Parameter-optimal unitary synthesis with flag decompositions

Abstract

We introduce the flag decomposition as a central tool for unitary synthesis. It lets us carve out a diagonal unitary with $2^n$ degrees of freedom in such a way that the remaining flag circuit is parametrized by the optimal number of $4^n-2^n$ rotations. This enables us to produce parameter-optimal quantum circuits for generic unitaries and matrix product state preparation. Our approach improves upon the state of the art, both when compiling down to the {Clifford + Rot} gate set with what we call selective de-multiplexing, and when using phase gradient resource states together with quantum arithmetic to implement multiplexed rotations. All of our synthesis methods are efficiently implementable in terms of recursive Cartan decompositions realized by standard linear algebra routines, making them applicable to all practically relevant system sizes.

Figures (4)

• Figure 1: In this work we discuss the so-called flag decomposition (top left) as a central tool for parameter-optimal unitary synthesis. It decomposes a unitary (magenta) with $4^n$ parameters into a flag circuit (orange) with $4^n-2^n$ parameters and a diagonal (blue) with $2^n$ parameters. Using this tool, we derive parameter-optimal synthesis results for unitaries in general (left column) and MPS matrices (right column). The shown circuit skeletons can then be decomposed further into a target gate set. We consider {Clifford + Rot} decompositions (top row), as well as phase gradient decompositions (bottom row). The multiplexer node with a dot inside refers to a symmetrized Möttönen decomposition; see \ref{['eq:symmetrized_Möttönen']}.
• Figure 2: Phase gradient decomposition of our unitary synthesis circuit. The unitary is first broken down into $(n-1)$-multiplexed single-qubit flags and a trailing diagonal via a recursive flag decomposition (top circuit). Each multiplexed flag can be implemented via loading of the flag angles $\theta$ and $\phi$, adders onto a resource phase gradient state $|\nabla_Z\rangle$ that are control-flipped by the flag target into subtractors, and angle unloading (orange, bottom left). The latter is implemented via measurement in the $X$ basis and a corrective diagonal, which depends on the measurement outcomes and can be merged into subsequent flags (dark green, bottom mid). The terminal diagonal is implemented as a multiplexed $R_Z$ gate on a fixed-state qubit (blue, bottom right); see \ref{['appendix:diagonal']} for details.
• Figure 3: Selective de-multiplexing (SDM) proposed in this work. A unitary operator is decomposed with a single QSD step in a parameter-optimal fashion (\ref{['sec:po_qsd:single_qsd_step']}, top circuit), producing flag circuits, multiplexed rotations, and a smaller unitary. These constituents are decomposed by selectively de-multiplexing the flag circuit (orange, lower left, \ref{['sec:selective_de-mux']}), a (symmetrized) Möttönen decomposition (green, lower left mid, \ref{['eq:symmetrized_Möttönen']}) and a next QSD step using SDM. Ultimately, we use known base case decompositions for two-qubit flags (blue, lower right mid, \ref{['eq:flag_decomp_n-2']}) and two-qubit unitaries (magenta, lower right, \ref{['eq:two-qubit-unitary']}). Multiplexed flags (purple) are decomposed with the recursive flag decomposition/CSD from \ref{['sec:flag_decomposition']} (not shown here).
• Figure 4: Decomposition of matrix product state (MPS) preparation using our unitary synthesis results and further optimizations. For each tensor of the MPS (red), we obtain a multiplexed $R_Y$ gate and a subcircuit representing a reduced multiplexed unitary operator. The exact structure of this subcircuit depends on the target gate set, i.e., on whether we compile for the {Clifford + Rot} decomposition (left) or the phase gradient decomposition (right). The resulting (multiplexed) flags and rotations, as well as the trailing diagonal gate $\Delta$ are then decomposed with the respective techniques from \ref{['fig:three', 'fig:two']}. Note that for the phase gradient decomposition, the unitary degrees of freedom on the auxiliary register are split into a flag circuit and a diagonal on either side.