Analysis and performance of three-mode qutrit rotationally symmetric bosonic codes

Abstract

Encoding information in bosonic quantum states has an advantage over traditional two level systems since they inhabit Hilbert spaces of infinite dimensions. This makes bosonic codes into very resourceful carriers of quantum information which can be exploited in error correcting codes. Thus, making bosonic codes a viable option in the realisation of fault tolerant quantum computers. In this report we begin by adapting a general formulation of multi-mode d-dimensional Rotational symmetrical bosonic (RSB) codes from Ref. [1] and by setting the dimension to d = 3 results in a general qutrit code space in three modes. Before the code is further specified, the operations of beam splitters, describing linear passive optical system, is analysed from definitions in Ref. [1] and solutions for how excitations transforms under such evolutions are found analytically from an algebraic structure. We then specify the coefficients of a code in two separate methods. First, the single mode RSB basis states are projected onto a state characterized by a binomial distribution. The result is rejected from a preliminary analysis of the symmetries in the resulting single-mode states. Another method is then carried out by arbitrary truncating the sums of general RSB states and enforcing conditions of orthonormality and equal mean photon number ⟨ˆn⟩ of each single-mode basis state. The result is considered to be a good candidate as an error correctable code. The code, denoted as "the arbitrary code" is further analysed with a theorem which defines error correctable codes known as the KL-conditions. From this analytical study, we find that the code is correctable for single photon loss errors but not for dephasing errors. An analytical study of the "arbitrary code", in terms of the KL conditions, is followed by a numerical estimation of its performance against noise in the form of photon loss and dephasing, individually. The performance is estimated in terms of near-optimal entanglement fidelity. Inspired by the analysis of symmetries in the single-mode basis states and the KL conditions for the full code, we find a third approach experimentally, leading to a new framework for the construction and classification of d-nomial RSB codes. The new frame work also makes it clear that the qutrit codes that has been studied, are in fact char acterized by states from trinomial distributions and are therefore denoted as trinomial codes.