Quantum-state preparation with universal gate decompositions

In quantum computation every unitary operation can be decomposed into quantum circuits—a series of single-qubit rotations and a single type entangling two-qubit gates, such as controlled-not (cnot) gates. Two measures are important when judging the complexity of the circuit: the total number of cnot gates needed to implement it and the depth of the circuit, measured by the minimal number of computation steps needed to perform it. Here we give an explicit and simple quantum circuit scheme for preparation of arbitrary quantum states, which can directly utilize any decomposition scheme for arbitrary full quantum gates, thus connecting the two problems. Our circuit reduces the depth of the best currently known circuit by a factor of $2$. It also reduces the total number of cnot gates from $2^n$ to $\frac{23}{24}{2}^n$ in the leading order for even number of qubits. Specifically, the scheme allows us to decrease the upper bound from $11$ cnot gates to $9$ and the depth from $11$ to $5$ steps for four qubits. Our results are expected to help in designing and building small-scale quantum circuits using present technologies.

Figure 1

Gate sequence for preparation of an arbitrary four-qubit state. Individual one-qubit rotations (not depicted) need to be applied between cnot gates. The four individual phases are described in the text. The two cnot gates in phase 2, as well as in phases 3 and 4, can be performed in parallel, as they address different qubits. Altogether one needs 9 cnot gates, 4 pairs of which can be performed in parallel.