Decompose a two-qubit unitary \(U\) in terms of elementary operations.
It is known that an arbitrary two-qubit operation can be implemented using a maximum of 3 CNOTs. This transform first determines the required number of CNOTs, then decomposes the operator into a circuit with a fixed form. These decompositions are based a number of works by Shende, Markov, and Bullock (1), (2), (3), though we note that many alternative decompositions are possible.
For the 3-CNOT case, we recover the following circuit, which is Figure 2 in reference (1) above:
where \(A, B, C, D\) are \(SU(2)\) operations, and the rotation angles are computed based on features of the input unitary \(U\).
For the 2-CNOT case, the decomposition is
For 1 CNOT, we have a CNOT surrounded by one \(SU(2)\) per wire on each side. The special case of no CNOTs simply returns a tensor product of two \(SU(2)\) operations.
This decomposition will not be differentiable in the
unitary_to_rottransform if the matrix being decomposed depends on parameters with respect to which we would like to take the gradient. See the documentation of
unitary_to_rot()for explicit examples of the differentiable and non-differentiable cases.
U (tensor) – A \(4 \times 4\) unitary matrix.
wires (Union[Wires, Sequence[int] or int]) – The wires on which to apply the operation.
A list of operations that represent the decomposition of the matrix U.
- Return type
Suppose we create a random element of \(U(4)\), and would like to decompose it into elementary gates in our circuit.
>>> from scipy.stats import unitary_group >>> U = unitary_group.rvs(4) >>> U array([[-0.29113625+0.56393527j, 0.39546712-0.14193837j, 0.04637428+0.01311566j, -0.62006741+0.18403743j], [-0.45479211+0.25978444j, -0.52737418-0.5549423j , -0.23429057+0.10728103j, 0.16061807-0.21769762j], [-0.4501231 +0.04065613j, -0.25558662+0.38209554j, -0.04143479-0.56598134j, 0.12983673+0.49548507j], [ 0.23899902+0.24800931j, 0.03374589-0.15784319j, 0.24898226-0.73975147j, 0.0269508 -0.49534518j]])
We can compute its decompositon like so:
>>> decomp = qml.transforms.two_qubit_decomposition(np.array(U), wires=[0, 1]) >>> decomp [Rot(tensor(-1.69488788, requires_grad=True), tensor(1.06701916, requires_grad=True), tensor(0.41190893, requires_grad=True), wires=), Rot(tensor(1.57705621, requires_grad=True), tensor(2.42621204, requires_grad=True), tensor(2.57842249, requires_grad=True), wires=), CNOT(wires=[1, 0]), RZ(0.4503059654281863, wires=), RY(-0.8872497960867665, wires=), CNOT(wires=[0, 1]), RY(-1.6472464849278514, wires=), CNOT(wires=[1, 0]), Rot(tensor(2.93239686, requires_grad=True), tensor(1.8725019, requires_grad=True), tensor(0.0418203, requires_grad=True), wires=), Rot(tensor(-3.78673588, requires_grad=True), tensor(2.03936812, requires_grad=True), tensor(-2.46956972, requires_grad=True), wires=)]