Autonomous Quantum Error Correction of Gottesman-Kitaev-Preskill States

The Gottesman-Kitaev-Preskill (GKP) code encodes a logical qubit into a bosonic system with resilience against single-photon loss, the predominant error in most bosonic systems. Here we present experimental results demonstrating quantum error correction of GKP states based on reservoir engineering of a superconducting device. Error correction is made fully autonomous through an unconditional reset of an auxiliary transmon qubit. We show that the lifetime of the logical qubit is increased from quantum error correction, therefore reaching the point at which more errors are corrected than generated.

Figure 1

(a) Schematic of hardware architecture. The storage mode, in which the GKP code is encoded, is the fundamental mode of a coaxial cavity dispersively coupled to an auxiliary transmon qubit. The auxiliary is also dispersively coupled to an on-chip resonator used for readout and reset. (b) Protocol for the initialization of GKP logical states with a depth-$N$ circuit. Each layer $n$ consists of an auxiliary rotation of parameter $R_n$ and a conditional displacement on the storage mode of parameter ${\beta }_n$. (c) Protocol for measurement of the characteristic function of initialized GKP logical states. Real part of the characteristic function $\mathrm{Re}(C_{|\pm \overline{\mu }⟩})$, of GKP logical states (d) $|-\overline{X}⟩$, (e) $|+\overline{X}⟩$, and (f) $|-\overline{Y}⟩$ for $\mathrm{\Delta }=0.36$ using a circuit with a depth $N=9$, leading to an initialization duration of $7.86\mathrm{\mu }\mathrm{s}$ excluding resets. (g) State fidelity estimated from state reconstruction given measured (d)–(f) and simulated characteristic functions.

Figure 2

(a) Dissipative swap between the auxiliary qubit and the resonator enabled by an effective $|f0{⟩}_{\mathrm{r}}\leftrightarrow |g1{⟩}_{\mathrm{r}}$ transition, followed by relaxation as $|g1{⟩}_{\mathrm{r}}\to |g0{⟩}_{\mathrm{r}}$. (b) Reset protocol with $N_{\mathrm{ds}}$ dissipative swaps. (c) Average reset error ${ϵ}_{\mathrm{rt}}$ as a function of $N_{\mathrm{ds}}$ measured experimentally (orange) and obtained from numerical simulations (teal). The horizontal dashed line indicates the experimental noise floor of $4\times {10}^{-4}$. (d) Logical fidelity $F_{\mathrm{L}}$ measured as a function of free evolution time (blue, top axis) or number of dissipative swaps (orange, bottom axis). Plain and dashed lines show fits to the data from which logical lifetimes $T_{\mathrm{L}}$ of 0.176 and 0.143 ms are obtained for free evolution without and with dissipative swaps, respectively. The limit $F_{\mathrm{L}}\to 0.5$ represents a fully decohered logical channel. Inset: enlarged to highlight the optimal value of $N_{\mathrm{ds}}=2$.

Figure 3

(a) Quantum error correction protocol based on the sBs protocol composed of auxiliary rotations, conditional displacements, and a reset of the auxiliary. (b) Protocol used to measure the logical lifetime. After initialization of a GKP logical state (encoding), the sBs protocol is either applied (with QEC) or not (without QEC) $N_{\mathrm{id}}$ times. An extra idle time $T_{\mathrm{id}}$ between each round is added in both cases. The GKP qubit Pauli expectation value $⟨{\widehat{\mu }}_0{⟩}_{\pm }$ is measured for states $|\pm \overline{\mu }⟩$ (decoding). (c) Logical fidelity $F_{\mathrm{L}}$ measured as a function of time without QEC (blue) and with default (red) or optimized sBs protocols (turquoise) for $T_{\mathrm{id}}=40\mathrm{\mu }\mathrm{s}$. The full lines show the decay of the logical fidelity obtained from a fitting procedure [31]. The data without QEC are the same as in Fig. 2. (d) Logical lifetime $T_{\mathrm{L}}$ obtained experimentally (dark bars) and numerically (pale bars). The horizontal dashed line corresponds to the logical lifetime without QEC.