How to transform e.g., a 4 by 4 unitary matrix into a quantum circuit

Question

I am new to Quantum Computing. I would really appreciate if some one could offer me some ideas, suggestions and/or useful references to solve the following problem about realizing an arbitrary operator by a quantum circuit.

If I start with a general two-qubit operator $\hat{U}$ (which corresponds to a 4 by 4 unitary matrix $U$ in the computational basis), what is the strategy to (1) decompose $U$ into the basic gates, such as the $CNOT$ gate, the Hadamard gate $H$, and the single-qubit rotational gates ($R_x$, $R_y$ and $R_z$); (2) design the corresponding quantum circuit to realize this two-qubit operator $\hat{U}$ (either with ancillary qubits or not)?

In other words, what is the strategy (1) to convert the following unitary matrix (two-qubit operator $\hat{U}$ in the computational basis $\{ |00\rangle, \ |01\rangle, \ |10\rangle, \ |10\rangle \}$) \begin{align} U= \begin{pmatrix} a & b & c & d \\ b^{\ast} & f & g & h \\ c^{\ast} & g^{\ast} & i & j \\ d^{\ast} & h^{\ast} & j^{\ast} & k \end{pmatrix} \ , \end{align} into the basic gates (e.g., $CNOT$, $H$, $R_x$, $R_y$ and $R_z$ gates); and (2) to realize the unitary matrix $U$ in a quantum circuit?

Thank you!

Answer 1

Every 2x2 unitary matrix can be decomposed to a matrix product of four matrices expresing rotations and phase shifts.

Every controlled operator (i.e. two qubit gate) can be decomposed to product of three single qubit gates acting on target qubit and two CNOT gates. Single qubit gates can be again decomposed to rotations and phase shifts are mentioned above.

In case your 4x4 matrix can be writen as Kronecker product of two 2x2 matrices, you can again apply decomposition from the first paragraph.

Based on these theorems, you can rewrite any two qubit operators in terms of CNOTs and single qubits rotations and phase shifts. All these gates are available on a quantum computer.

Please see details how to do so in Elementary gates for quantum computing.

Answer 2

A universal method to decompose 2-qubit unitaries into primitive gates is sometimes referred to as "Krauss-Cirac decomposition". Here are several sources:

• Optimal Quantum Circuits for General Two-Qubit Gates by Vatan and Williams,
• Optimal Creation of Entanglement Using a Two–Qubit Gate by Kraus and Cirac.
• “Explorations in Quantum Computing” by Williams, chapter 2.

For gates on multiple qubits, this question offers several more sources.