Tensor networks in physics can be traced back to a 1971 paper by Penrose (1). Such network diagrams appear in digital circuit theory, and they form the foundations of quantum computing—starting with the work of Feynman and others in the 1980s (2) and further extended by Deutsch in his ’’quantum computational network model’’ (3). Building on a series of results (4⇓–6), Liu et al. (7) recently developed a topological variant of tensor networks that, among other results, led to their discovery of an elegant charged string braiding for the controlled NOT gate (also known as the Feynman gate).
Fig. 1.
Lafont’s 2003 categorical model of quantum circuits included the bialgebra ( D ) and Hopf ( G ) relations between the building blocks needed to form a controlled NOT gate. The practical utility of using Baez–Dolan † categories (10) to describe quantum circuits (6, 9) is that category theory provides a graphical language that fully dictates the types of admissible relationships and transformations to reason about the interaction of network components. ( A ) Associativity; ( B ) gate unit laws; ( C ) symmetry; ( E ) copy laws; ( F ) unit scalar given as a blank on the page. Redrawn from ref. 9.
Category theory is a branch of mathematics well suited to describe a wide range of networks (8). Quantum circuits were first given a ’’categorical model’’ in pioneering work by Lafont in 2003 (9), and dagger compact closed categories (10), also called Baez–Dolan † categories, were first derived to describe standard quantum theory as well as classes of topological quantum field theories in seminal work published in 1995 (10). (See ref. 8 for a well-written review of categorical quantum mechanics.) Liu et al. (7) formulated their topological model, in part, using category theory. For practical purposes, the graphical language turns out to be mathematically equivalent to …
[↵][1]1Email: jacob.biamonte{at}qubit.org.
[1]: #xref-corresp-1-1
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