Implementation of Quantum Fourier Transform and Quantum Hashing for Quantum Devices with Arbitrary Qubit Connectivity Graphs

In the paper, we consider quantum circuits for quantum fingerprinting (quantum hashing) and quantum Fourier transform (QFT) algorithms. Quantum fingerprinting (quantum hashing) is a well-known technique for comparing large objects using small images. The QFT algorithm is a very popular technique used in many algorithms. We present a generic method for constructing quantum circuits for these algorithms for quantum devices with restrictions. Many quantum devices (e.g., based on superconductors) have restrictions on applying two-qubit gates. The restrictions are presented by a qubit connectivity graph. Typically, researchers consider only the linear nearest neighbor (LNN) architecture, but current devices have more complex graphs. We present a method for arbitrary connected graphs that minimizes the number of CNOT gates in the circuit. The heuristic version of the method is fast enough and works with O(n⁶) time complexity, where n is the number of qubits. The certain version of the algorithm has an exponential time complexity that is O(n²2ⁿ). We compare quantum circuits built by our algorithm with quantum circuits optimized for specific graphs that are Linear-nearest-neighbor (LNN) architecture, “sun” (a cycle with tails, presented by the 16-qubit IBMQ device) and “two joint suns” (two joint cycles with tails, presented by the 27-qubit IBMQ device). Our generic method gives similar results with a few more CNOT gates. At the same time, our method allows us to construct a circuit for arbitrary connected graphs.