Quantum Circuit Design for Decoded Quantum Interferometry

Decoded Quantum Interferometry (DQI) is a recently proposed quantum algorithm for approximating solutions to combinatorial optimization problems by reducing instances of linear satisfiability to bounded-distance decoding over superpositions of quantum states. A central challenge in realizing DQI is the design of a decoder that operates coherently on quantum superpositions. In this work, we present a concrete quantum circuit implementation of DQI, with a focus on the decoding subroutine. Our design leverages a reversible Gauss-Jordan elimination circuit for the decoding stage. We analyze the circuit's depth and gate complexity and validate its performance through simulations on systems with up to 30 qubits. These results establish a concrete foundation for scalable implementations of DQI and open the door to future algorithmic refinements and hardware-level realizations.