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Foundations of Physics

The Algebraic Structure of an Approximately Universal System of Quantum Computational Gates

María Luisa Dalla Chiara, Roberto Giuntini, Héctor Freytes, Antonio Ledda, Giuseppe Sergioli

Published Mar 31, 2009• 17 citations
Publisher
DOI
10.1007/s10701-009-9302-0

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Shi and Aharonov have shown that the Toffoli gate and the Hadamard gate give rise to an approximately universal set of quantum computational gates. We study the basic algebraic properties of this system by introducing the notion of Shi-Abaronov quantum computational structure. Finally, we investigate the possibility of an algebraic abstraction from Hilbert-space structures, by proposing the notion of Toffoli-Hadamard algebra. From an intuitive point of view, such abstract algebras represent a natural quantum generalization of both classical and fuzzy-like structures.
Universal Computation via T-Gate Synthesis in S4 Subgroups We investigate the computational complexity class of the finite monoid M_ext. We demonstrate that the algebraic structure supports a framework for universal quantum computation. 1. Clifford Group Mapping: The base structure (S4, Order 24) is shown to be isomorphic to the Clifford group operations acting on qubits. 2. Universal Resource: The symmetry-breaking boundary condition provides the necessary non-Clifford resource (analogous to Magic State injection or T-Gate) required for universality. 3. Turing Completeness: The braiding of algebraic generators allows for the approximation of unitary target gates, defining a complete computational model.
Any quantum computational network can be constructed with a sequence of the two-qubit diagonal quantum gates and one-qubit gates in two-state quantum systems. The universal construction of these quantum gates in the quantum systems and of the quantum computational networks with these gates may be achieved with the help of the operator algebra structure of Hamiltonians of the systems and the properties of the multiple-quantum operator algebra subspaces of the Liouville operator space and the specific properties of the quantum algorithm corresponding to the quantum network. As an example, the two-qubit diagonal gates are exactly prepared in detail in superconducting Josephson junctions.