The number of qubits on the ancilla determines the number of V i s and hence the size of V in Eq. (2).
Block circuit diagram to simulate U by modifying the input |0⟩|ψ⟩ to and constructing V in two steps: the formation of the elements of U in V and bringing the same row elements in U to the first rows of V i s in V, combination. The necessary gates to form V and to also transform |0⟩|ψ⟩ to will generate the circuit.
The first circuit design for a given general matrix: the initial Hadamards and the SWAPs are to modify the input, and the last Hadamards carry the elements to the first rows of V i s (combination step). The uniformly controlled quantum gates in the middle form all elements of U on the diagonal of V (formation step).
The second circuit with 4 × 4 initial blocks: The differently controlled quantum gates in the networks, after the V i blocks, combine small blocks and build the N × N blocks at the end. The initial Hadamards are for the modification of the input. The V i blocks are for the formation step.
(a) A gray-coded multi-control network. (b) The decomposition of the gray-coded network in (a) into CNOT and single quantum gates.
The circuit in (a) with 4 × 4 initial blocks can be represented as in (b) by using the circuit given in Fig. (7). Without changing the order of the gates having the same control state, the gates can be moved to form uniformly controlled networks as in (c): If a gate has the same angle value for all control states such as the control X gates in the circuit, they are equal to a single gate (in the case of X gates in the circuit, only one CNOT is required).
Quantum circuit which is found by following the Schmidt decomposition and can generate any vector of dimension 4 as the first row of its matrix representation.
The circuit for the simulation of the hydrogen molecule. The angle values for the rotation gates are determined to create the elements of : There are only 19 rotation gates, the rest are X gates in order to get the right order for the elements after the combination. For diagonal elements of , these rotations are only around z-axis. For nonzero-diagonal elements, rotation about z-axis followed by rotations about y-axis. The angles for these gates are given in Table I.
Parameters for the rotation gates.
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