Arbitrary d -dimensional Pauli X gates of a flying qudit

Abstract

High-dimensional degrees of freedom of photons can encode more quantum information than their two-dimensional counterparts. While the increased information capacity has advantages in quantum applications (such as quantum communication), controlling and manipulating these systems has been challenging. Here we show a method to perform lossless arbitrary high-dimensional Pauli-X gates for single photon. The X-gate consists of a cyclic permutation of qudit basis vectors, and, together with the Z gate, forms the basis for performing arbitrary transformations. We propose an implementation of such gates on the orbital angular momentum of photons. The proposed experimental setups only use two basic optical elements such as mode-sorters and mode-shiftersthus could be implemented in any system where these experimental tools are available. Furthermore the number of involved interferometers scales logarithmically with the dimension, which is important for practical implementation.

High-dimensional quantum gates using full-field spatial modes of photons

“…which simply corresponds to a cyclic operation where each mode gets transformed to its m-th nearest neighbour-mode, modulo the number of modes d. Interestingly, it was shown only recently and only for the OAM degree of freedom that this operation can be implemented for arbitrary dimensions using a complex arrangement of linear optical elements and free-space propagation [48]. For full-field modes, i.e.…”

“…C. High-dimensional gates for radial modes While the OAM transformations performed above can be (at least in theory) realised using bulk optical elements [48], we now turn to mode transformations of radial modes, i.e. p-modes, a task for which no other implementation is known so far.…”

“…a cyclic permutation of the input modes [46,47]. It has also been shown how to extend this to arbitrary dimensions using only linear optical elements arranged in complex interferometric setups and free-space propagation [48]. Thus, together with a mode-dependent phase operation (simply performed by a Dove prism) any unitary operation on the OAM subspace could be performed, at least theoretically.…”

Parallelization of high-dimensional single-photon quantum gates

“…In the scheme, the use is made of O(d 2 ) beam splitters, O(d 2 ) phase shifters, O(d) Dove prisms, and O(d) holograms as can be deduced from the structure of the OAM sorter [26,27] and the Reck et al scheme [14].…”

Outcome determinism in measurement-based quantum computation with qudits

“…The added flexibility of increased dimension allows, for example, for shorter circuits in computation [Kik+20], asymptotic improvement in circuit depths [Gok+19], optimal error correcting codes [CGL99] and noise tolerance in quantum key distribution [BT00;Cer+02]. Furthermore many physical systems exist which naturally encode qudits [BW08;Erh+18;Gao+19]. This has motivated the translation of MBQC into qudits [Zho+03], which naturally leads to the question: can the flow techniques above be extended to the qudit setting?…”