Cyclic permutations for qudits in $d$ dimensions

Abstract

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 $X_d$ in $d$ dimensions, a major primitive for constructing arbitrary qudit gates. Using orbital angular momentum states as a qudit, the simplest implementation of the $X_d$ gate in $d$ dimensions requires a single quantum sorter $S_d$ and two spiral phase plates. We then extend this construction to a generalised $X_d(p)$ gate to perform a cyclic permutation of a set of $d$, equally spaced values $\{|{\ell }_0⟩,|{\ell }_0+p⟩,\dots ,|{\ell }_0+(d-1)p⟩\}\mapsto \{|{\ell }_0+p⟩,|{\ell }_0+2p⟩,\dots ,|{\ell }_0⟩\}$. We find compact implementations for the generalised $X_d(p)$ gate in both Michelson (one sorter $S_d$, two spiral phase plates) and Mach-Zehnder configurations (two sorters $S_d$, two spiral phase plates). Remarkably, the number of spiral phase plates is independent of the qudit dimension $d$. Our architecture for $X_d$ and generalised $X_d(p)$ gate will enable complex quantum algorithms for qudits, for example quantum protocols using photonic OAM states.

References in corpus (21)

• Quantum Cryptography
• Elementary gates for quantum computation
• Entanglement of Orbital Angular Momentum States of Photons
• The Physical Implementation of Quantum Computation
• Quantum Entanglement of High Angular Momenta
• Efficient separation of the orbital angular momentum eigenstates of light
• Free-space quantum key distribution by rotation-invariant twisted photons
• Bloch vectors for qudits
• Complete experimental toolbox for alignment-free quantum communication
• Generation and Confirmation of a (100x100)-dimensional entangled Quantum System
• Quantum entanglement of angular momentum states with quantum numbers up to 10010
• High-Dimensional Single-Photon Quantum Gates: Concepts and Experiments
• Mesoscopic Stern-Gerlach device to polarize spin currents
• Characterization of high-dimensional entangled systems via mutually unbiased measurements
• Gouy Phase Radial Mode Sorter for Light: Concepts and Experiments
• Cyclic transformation of orbital angular momentum modes
• Arbitrary d-dimensional Pauli X-Gates of a flying Qudit
• Sorting quantum systems efficiently
• Single spin measurement using spin-orbital entanglement
• Quantum Key Distribution with High Order Fibonacci-like Orbital Angular Momentum States
• Recursive multiport schemes for implementing quantum algorithms with photonic integrated circuits

Cited by in corpus (7)

• Advances in High Dimensional Quantum Entanglement
• Quantum communication networks with optical vortices
• OAM tomography with Heisenberg-Weyl observables
• Generation of minimum energy entangled states
• Polarization selective Dove prism
• Optimized architectures for universal quantum state transformations using photonic path and polarization
• Arbitrary unitaries in orbital angular momentum of single photons