Improved quantum ternary arithmetic

Author:

Bocharov Alex,Cui Shawn X.,Roetteler Martin,Svore Krysta M.

Abstract

Qutrit (or ternary) structures arise naturally in many quantum systems, notably in certain non-abelian anyon systems. We present efficient circuits for ternary reversible and quantum arithmetics. Our main result is the derivation of circuits for two families of ternary quantum adders. The main distinction from the binary adders is a richer ternary carry which leads potentially to higher resource counts in universal ternary bases. Our ternary ripple adder circuit has a circuit depth of O(n) and uses only 1 ancilla, making it more efficient in both, circuit depth and width, when compared with previous constructions. Our ternary carry lookahead circuit has a circuit depth of only O(log n), while using O(n) ancillas. Our approach works on two levels of abstraction: at the first level, descriptions of arithmetic circuits are given in terms of gates sequences that use various types of non-Clifford reflections. At the second level, we break down these reflections further by deriving them either from the two-qutrit Clifford gates and the non-Clifford gate C(X) : |i, ji 7→ |i, j + δi,2 mod 3i or from the two-qutrit Clifford gates and the non-Clifford gate P9 = diag(e −2π i/9 , 1, e 2π i/9 ). The two choices of elementary gate sets correspond to two possible mappings onto two different prospective quantum computing architectures which we call the metaplectic and the supermetaplectic basis, respectively. Finally, we develop a method to factor diagonal unitaries using multi-variate polynomials over the ternary finite field which allows to characterize classes of gates that can be implemented exactly over the supermetaplectic basis.

Publisher

Rinton Press

Subject

Computational Theory and Mathematics,General Physics and Astronomy,Mathematical Physics,Nuclear and High Energy Physics,Statistical and Nonlinear Physics,Theoretical Computer Science

Cited by 6 articles. 订阅此论文施引文献 订阅此论文施引文献,注册后可以免费订阅5篇论文的施引文献,订阅后可以查看论文全部施引文献

1. Exact Synthesis of Multiqutrit Clifford-Cyclotomic Circuits;Electronic Proceedings in Theoretical Computer Science;2024-08-12

2. Quantum circuit model for discrete-time three-state quantum walks on Cayley graphs;Physical Review A;2024-07-23

3. Building Qutrit Diagonal Gates from Phase Gadgets;Electronic Proceedings in Theoretical Computer Science;2023-11-16

4. Scaling W State Circuits in the qudit Clifford Hierarchy;Companion Proceedings of the 7th International Conference on the Art, Science, and Engineering of Programming;2023-03-13

5. Quantum indistinguishability by path identity and with undetected photons;Reviews of Modern Physics;2022-06-21

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