Erasure Detection of a Dual-Rail Qubit Encoded in a Double-Post Superconducting Cavity

Qubits with predominantly erasure errors present distinctive advantages for quantum error correction (QEC) and fault-tolerant quantum computing. Logical qubits based on dual-rail encoding that exploit erasure detection have been recently proposed in superconducting circuit architectures, with either coupled transmons or cavities. Here, we implement a dual-rail qubit encoded in a compact, double-post superconducting cavity. Using an auxiliary transmon, we perform erasure detection on the dual-rail subspace. We characterize the behavior of the code space by a novel method to perform joint-Wigner tomography. This is based on modifying the cross-Kerr interaction between the cavity modes and the transmon. We measure an erasure rate of $3.981\pm 0.003(\mathrm{ms}{)}^{-1}$ and a residual, postselected dephasing error rate up to $0.17(\mathrm{ms}{)}^{-1}$ within the code space. This strong hierarchy of error rates, together with the compact and hardware-efficient nature of this novel architecture, holds promise in realizing QEC schemes with enhanced thresholds and improved scaling.

Figure 1

(a) Schematic of the double-post cavity, made of 5N aluminum with a transmon qubit, readout, and Purcell filter fabricated on a sapphire chip. (b) Field distributions of the symmetric (Alice) and antisymmetric (Bob) eigenmodes of the system, encoding the dual-rail qubit. (c) Energy-level diagram of the combined Bob-transmon system showing dispersive shifts. The purple arrows connect the pairs of levels $|n-1,h⟩$ and $|n,e⟩$ being coupled via the cross-Kerr tuning drive. (d) Avoided crossing observed when preparing $|n,e⟩$ states in Bob-transmon, sweeping pump detuning $\mathrm{\Delta }$ with fixed amplitude $\mathrm{\Omega }/2\pi =0.5\mathrm{MHz}$. No drive is applied to affect Alice-transmon coupling. Solid horizontal lines are a visual aid to the bare Fock state energies, and simulation results are overlaid in black lines. (e) Number-split peaks of the transmon when either Alice (blue) or Bob (red) is populated with a coherent state of amplitude $\alpha =1.5$, without any pump. (f) Bob’s peaks align with that of Alice in the presence of the cross-Kerr tuning pump with parameters $(\mathrm{\Omega }/2\pi ,\mathrm{\Delta }/2\pi )=(0.5,-5.4)\mathrm{MHz}$.

Figure 2

(a) Dual-rail Bloch sphere where the code words are two-mode Fock states $|+Z{⟩}_L=|01⟩$ and $|-Z{⟩}_L=|10⟩$ (the first mode is Alice, and the second mode is Bob). (b) Erasure detection using a selective $\pi$ pulse centered at the transmon frequency $({\widehat{X}}_{\pi }^0)$. After $n$ detection rounds, we perform joint-Wigner tomography on the system with the cross-Kerr matched Hamiltonian ${\widehat{H}}_{\mathrm{\Sigma }}$ represented as the $\mathrm{\Sigma }$ gate. (c) Experimental results of erasure detection for $n=75$ rounds after preparing the $|+X{⟩}_L$ state. Each row is an separate run of the circuit. Ideally, state transition from $|g⟩\to |e⟩$ heralds an erasure for the dual-rail qubit. But readout infidelities and transmon errors cause deviation from this behavior in the form of isolated “$e$” or “$f$” outcomes. (d) Experimental result of the full circuit shown in (b). Erasure detection is performed on the $|+X{⟩}_L$ state for $n$ rounds before measuring the $\mathrm{Re}(\alpha )\text{−}\mathrm{Re}(\beta )$ joint-Wigner cut of the state. The experiment is repeated for $n=0$, 10, 20, 30, 40 rounds. The initial state clearly decays to the ground state due to cavity photon loss. Postselecting on a qubit being in “$g$” in the trajectory data, we can reconstruct the encoded information, visually proving faithful conversion of photon loss to erasures.

Figure 3

(a) Expectation values of the logical Pauli operators $\widehat{I}$, $\widehat{X}$, $\widehat{Y}$, and $\widehat{Z}$ for the dual-rail subspace as a function of detection rounds, measured using the circuit in Fig. 2. Triangles indicate data after postselecting on no erasures from the dots. For $⟨\widehat{X}⟩$, the experiment is performed after preparing the $|+X⟩$ (darker color at the top) and $|-X⟩$ (lighter color at the bottom) states. The same is repeated for $⟨\widehat{Y}⟩$ and $⟨\widehat{Z}⟩$. Postselection preserves the expectation values for far longer. The fits (shown in solid lines) for the postselected data account for the no-jump evolution, due to the difference in lifetimes between Alice and Bob, and residual Pauli errors within the subspace. (b) Survival probability, i.e., the fraction of experimental shots that survive after $n$ rounds.