A Logarithmic-Depth Quantum Multiplier

We present a quantum algorithm for multiplying two n-bit integers with overall circuit depth and T-depth both bounded by O(n), while using O(n 2) gates and ancillary qubits. Our construction generates partial products via indicator-controlled copying and adds them using a tree of reversible 3-to- 2 adders followed by one quantum carry-lookahead adder. Building on the carry-save adders of Kim et al., we combine full input fanout, workspace reuse, and phase-corrected gate implementations to reduce T-depth. We provide explicit circuit constructions, proofs of correctness, and concrete closed-form resource bounds.