We present geometric methods for uniformly discretizing the continuous N-qubit Hilbert space H N . When considered as the vertices of a geometrical figure, the resulting states form the equivalent of a Platonic solid. The discretization technique inherently describes a class of π/2 rotations that connect neighboring states in the set, i.e., that leave the geometrical figures invariant. These rotations are shown to generate the Clifford group, a general group of discrete transformations on N qubits. Discretizing H N allows us to define its digital quantum information content, and we show that this information content grows as N 2. While we believe the discrete sets are interesting because they allow extra-classical behavior—such as quantum entanglement and quantum parallelism—to be explored while circumventing the continuity of Hilbert space, we also show how they may be a useful tool for problems in traditional quantum computation. We describe in detail the discrete sets for one and two qubits.
Quantum Information Processing – Springer Journals
Published: Sep 21, 2004
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