Cyclic permutations for qudits in d dimensions

One of the main challenges in quantum technologies is the ability to control individual quantum systems. This task becomes increasingly difficult as the dimension of the system grows. Here we propose a general setup for cyclic permutations Xd in d dimensions, a major primitive for constructing arbitrary qudit gates. Using orbital angular momentum states as a qudit, the simplest implementation of the Xd gate in d dimensions requires a single quantum sorter Sd and two spiral phase plates. We then extend this construction to a generalised Xd(p) gate to perform a cyclic permutation of a set of d, equally spaced values {|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\boldsymbol{\ell }}}_{{\bf{0}}}$$\end{document}ℓ0〉, |\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\boldsymbol{\ell }}}_{{\bf{0}}}$$\end{document}ℓ0 + p〉, …, |\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\boldsymbol{\ell }}}_{{\bf{0}}}$$\end{document}ℓ0 + (d − 1)p〉} \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mapsto $$\end{document}↦ {|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\boldsymbol{\ell }}}_{{\bf{0}}}$$\end{document}ℓ0 + p〉, |\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\boldsymbol{\ell }}}_{{\bf{0}}}$$\end{document}ℓ0 + 2p〉, …, |\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\boldsymbol{\ell }}}_{{\bf{0}}}$$\end{document}ℓ0〉}. We find compact implementations for the generalised Xd(p) gate in both Michelson (one sorter Sd, two spiral phase plates) and Mach-Zehnder configurations (two sorters Sd, two spiral phase plates). Remarkably, the number of spiral phase plates is independent of the qudit dimension d. Our architecture for Xd and generalised Xd(p) gate will enable complex quantum algorithms for qudits, for example quantum protocols using photonic OAM states.

 0 corresponding to the number of helices. Photonic OAM states have been used in entanglement generation [7][8][9] and alignment-free quantum key distribution 10,11 . Thus OAM is attractive since it allows us to use a larger alphabet to transmit quantum information with a single photon. However, without the appropriate tools, a larger alphabet for encoding information has only a limited functionality. This brings us to the problem of how to implement efficiently the generalised Pauli operators X d and Z d for qudits 12 .
For photonic OAM states, Z d can be implemented with Dove prisms. An open question is how to implement a cyclic permutation X d for any dimension d. Experimentally, cyclic X d gates for OAM states have been realised only for = d 4 13 and = d 5 14 . In this article we propose a general scheme to perform cyclic permutations X d for any set of d consecutive states. We then generalise it for cyclic permutations X d (p) of an arbitrary set of d, equally spaced states . For any dimension d, the minimal implementation of both X d and X d (p) requires a single sorter S d and two spiral phase plates (SPPs) [15][16][17][18] . To arrive at this setup, we use quantum information methods and quantum network analysis. This approach has been employed previously to design a universal quantum sorter 19 and spin measuring devices 20,21 .
We focus on OAM encoded qudits, as several experimental tools are already available [22][23][24][25][26][27][28] (2) d j with ⊕ addition mod d and ω = π e i d 2 / a root of unity of order d. The gate X d performs a cyclic permutation of the basis states, i.e., maps the set | 〉 | 〉 … | − 〉 d { 0 , 1 , , 1 } to | 〉 | 〉 … | 〉 { 1 , 2 , , 0 }. Our scheme for the X d gate is shown in Fig. 1. The main element of our proposal is a d-dimensional sorter S d introduced in refs 19,29 . A quantum sorter S d is a device which directs an incoming particle into different outputs (i.e., sorts) according to the value of an internal degree of freedom Σ. In the following we take Σ to be orbital angular momentum (OAM). Nevertheless, the setup is general and can be implemented for other variables as well, like wavelength 19 or radial quantum number 30,31 .
The quantum sorter S d is formally equivalent to a controlled-X d gate between the degree of freedom we want to sort (OAM, Σ etc) and spatial modes m, see Fig. 2: where | 〉 i OAM , | 〉 j m are OAM and mode qudits, respectively. Thus a photon in OAM state | 〉 i incident on port (mode) 0 will exit on port i d ( mod ) with unit probability. Apart from the sorter S d another ingredient are spiral phase plates 15-18 of order n. The action of the SPP on OAM states is: integer. This transformation adds (or subtracts) n units of OAM. Since this is normal addition, it shifts the whole  axis by n units.
We now discuss how the X d gate in Fig. 1 works. The first SPP adds +1 to all OAM states. Then the sorter S d directs each OAM state | 〉 i OAM to the corresponding output | 〉 i d mod m , eq. (3). Since sorting on modes is done modulo d, the state | 〉 d OAM will exit on mode 0. Consequently, only the state on mode 0 needs to be shifted by −d; in terms of quantum networks, this is equivalent to a controlled-SPP(−d) gate, with the control on the mode = k 0 (open circle on control qudit in Fig. 1). After this operation the states from all spatial modes are recombined on mode 0 by the gate C(X d ) −1 , which is nothing else but a sorter run in reverse − S d 1 . This decouples the OAM and mode qudits, such that the final state is factorised and the photon always exits on mode 0 with unit probability. Thus the gate in Fig. 1 performs the following sequence: www.nature.com/scientificreports www.nature.com/scientificreports/ Since the ancilla is decoupled after the gate, an arbitrary superposition of OAM states transforms under the cyclic gate X d as Consequently, our scheme preserves coherence and can be used in arbitrary quantum algorithms.  Fig. 2. Fourier gates for spatial modes [32][33][34][35] can be implemented in several ways.

