Design of Quantum Circuits for Cryptanalysis and Image Processing Applications

Quantum circuits for arithmetic functions over Galois fields such as squaring are required to implement quantum cryptanalysis algorithms. Quantum circuits for integer arithmetic such as multiplication are required to implement scientific computing algorithms and quantum image processing algorithms on quantum computers. Reliable quantum circuits require error correcting codes and gates that are fault tolerant in nature. Quantum circuits of many qubits are challenging to implement making designs with low qubit cost desirable. In this work, we present quantum arithmetic circuits for applications in quantum cryptanalysis and quantum image processing. We present a proposed algorithm for synthesizing gate cost, qubit cost and depth optimized Galois field (GF(2^m)) squaring circuits for quantum cryptanalysis applications. In addition, these squaring circuits are incorporated into a proposed quantum circuit for inversion in GF(2^m). This work also presents a proposed quantum integer conditional addition circuit and a quantum integer multiplication circuit optimized for T-count and qubit cost. The quantum conditional addition circuit and quantum multiplier are incorporated into proposed quantum circuits for bilinear interpolation optimized for T-count cost that can be used in quantum image processing applications.