qml.ops.two_qubit_decomposition¶

two_qubit_decomposition(U, wires)[source]¶

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:

../../_images/two_qubit_decomposition_3_cnots.svg

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 currently not supported and will instead produce a 3-CNOT circuit like above.

For a single 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 can be applied automatically to all two-qubit QubitUnitary operations using the unitary_to_rot() transform.

Warning

This decomposition will not be differentiable in the unitary_to_rot transform 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.

Parameters:

U (tensor) – A \(4 \times 4\) unitary matrix.
wires (Union[Wires, Sequence[int] or int]) – The wires on which to apply the operation.

Returns:

A list of operations that represent the decomposition of the matrix U.

Return type:

list[Operation]

Example

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.ops.two_qubit_decomposition(np.array(U), wires=[0, 1])
>>> decomp
[QubitUnitary(array([[ 0.02867704+0.82548843j,  0.5568274 -0.08769111j],
       [-0.5568274 -0.08769111j,  0.02867704-0.82548843j]]), wires=[0]),
QubitUnitary(array([[ 0.32799033-0.78598401j,  0.40660725+0.33063881j],
       [-0.40660725+0.33063881j,  0.32799033+0.78598401j]]), wires=[1]),
CNOT(wires=[1, 0]),
RZ(0.259291854677022, wires=[0]),
RY(-0.05808874413267284, wires=[1]),
CNOT(wires=[0, 1]),
RY(-1.6742322786950354, wires=[1]),
CNOT(wires=[1, 0]),
QubitUnitary(array([[ 0.91031205-0.21930866j,  0.20674186-0.28371375j],
       [-0.20674186-0.28371375j,  0.91031205+0.21930866j]]), wires=[1]),
QubitUnitary(array([[-0.81886788-0.02979899j,  0.53279787-0.21140919j],
       [-0.53279787-0.21140919j, -0.81886788+0.02979899j]]), wires=[0]),
GlobalPhase(0.1180587403699308, wires=[])]

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qml.ops.two_qubit_decomposition¶

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