Encoding an Oscillator into Many Oscillators

An outstanding challenge for quantum information processing using bosonic systems is Gaussian errors such as excitation loss and added thermal noise errors. Thus, bosonic quantum error correction is essential. Most bosonic quantum error correction schemes encode a finite-dimensional logical qubit or qudit into noisy bosonic oscillator modes. In this case, however, the infinite-dimensional bosonic nature of the physical system is lost at the error-corrected logical level. On the other hand, there are several proposals for encoding an oscillator mode into many noisy oscillator modes. However, these oscillator-into-oscillators encoding schemes are in the class of Gaussian quantum error correction. Therefore, these codes cannot correct practically relevant Gaussian errors due to the established no-go theorems that state that Gaussian errors cannot be corrected by using only Gaussian resources. Here, we circumvent these no-go results and show that it is possible to correct Gaussian errors by using Gottesman-Kitaev-Preskill (GKP) states as non-Gaussian resources. In particular, we propose a non-Gaussian oscillator-into-oscillators code, namely the GKP two-mode squeezing code, and demonstrate that it can quadratically suppress additive Gaussian noise errors in both the position and momentum quadratures except for a small sublogarithmic correction. Furthermore, we demonstrate that our GKP two-mode squeezing code is near optimal in the weak noise limit by proving via quantum information theoretic tools that quadratic noise suppression is optimal when we use two physical oscillator modes. Lastly, we show that our non-Gaussian oscillator encoding scheme can also be used to correct excitation loss and thermal noise errors, which are dominant error sources in many realistic bosonic systems.

Figure 1

(a) An approximate GKP state with an average photon number $\overline{n}=5$. (b) Measurement of the position or momentum operator modulo $\sqrt{2\pi }$. The controlled $\oplus$ and $\ominus$ symbols, respectively, represent the SUM and the inverse-SUM gates where ${\mathrm{SUM}}_{j\to k}\equiv \exp [-i{\widehat{q}}_j{\widehat{p}}_k]$. The final measurement in the bottom mode is the homodyne measurement of the position or the momentum quadrature operator.

Figure 2

(a) Encoding circuit of the GKP two-mode squeezing code subject to independent and identically distributed additive Gaussian noise errors. (b) Decoding circuit of the GKP two-mode squeezing code. Note that the circuits for the measurements of the position and the momentum operators modulo $\sqrt{2\pi }$ (at the end of the decoding) are defined in Fig. 1.

Figure 3

(a) The minimum standard deviations of the output logical quadrature noise ${\sigma }_q={\sigma }_p={\sigma }_L^{\star }$ as a function of the input standard deviation $\sigma$ for the GKP two-mode squeezing code and (b) the optimal two-mode squeezing gain $G^{\star }$ that achieves ${\sigma }_L^{\star }$, translated to the required single-mode squeezing in the unit of decibel $20{\log }_{10}{\lambda }^{\star }$, where ${\lambda }^{\star }\equiv \sqrt{G^{\star }}+\sqrt{G^{\star }-1}$. The black dotted line, the green dashed line, and the grey dotted line in (a) represent the breakeven point ${\sigma }_L=\sigma$, the asymptotic expression ${\sigma }_L^{\star }=(2{\sigma }^2/\sqrt{\pi })\sqrt{{\log }_e[{\pi }^{3/2}/2{\sigma }^4]}$, and the lower bound on ${\sigma }_L$ in Eq. (12), respectively. Also, the yellow dashed line in (b) represents $G^{\star }=(\pi /8{\sigma }^2)({\log }_e[{\pi }^{3/2}/2{\sigma }^4]{)}^{-1}+\frac{1}{2}$.