There is ongoing work to develop quantum computers using qutrits and qubits with multiple states.
A qutrit has three orthonormal basis states or vectors, often denoted , , and in Dirac or bra–ket notation. These are used to describe the qutrit as a superposition state vector in the form of a linear combination of the three orthonormal basis states:
where the coefficients are complex probability amplitudes, such that the sum of their squares is unity (normalization):
The qubit's orthonormal basis states span the two-dimensional complex Hilbert space , corresponding to spin-up and spin-down of a spin-1/2 particle. Qutrits require a Hilbert space of higher dimension, namely the three-dimensional spanned by the qutrit's basis , which can be realized by a three-level quantum system. However, not all three-level quantum systems are qutrits.
Qutrits have several peculiar features when used for storing quantum information. For example, they are more robust to decoherence under certain environmental interactions. In reality, manipulating qutrits directly might be tricky, and one way to do that is by using an entanglement with a qubit.
Qutrit quantum gates
The quantum logic gates operating on single qutrits are unitary matrices and gates that act on registers of qutrits are unitary matrices (the elements of the unitary groups U(3) and U(3n) respectively).
The rotation operator gates[a] for SU(3) are , where is the a'th Gell-Mann matrix, is a real value (with period ), and . The Lie algebra of the matrix exponential is provided here. The same rotation operators are used for gluon interactions, where the three basis states are the three colors () of the strong interaction.[b]
- This can be compared with the three rotation operator gates for qubits. We get eight linearly independent rotation operators by selecting appropriate . For example, we get the 1st rotation operator for SU(3) by setting and all others to zero.
- Note: Quarks and gluons have color charge interactions in SU(3), not U(3), meaning their color charge can not have global phase. If they could have global phase, it would mean that there would be a 9th gluon, but there is only 8. Qutrits can however have global phase.
- Comparable with the global phase shift gate for qubits.
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