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arXiv:1902.06229

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

Kevin Jin, Tahsin Saffat, Marek A. Perkowski

Published Feb 17, 2019• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.1902.06229

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Previous work has provided methods for decomposing unitary matrices to series of quantum multiplexers, but the multiplexer circuits created in this way may be highly non-minimal. This paper presents a new approach for optimizing quantum multiplexers with arbitrary single-qubit quantum target functions and ternary controls. For multivalued quantum multiplexers, we define standard forms and two types of new forms: Fixed Polarity Quantum Forms (FPQFs) and Kronecker Quantum Forms (KQFs). Drawing inspiration from the usage of butterfly diagrams, we devise a method to exhaustively construct new forms. In contrast to previous butterfly-based methods, which are used with classical Boolean functions, these new forms are used to optimize quantum circuits with arbitrary target unitary matrices. Experimental results on the new forms applied to various target gates such as NOT, V, V+, Hadamard, and Pauli rotations, demonstrate that these new forms greatly reduce the gate costs of ternary quantum multiplexers.
This paper introduces a new logic structure for reversible quantum circuit synthesis. Our synthesis method aims to minimize the quantum cost of reversible quantum circuits with decoders. In this method, multi-valued input, binary output (MVI) functions are utilized as a mathematical concept only, but the circuits are binary. We introduce the new concept of ``Multi-Valued Input Fixed Polarity Reed-Muller (MVI-RM)" forms. Our decoder-based circuit uses three logical levels in contrast to commonly-used methods based on Exclusive-or Sum of Products (ESOP) with two levels (AND-XOR expressions), realized by Toffoli gates. In general, the high number of input qubits in the resulting Toffoli gates is a problem that greatly impacts the quantum cost. Using decoders decreases the number of input qubits in these Toffoli gates. We present two practical algorithms for three-level circuit synthesis by finding the MVI-FPRM: products-matching and the newly developed butterfly diagrams. The best MVI-FPRM forms are factorized and reduced to approximate Multi-Valued Input Generalized Reed-Muller (MVI-GRM) forms.
This paper presents a new approach to optimize arbitrary quantum circuits based on multi-valued Quantum Multiplexers. We define standard, and fixed polarity forms for binary valued quantum multiplexers that are analogous to the disjoint sum of product and Fixed Polarity Reed-Muller Forms for classical logic functions. Then, the method is extended to logic with an arbitrary radix. The algorithm produced requires O(mqm) butterfly transformations, where m is the number of control variables, and q is the radix of logic. A software script is then added and described to facilitate in the computation of larger and more complex quantum multiplexers.
Paper introduces a new spectral expansion of ternary-input binary-output functions that generalizes the binary Kronecker-Reed-Muller forms. Two binary Davio Expansions are generalized to 27 Ternary-Input Davio Expansions. New KRM spectrum has 28nexpansions for n ternary variables. By creating an oracle for this problem we generalize the quantum Grover-based algorithms presented for the binary FPRM and KRM forms in the past. Because the method finds solutions also to incompletely specified functions, it can be used to both quantum circuit design and Machine Learning classifier design.
Optimization problems in machine-learning (ML) applications can be computationally challenging for classical methods, particularly when they involve large, unstructured search spaces. Quantum computing offers a promising approach to combinatorial optimization by utilizing quantum superposition and amplitude amplification. This paper presents a new methodology using logic-based quantum machine learning (QML) as a complete framework for employing a multi-solution ternary Grover’s algorithm to minimize incomplete binary functions represented by Pseudo-Kronecker Reed–Muller (PKRO) expansions, consistent with Occam’s razor principle. Unlike previous quantum approaches based on Kronecker Reed–Muller (KRO) or fixed-polarity Reed–Muller (FPRM) representations, our work introduces the first Grover-based optimization framework for PKRO forms. This framework formulates logic minimization as a quantum search problem to identify minimum-cost AND-XOR representations with the fewest nonzero coefficients. A hybrid binary–ternary quantum architecture is developed to explore the enlarged PKRO search space, enabling optimization of both completely and incompletely specified binary functions. Owing to the greater flexibility of PKRO expansions, our framework produces more compact AND–XOR representations than KRO- and FPRM-based approaches. Across all 256 3-variable binary functions, PKRO provides less nonzero coefficients for 4.688% of functions compared with KRO and for 37.5% compared with FPRM, with average reductions of 0.047 and 0.484 nonzero coefficients, respectively. Our multi-solution ternary Grover search reduces the number of Grover iterations by more than 91% compared with the single-solution approach.