Some remarks on arithmetic physics

Yvon Gauthier

Michael Kirkedal Thomsen, Robert Glück, Holger Bock Axelsen

This communication presents the complete design of a reversible arithmetic logic unit (ALU) that can be part of a programmable reversible computing device such as a quantum computer. The presented ALU is garbage free and uses reversible updates to combine the standard reversible arithmetic and logical operations in one unit. Combined with a suitable control unit, the ALU permits the construction of an r-Turing complete computing device. The garbage-free ALU developed in this communication requires only 6 n elementary reversible gates for five basic arithmetic–logical operations on two n -bit operands and does not use ancillae. This remarkable low resource consumption was achieved by generalizing the V-shape design first introduced for quantum ripple-carry adders and nesting multiple V-shapes in a novel integrated design. This communication shows that the realization of an efficient reversible ALU for a programmable computing device is possible and that the V-shape design is a very versatile approach to the design of quantum networks.

Gil Jannes

Siyi Wang, Xiufan Li, Wei Lee +3 more

In recent decades, the field of quantum computing has experienced remarkable progress. This progress is marked by the superior performance of many quantum algorithms compared with their classical counterparts, with Shor's algorithm serving as a prominent illustration. Quantum arithmetic circuits, which are the fundamental building blocks in numerous quantum algorithms, have attracted much attention. Despite extensive exploration of various designs in the existing literature, researchers remain keen to develop novel designs and improve existing ones. In this review article, we aim to provide a systematically organized and easily comprehensible overview of the current state of the art in quantum arithmetic circuits. Specifically, this study covers fundamental operations such as addition, subtraction, multiplication, division and modular exponentiation. We delve into the detailed quantum implementations of these prominent designs and evaluate their efficiency considering various objectives. We also discuss potential applications of the presented arithmetic circuits and suggest future research directions.This article is part of the theme issue 'Emerging technologies for future secure computing platforms'.

Siyi Wang, Anupam Chattopadhyay

Craig Edwin Holdway

We isolate a self-contained three-frequency effective subsystem and prove a uniform upper bound relating the complementary return amplitude to the interior coherence factor. The suppression coefficients 1/320, 1/318, 1/316 form an arithmetic family with common difference -2. The central result is the bound |A_return(t)| ≤ (1/318)|R_int(t)| + 1/50244, holding for all t ∈ ℝ under unit normalization. A corollary establishes that when at least two fibres are occupied, and frequencies are distinct, dephasing forces the return amplitude to remain small at generic post-coherence times, up to isolated recurrences. An interpretive remark notes that any added decoherence mechanism would convert bounded return into sustained suppression. The theorem is stated as pure mathematics: no physical interpretation is assumed.