depth, enhancing the overall computational efficiency of the FIR filter. Extensive simulations demonstrate significant improvements in power consumption, area, and delay, without compromising filter performance. This work offers a promising solution for energy-efficient digital signal processing in noise reduction for audio and image applications in quantum-inspired computing environments. The proposed DF-TDQA-RP-ALPM filters are intended and implemented in the Verilog programming language, in addition to simulation is on Xilinx ISE 14.5 design tackles. The untried performances of the projected DF-TDQA-RP-ALPM filter are predicted to have a lower delay of 10 ns, and 25.4 mW lower power consumption is associated with the existing filters. The projected Filter is triggered in MATLAB/Simulink for understanding input signals. The experimental performances of the proposed DF-TDQA-RP-ALPM filter are calculated as 85.6 % higher SNR is associated with the standing filters.