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I've recently started doing some reading on the subject of qudit codes. In particular, i'm interested in the frequently used clock and shift operators.

Can these operators be physically realized? Or, even better, can they be expressed in terms of sums and products of spin operators?

I have found one reference that shows that any 3x3 matrix can be expressed as a (complicated) set of sums and products of the spin-1 operators:

http://arxiv.org/abs/0806.1174

But i'm not sure if this holds for higher spins.

Any ideas?

EDIT: I realize my initial question was slightly misleading. Sorry. I am looking at higher spin generalizations of the Toric code in terms of Qudits, whose hamiltionian is written in terms of these clock and shift operators. So my sense of "physically realizable" would be expressing these operators in terms of something I could find in a condensed matter system.

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  • $\begingroup$ Pardon for my not appreciating the technical language. Any NxN matrix can be represented by the clock and shift matrices of Sylvester (1882), since they span SU(N), or GL(N). Spin matrices are only sparse N-dimensional reps of SU(2). Could you tweak your wording to appeal to a less QI audience? $\endgroup$ – Cosmas Zachos Jan 12 '17 at 20:15
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Yes, they're realizable. All unitary operations on qubits (or qudits) are physically realizable (with negligible error) (unless you're working with an extremely limited gate set).

The 'shift' operator is just adding a constant number to the little-endian number represented by the qubits (as if they were bits). Here's a blog post about efficiently implementing an increment gate when you have one uninvolved bit and only have access to NOT, CNOT, and CCNOT gates.

The 'clock' operator creates a phase gradient, and is an important part of the Quantum Fourier Transform circuit. The clock operator is much simpler to implement than the shift operator, so I can just show it to you:

Clock circuit

You don't even need to have the spins interact! Just rotate one by 180 degrees, the next by 90°, then 45°, then 22.5°, and so forth until you've gone through all the qubits.

(Of course on an actual quantum computer, where you're unlikely to have any gates finer-grained than the 45 degree rotation, you'd have to use constructions to approximate the very small Z gates.)


If you want to play with these operators, my toy in-browser circuit simulator 'Quirk' has built-in gates for incrementing, adding, and making phase gradients:

Circuit with increments and phase gradients

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  • $\begingroup$ Thanks for the detailed reply! Unfortunately I think my question was a bit poorly phrased, i've gone back and edited it. $\endgroup$ – Mdupont Jun 10 '16 at 21:55

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