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In general, all the quantum algorithms which I have read so far use qubits (so the space is $\mathbb{C}^2$) and the tensor products of the qubit spaces (space is ${\mathbb{C}^2}^{\otimes n}$). So my question is, are we going to get any advantage if we take space of other dimensions. For instance, what if we do the similar works with qutrits (and things in higher dimensions as basic units).

My guess is that, it may have some advantages for error correcting codes. Can somebody point out some reference in this direction (coding) or in something else, of which I am not sure or.

Further, are we going to get any advantage from the point of experimental realization? I am form Maths and I do not have much idea regarding this. Advanced thanks for any help/ suggestion/references.

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3 Answers 3

Using higher dimension (d>2) quantum systems, or qudits, indeed provides an advantage through greater control of the Hilbert space. In quantum key distribution (QKD) for example, qudits can enhance the average raw key rates as one can encode more bits per symbol. Even more, they improve the robustness or the noise tolerance of the QKD protocol -- Alice and Bob can distill a secret key in (noisier) conditions where using only qubits would have failed.

Using qutrits, one can address fundamental problems such as the Byzantine Agreement, enhance the efficiency of Toffoli gates, boost the classical channel capacity of a quantum channel (i.e. demonstrate the idea of superadditivity), etc.

I am not sure if I correctly understand the part about the 'advantage from the point of experimental realization', but what may be stated is that producing qudits is almost invariably more difficult than qubits. Therefore, cases where one may obtain an advantage because the experimental setup is easier to build or is somehow naturally more suited to producing and manipulating qudits than qubits are rather rare. In that sense, the total cost of a qutrit-based implementation & operation may not always surpass that of its lower-dimensional version (i.e. with qubits).

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Presently there is no known general argument to say wehther some qudit dimensions are better than others for implementing fault tolerant quantum computation schemes. (I know of no paper showing something like that.)

However, it is true that sometimes you can gain something by using qudits. (Meaning that some particular codes work better if you use qudits.)

In particular, there is an interesting paper by Campbell, Anwar and Browne where the authors show that for some particular families of quantum error correcting codes$^1$, a quite popular method to implement fault-tolerant quantum computations (known as Magic State Distillation) becomes more efficient if you are using qudits have certain dimensions: namely, they show that the fault-tolerant protocol they study performs better when the qudits have dimension five. (Several figures of merit are carefully compared in the paper.)

$^1$ These are Reed-Müller codes with with transversal non-Clifford gates.

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To add to the answers above, an advantage to the quantum information theorist is that finite fields of ODD prime characteristic have nice properties, hence when using tools such as discrete phase spaces, qutrits can have properties that are otherwise hard to generalize to qubits.

For example see arXiv:quant-ph/0602001.

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