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arXiv · 2510.01455

Visualizing the state space and transformations of higher order quantum logics via toric geometry

Steven Bleiler·Shanyan Chen·Emma O'Neil·J. Eliot Reich·Julia Rezvani·Elijah Whitham-Powell·Ali Al-Bayaty·Jerzy Jegier·Sonia Yang·Marek Perkowski
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Abstract

We propose some new uses of toric variety structures in the study of quantum computation for small radices. In particular, we observe the concurrence of the equivalence classes of quantum states under quantum measurement and the orbits of the toric geometric structure of the state space. Visualizations of these state spaces and of certain fundamental unitary transformations in binary and ternary quantum logic and a method to develop new transformations based on these visualization techniques are presented. Transformations discussed included minimal universal sets for permutative ternary quantum circuits. In addition, general structures and synthesis methods based on quantum multiplexers are presented. A general framework for the design of optimal ternary quantum transformations and circuits is additionally presented. Finally, a number of open research areas that are extensions of the work presented herein are given.

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Steven Bleiler, Shanyan Chen, Emma O'Neil, J. Eliot Reich, Julia Rezvani, Elijah Whitham-Powell, Ali Al-Bayaty, Jerzy Jegier, Sonia Yang, Marek Perkowski. 2025-10-01. Visualizing the state space and transformations of higher order quantum logics via toric geometry. https://arxiv.org/abs/2510.01455

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