Figure 3: Quantum circuit for the BCCKS algorithm in single output duality quantum computing.
From: Duality quantum algorithm efficiently simulates open quantum systems
In Part A of Fig. 3, |Ψ〉 is the initial state of work qubit and there are K numbers of |0〉 auxiliary controlling qubits and Knumbers of L level |0〉L auxiliary controlling qudits in the auxiliary system. K numbers of |0〉 auxiliary controlling qubits control the Knumbers of L level |0〉L auxiliary controlling qudits and the unitary operations U0 are activated only when the L level |0〉L auxiliary controlling qudits hold the respective values indicated in circles. Part B of Fig. 3 is to illustrate that each unitary operation U0 is composed of H1, H2, …, HL−1, HL. We only “readout” the result with the auxiliary system in state 
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