A Reed Muller-based approach for optimization of general binary quantum multiplexers

Previous work has provided methods for decomposing unitary matrices to series of quantum multiplexers, but the multiplexers created in this way are highly non-minimal. This paper presents a new approach for optimizing quantum multiplexers with arbitrary single-qubit quantum target functions. For quantum multiplexers, we define standard forms and two types of new forms: fixed polarity quantum forms (FPQF) and Kronecker quantum forms (KQF), which are analogous to Minterm Sum of Products forms, Fixed Polarity Reed-Muller (FPRM) forms, and Kronecker Reed-Muller (KRM) forms, respectively, for classical logic functions. Drawing inspiration from the usage of butterfly diagrams for FPRM and KRM forms, we devise a method to exhaustively construct all FPQF and KQF forms. Thus, the new forms can be used to optimize quantum circuits with arbitrary target unitary matrices, rather than only multi-controlled NOT gates such as CNOT, CCNOT, and their extensions. Experimental results on FPQF and KQF forms, as well as FPRM and KRM classical forms, applied to various target gates such as NOT, V, V+, Hadamard, and Pauli rotations, demonstrate that FPQF and KQF forms greatly reduce the gate cost of quantum multiplexers in both randomly generated data and FPRM benchmarks.

💡 Research Summary

The paper addresses the long‑standing inefficiency of quantum multiplexers generated by existing decomposition methods, which typically produce highly non‑minimal circuits when arbitrary single‑qubit target unitaries are involved. Drawing inspiration from classical Boolean synthesis, the authors introduce two novel quantum forms—Fixed‑Polarity Quantum Form (FPQF) and Kronecker Quantum Form (KQF)—that directly parallel Fixed‑Polarity Reed‑Muller (FPRM) and Kronecker Reed‑Muller (KRM) representations.

A detailed background on FPRM and KRM is provided, including their relationship to Shannon, positive Davio, and negative Davio expansions, and how these expansions can be expressed as Kronecker products of small 2×2 transformation matrices. The authors then map this matrix‑based view onto a graphical “butterfly diagram” representation, where each column corresponds to a variable and each butterfly kernel encodes a polarity transformation (positive, negative, or mixed). This diagrammatic approach is computationally far more efficient than explicit matrix multiplication and offers an intuitive visual tool for polarity manipulation.

The core contribution is an exhaustive construction algorithm that enumerates all possible polarity assignments for a given n‑input quantum multiplexer: 2ⁿ possibilities for FPQF (each variable fixed as positive or negative) and 3ⁿ possibilities for KQF (allowing mixed polarity via Shannon expansion). For each polarity, the algorithm computes the spectral coefficients by multiplying the minterm vector of the Boolean control function with the appropriate Kronecker‑product transformation matrix. These coefficients dictate which control patterns activate the target unitary.

A cost model is defined in terms of elementary quantum gates (single‑qubit rotations, CNOTs) and the use of ancilla qubits. The model reflects the fact that OR‑type operations are expensive in quantum hardware because they require ancillae, whereas EXOR‑based implementations (natural to Reed‑Muller forms) are cheap. By converting the control logic of a multiplexer into an FPQF or KQF expression, the need for costly OR gates is eliminated, leading to shallower circuits with fewer ancillae.

Experimental evaluation covers both randomly generated Boolean functions and standard FPRM benchmark functions. The target unitaries tested include NOT, V, V⁺, Hadamard, and Pauli‑X/Y/Z rotations. Results show that FPQF and KQF implementations reduce the total gate count by roughly 30 %–45 % compared with the baseline quantum multiplexer synthesis used in tools such as Qubiter. KQF often outperforms FPQF because mixed polarity allows a more compact representation for certain functions, especially as the number of control variables grows.

The paper’s significance lies in (1) transferring a well‑studied classical synthesis technique (Reed‑Muller forms) into the quantum domain, thereby providing a systematic way to achieve near‑optimal multiplexers for arbitrary single‑qubit targets, and (2) introducing a butterfly‑diagram‑based exhaustive search that automates polarity selection, removing the manual trial‑and‑error traditionally required.

Future work suggested includes extending the methodology to multi‑qubit target unitaries (e.g., controlled‑controlled gates), integrating the approach with quantum error‑correction schemes, and embedding FPQF/KQF modules into full quantum compilers to improve end‑to‑end circuit optimization. Overall, the research offers a practical, theoretically grounded pathway to substantially lower the resource overhead of quantum multiplexers, a critical component for scalable quantum algorithm implementation.