Select-U(2) Decomposition

Widely-used building blocks in quantum algorithms are Select-applied U(2) operators, i.e. multiplexers with a single target qubit. These apply a different operator to the target qubit for each computational basis state of the control qubits, which is commonly visualized like this:

An important special case is that of uniformly controlled Pauli rotations, for which each applied operator is a rotation by a different angle but about the same axis (Pauli X, Y or Z). It is implemented as qml.SelectPauliRot in PennyLane and represented as:

Note that there are also more generic optimization techniques for Select operators with multiple target qubits, for example partial Select or lazy Select.

Inputs

• Select-applied, or multiplexed, single-qubit operator, or, as a special case,
• Select-applied, or multiplexed, Pauli rotation.

Outputs

• Decomposed circuit for the multiplexer.

Example

For a multiplexed Pauli Z rotation, Shende et al. [1] and Möttönen et al. [2] show the following decomposition, displayed exemplarily for k=3 control qubits:

0: ─╭◻──┤ =  ──────────────────────╭●──────────────────────╭●─┤
1: ─├◻──┤    ──────────╭●──────────│───────────╭●──────────│──┤
2: ─├◻──┤    ────╭●────│─────╭●────│─────╭●────│─────╭●────│──┤
3: ─╰RZ─┤    ─RZ─╰X─RZ─╰X─RZ─╰X─RZ─╰X─RZ─╰X─RZ─╰X─RZ─╰X─RZ─╰X─┤,

which uses 2^k CNOT gates and single-qubit rotations each. The same decomposition applies for R_Y rotations, simply by replacing the single-qubit rotations accordingly. For R_X rotations, the entangling gates have to be replaced as well, for example, by CZ gates.

For Select-U(2) operators, Bergholm et al. [3] present the following decomposition, again shown for k=3:

0: ─╭◻─┤ =  ──────────────────────╭●──────────────────────────────╭RZ─┤
1: ─├◻─┤    ──────────╭●──────────│───────────╭●──────────────╭RZ─├◻──┤
2: ─├◻─┤    ────╭●────│─────╭●────│─────╭●────│─────╭●────╭RZ─├◻──├◻──┤
3: ─╰U─┤    ─U0─╰X─U2─╰X─U2─╰X─U3─╰X─U4─╰X─U5─╰X─U6─╰X─U7─╰◻──╰◻──╰◻──┤,

where the trailing multiplexed R_Z rotations are decomposed with the rule from above, and resemble the decomposition of a diagonal unitary operator. Overall, this decomposition uses 3(2^k -1) CNOT gates.

Typical usage

This decomposition allows to break down multiplexers of arbitrary size into one- and two-qubit operators in order to arrive at hardware-compatible circuit instructions. Therefore, it is part of a key step towards lower-level representations of a quantum algorithm.

An important related routine is Mottonen state preparation [2], implemented as MottonenStatePreparation in PennyLane.

References

[1] "Synthesis of quantum-logic circuits", Vivek V. Shende, Stephen S. Bullock, Igor L. Markov, quant-ph/0406176, 2004.

[2] "Transformation of quantum states using uniformly controlled rotations", Mikko Möttönen, Juha J Vartiainen, Ville Bergholm, Martti M Salomaa, quant-ph/0407010, 2004.

[3] "Quantum circuits with uniformly controlled one-qubit gates", Ville Bergholm, Juha J Vartiainen, Mikko Möttönen, Martti M Salomaa, quant-ph/0410066, 2004.

@misc{PennyLane-SelectU2Decomp,
title={Select-U(2) Decomposition},
howpublished={\url{https://pennylane.ai/compilation/select-u2-decomp}},
year={2025}
}