Geometry of Quantum Computation with Qutrits

Abstract

Determining the quantum circuit complexity of a unitary operation is an important problem in quantum computation. By using the mathematical techniques of Riemannian geometry, we investigate the efficient quantum circuits in quantum computation with $n$ qutrits. We show that the optimal quantum circuits are essentially equivalent to the shortest path between two points in a certain curved geometry of $SU(3^n)$. As an example, three-qutrit systems are investigated in detail.

14 pages

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• Quantum Computation as Geometry
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• Geometrical, topological and dynamical description of $\mathcal{N}$ interacting spin-$\mathtt{s}$ under long-range Ising model and their interplay with quantum entanglement
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• Neutrinos as qubits and qutrits
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• Efficient characterization of qudit logical gates with gate set tomography using an error-free Virtual-Z-gate model
• Geometry and speed of evolution for a spin-s system with long-range zz-type Ising interaction
• Polynomial Equivalence of Complexity Geometries
• Exact solutions for time-optimal control of spin I=1 by NMR
• Geometry of quantum state manifolds generated by the Lie algebra operators
• An Inequality for Entangled Qutrits in SU(3) basis
• Qudit low-density parity-check codes
• Time Evolution Operators for Periodic SU(3)
• Physically motivated decompositions of single-qutrit gates
• Measuring distance between quantum states on a quantum computer
• Gaussian Amplitude Amplification for Quantum Pathfinding
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