“…Duality quantum algorithm was proposed in 2002 [ 71 ] for the first time, and it developed fast [ 72 , 73 , 74 , 75 ]. Because both the products of unitary operations and linear combinations of unitary (LCU) operations can be realized by duality quantum algorithm, it has a lot of applications to investigate novel quantum systems [ 16 , 17 , 18 , 19 , 20 , 21 , 76 , 77 , 78 , 79 , 80 ], e.g., systems of large-scale silicon quantum photonics, efficient quantum simulations of open quantum systems, quantum secure computing, passive quantum error correction, etc. Duality quantum algorithm has become one of the strongest tool in designing quantum algorithms [ 81 ].…”