Resources. Our implementation of the cyclic
In integrated optics the Fourier gate is performed by a single slab coupler 36 , e.g., as used in arrayed waveguide gratings (AWG) 37 . Commercially available AWGs containing Fourier gates have tens to hundreds of spatial modes (channels).
In bulk optics the Fourier transform can be implemented with a pair of confocal lenses with waveguides attached 36  linear optical elements (beamsplitters and phase-shifters) 32,40 . The resource scaling for the X d gate is summarised in Table 1.
We now briefly discuss possible implementations. Given the scaling discussed above and the current advances in photonics, we expect to have sizes d ~ 5 − 10 in bulk optics and d ~ 100 in integrated photonics.
Generalised X d gate. The  . From the general transformation of the sorter, we know that a state with OAM  0 , incident on spatial mode 0, will exit on spatial mode =  k d mod 0 . Thus the only modification of the scheme in Fig. 1 is to move SPP(−d) from mode 0 to mode k, Fig. 3(a).
A second method to perform the cyclic X d gate on the set | 〉 | + is to first shift all OAM states with −k, then perform the usual X d gate, and finally shift back all states with +k, Fig. 3(b).
Importantly, the value of the shift k is bounded by the dimension d of the qudit and not by  0 , since ≤ < k d 0 . This is noteworthy -although OAM with large values =  10010 have been experimentally prepared 41 , SPPs with such large values are very difficult to manufacture. Therefore, for d we need to shift the state only with a much smaller value , irrespective of the magnitude of  0 . Thus the same device can be used to perform the cyclic X d gate on any set of d consecutive OAM states. In this case the only change is the position of SPP(−d) (for the first method), or adding two extra SPP(±k) (for the second method).

(ii) Cyclic permutation for d, equally spaced values. Let
be a positive integer. We now show how to implement a cyclic permutation with step p We define the generalised gate X d (p) and ⊕ addition mod d. The gate is characterised by two parameters: the qudit dimension d and the step p between two consecutive values; clearly The implementation of X d (p) uses the same architecture as before, but with a different sorter S d p   Fig. 4. The first part of the scheme is identical to the one discussed previously. The state | 〉 dp OAM exits on spatial mode k and, after a reflection on SPP(−pd), becomes | 〉 0 OAM . All other OAM states undergo a double reflection on the retro-reflector R and remain unchanged. Finally, all states re-enter the sorter from the opposite direction, thus performing − S d p 1/ , and end up in the same spatial mode |0〉 m . A circulator C separates the output from the input. Note that a spiral phase plate acts as its inverse if the photon enters from the opposite direction; thus in Fig. 4(b) we need only a single SPP(−k).  . There are two ways to correct for the initial state  0 : (a) by moving SPP(−pd) to spatial mode k, with =  k d mod 0 ; (b) by inserting before and after the sorters two SPPs, SPP(−k) and SPP(k). Note that ≤ < k d 0 .

Discussion
The ability to control higher-dimensional quantum systems is essential for developing useful quantum technologies. Due to coherence constraints, efficiency will play a key role in the success of real-life quantum protocols. In this article we designed implementations for cyclic permutations X d in d dimensions, one of the building blocks for constructing arbitrary single-qudit gates. The scheme is deterministic, works at single-particle level and can be applied to arbitrary superpositions of qudit states. Regarding the resource scaling, both X d and X d (p) gates require a linear number of phase-shifts Z d . Remarkably, for any dimension d the number of spiral phase plates SPPs is two, thus constant.
Although our focus has been on orbital angular momentum, the method is general and can be adapted to other degrees of freedom. This will require a sorter S d and shift gates (the equivalent of SPP) for the respective degree of freedom. Since a general scheme for a universal quantum sorter exists 19 , a future challenge to implement the cyclic gate X d for a particular degree of freedom Σ is to find appropriate implementations for shift gates | 〉 | + 〉 Σ Σ  i i n . A possible application of the generalised cyclic permutation X d (p) is quantum communication, e.g., in QKD protocols with Fibonacci coding for key distribution 42,43 . Note added. While finishing this article we became aware of another method for performing X d gates for OAM 44 .  Figure 4. Michelson interferometer configuration for X d (p) gate. This is the folded version of the Mach-Zehnder setup in Fig. 3. The state | 〉 dp OAM exits on spatial mode 0 and after a reflection on SPP(−pd) becomes | 〉 0 OAM . All other OAM states | 〉 jp OAM , ≠ j d, are left invariant by a double reflection on retro-reflectors R. Subsequently, all states re-enter the sorter from the opposite direction, thus performing − S d p 1/ , and end up in the same spatial mode |0〉 m . A circulator C separates the output from the input. Cases (a and b) are equivalent to Fig. 3(a,b), respectively.