A Comprehensive Review of Bosonic Quantum Error Correcting Codes

Quantum error correction is essential for reliable fault-tolerant quantum computing, necessitating the encoding of information redundantly into physical degrees of freedom to safeguard it against noise. A prominent approach involves continuous variable quantum information processing using bosonic modes. This technique encodes information within the harmonic oscillator's occupation number space, expressed through number states ${\ket{n}}^\infty_{n=0}$, position and momentum eigenstates and , or a selection of coherent states (for a finite set $S$) .

The initial continuous variable scheme involving bosonic modes is the two-mode "dual-rail" encoding, introduced in 1995. Presently, numerous bosonic codes are under assessment for their potential in fault-tolerant quantum computation. This review will focus on key contenders: firstly, establishing a pragmatic bosonic error model; proceeding to explore three prominent single-mode codes renowned for their robust protection against this model; evaluating the performance of these codes, considering relevant theoretical aspects based on the work by Albert, Noh, Duivenvoorden, Young, Brierley, Reinhold; and finally, delving into hardware-efficient multi-mode extensions, notable for their strides towards feasible physical implementation. These extensions will be situated within the evolving realm of bosonic quantum error-correcting codes.