Bosonic quantum error correction

In many bosonic quantum error correction (QEC) schemes proposed so far, a finite-dimensional discrete-variable (DV) system (consisted of qubits) is encoded into an oscillator or into many oscillators. In such qubit(s)-into-oscillator(s) schemes, the bosonic nature of the physical oscillator modes is lost at the logical level because the error-corrected logical system is described by discrete quantum variables such as Pauli operators. Therefore, the error-corrected logical DV system is not itself tailored to continuous-variable (CV) quantum information processing tasks such as boson sampling.

We propose a broad class of bosonic quantum error-correcting codes, GKP-stabilizer codes, that encode logical oscillator modes into physical oscillator modes. Our codes can correct Gaussian errors such as excitation losses, thermal noise and additive Gaussian noise errors. Since the bosonic nature of the physical modes is still retained at the logical level, our oscillator(s)-into-oscillators codes are tailored to CV quantum information processing tasks.

In particular, we show that there exists a highly hardware-efficient GKP-stabilizer code, the GKP-two-mode-squeezing code (shown above), that can quadratically suppress additive Gaussian noise errors in both the position and momentum quadratures using only two bosonic modes (one data mode and one ancilla mode).

Our codes are the first instances of oscillator(s)-into-oscillators codes that can correct Gaussian errors. Specifically, we circumvent the established no-go theorems that state Gaussian errors cannot be corrected by Gaussian QEC schemes, by using Gottesman-Kitaev-Preskill (GKP) states as non-Gaussian resources in our codes.

Various single-mode bosonic quantum error-correcting codes such as cat, binomial, and GKP codes have been implemented experimentally in circuit QED and trapped ion systems. Moreover, there have been many theoretical proposals to scale up such single-mode bosonic codes to realize large-scale fault-tolerant quantum computation. Here, we consider the concatenation of the single-mode GKP code with the surface code, namely, the surface-GKP code. In particular, we thoroughly investigate the performance of the surface-GKP code by assuming realistic GKP states with a finite squeezing and noisy circuit elements due to photon losses.

By using a minimum-weight perfect matching decoding algorithm on a 3D space-time graph, we show that fault-tolerant quantum error correction is possible with the surface-GKP code if the squeezing of the GKP states is higher than 11.2dB in the case where the GKP states are the only noisy elements. We also show that the squeezing threshold changes to 18.6dB when both the GKP states and circuit elements are comparably noisy. At this threshold, each circuit component fails with probability 0.69%. Finally, if the GKP states are noiseless, fault-tolerant quantum error correction with the surface-GKP code is possible if each circuit element fails with probability less than 0.81%.

As shown in several earlier works, finding the optimal pair of encoding and decoding operations that maximizes entanglement fidelity is a biconvex optimization problem. To find the best pair of encoding and decoding operations for the Gaussian thermal loss channels, we solved the biconvex optimization heuristically by an alternating semidefinite programming method and found that, surprisingly, the hexagonal-lattice Gottesman-Kitaev-Preskill (GKP) code emerges as an optimal encoding from a Haar-random initial code.

In addition to (heuristically) showing that the hexagonal-lattice GKP code is the optimal single-mode bosonic code for Gaussian thermal loss channels, we also proved that a family of multi-mode GKP codes (defined over an optimal symplectic lattice) achieves the quantum capacity of Gaussian thermal loss channels up to at most a constant number (~log_{2}e = 1.44...) of qubits per channel use.