A mid-circuit erasure check on a dual-rail cavity qubit using the joint-photon number-splitting regime of circuit QED
Stijn J. de Graaf, Sophia H. Xue, Benjamin J. Chapman, James D. Teoh, Takahiro Tsunoda, Patrick Winkel, John W. O. Garmon, Kathleen M. Chang, Luigi Frunzio, Shruti Puri, Robert J. Schoelkopf
Quantum control of a linear oscillator using a static dispersive coupling to a nonlinear ancilla underpins a wide variety of experiments in circuit QED. Extending this control to more than one oscillator while minimizing the required connectivity to the ancilla would enable hardware-efficient multi-mode entanglement and measurements. We show that the spectrum of an ancilla statically coupled to a single mode can be made to depend on the joint photon number in two modes by applying a strong parametric beamsplitter coupling between them. This ‘joint-photon number-splitting’ regime extends single-oscillator techniques to two-oscillator control, which we use to realize a hardware-efficient erasure check for a dual-rail qubit encoded in two superconducting cavities. This scheme leverages the high-fidelity beamsplitter coupling already required for single- and two-qubit gates while permitting minimal crosstalk between circuit elements. Furthermore, the flexibility to choose the pulse shape allows us to limit the susceptibility to different error channels. We use this scheme to detect leakage errors with a missed erasure fraction of (9.0 ± 0.5) × 10 −4 while incurring an erasure rate of 2.92 ± 0.01% and a Pauli error rate of 0.31 ± 0.01%, both of which are dominated by cavity errors.
Quantum control of a linear oscillator using a static dispersive coupling to a nonlinear ancilla underpins a wide variety of experiments in circuit QED. Extending this control to more than one oscillator while minimizing the required connectivity to the ancilla would enable hardware-efficient multi-mode entanglement and measurements. We show that the spectrum of an ancilla statically coupled to a single mode can be made to depend on the joint photon number in two modes by applying a strong parametric beamsplitter coupling between them. This ‘joint-photon number-splitting’ regime extends single-oscillator techniques to two-oscillator control, which we use to realize a hardware-efficient erasure check for a dual-rail qubit encoded in two superconducting cavities. This scheme leverages the high-fidelity beamsplitter coupling already required for single- and two-qubit gates while permitting minimal crosstalk between circuit elements. Furthermore, the flexibility to choose the pulse shape allows us to limit the susceptibility to different error channels. We use this scheme to detect leakage errors with a missed erasure fraction of (9.0 ± 0.5) × 10 −4 while incurring an erasure rate of 2.92 ± 0.01% and a Pauli error rate of 0.31 ± 0.01%, both of which are dominated by cavity errors.
Erasure qubits are a promising platform for implementing hardware-efficient quantum error correction. Realizing the error-correction advantages of this encoding requires frequent mid-circuit erasure checks that are fast, high-fidelity, and scalable. Here, we realize erasure detection with a hardware-efficient circuit consisting of a single readout resonator dispersively and symmetrically coupled to both transmons of a dual-rail qubit. We use this circuit to demonstrate single-shot erasure detection in 384 ns with minimal impact on the dual-rail logical manifold, achieving a residual error per check of 6.0(2)×10−4, with only 8(3)×10−5 induced dephasing per check, and an erasure error per check of 2.54(1)×10−2. The high degree of matched dispersive readout coupling (χ-matching) within the dual-rail qubit code space also allows us to realize a new modality: time-continuous erasure detection performed in parallel with single-qubit gates. Here we achieve a median 7.2×10−5 error per gate with <1×10−5 error induced by erasure detection. This demonstrates a reduction in erasure detection overhead as well as a crucial ingredient for soft information quantum error correction. Together, these results establish symmetrically coupled dispersive readout as a fast, hardware-efficient, and scalable component for erasure-based quantum error correction using transmon dual-rail qubits.
The design of quantum hardware that reduces and mitigates errors is essential for practical quantum error correction (QEC) and useful quantum computation. To this end, we introduce the circuit-Quantum Electrodynamics (QED) dual-rail qubit in which our physical qubit is encoded in the single-photon subspace, [Formula: see text], of two superconducting microwave cavities. The dominant photon loss errors can be detected and converted into erasure errors, which are in general much easier to correct. In contrast to linear optics, a circuit-QED implementation of the dual-rail code offers unique capabilities. Using just one additional transmon ancilla per dual-rail qubit, we describe how to perform a gate-based set of universal operations that includes state preparation, logical readout, and parametrizable single and two-qubit gates. Moreover, first-order hardware errors in the cavities and the transmon can be detected and converted to erasure errors in all operations, leaving background Pauli errors that are orders of magnitude smaller. Hence, the dual-rail cavity qubit exhibits a favorable hierarchy of error rates and is expected to perform well below the relevant QEC thresholds with today's coherence times.
Erasure qubits are beneficial for quantum error correction due to their relaxed threshold requirements. While dual-rail erasure qubits have been demonstrated with a strong error hierarchy in circuit quantum electrodynamics, biased-erasure qubits -- where erasures originate predominantly from one logical basis state -- offer further advantages. Here, we realize a hardware-efficient biased-erasure qubit encoded in the vacuum and two-photon Fock states of a single microwave cavity. The qubit exhibits an erasure bias ratio of over 265. By using a transmon ancilla for logical measurements and mid-circuit erasure detections, we achieve logical state assignment errors below 1% and convert over 99.3% leakage errors into detected erasures. After postselection against erasures, we achieve effective logical relaxation and dephasing rates of (6.2ms)−1 and (3.1ms)−1, respectively, which exceed the erasure error rate by factors of 31 and 15, establishing a strong error hierarchy within the logical subspace. These postselected error rates indicate a coherence gain of about 6.0 beyond the break-even point set by the best physical qubit encoded in the two lowest Fock states in the cavity. Moreover, randomized benchmarking with interleaved erasure detections reveals a residual logical gate error of 0.29%. This work establishes a compact and hardware-efficient platform for biased-erasure qubits, promising concatenations into outer-level stabilizer codes toward fault-tolerant quantum computation.
Quantum error correction using erasure qubits offers higher fault-tolerant thresholds and improved scaling by converting dominant physical errors into detectable erasures. In superconducting circuits, erasure qubits can be constructed using the dual-rail approach, which, however, requires additional qubit-count overhead and tailored coupling elements. Here, we demonstrate a hardware-efficient scheme that operates transmon qutrits as erasure qubits, which is compatible with standard superconducting circuit-QED hardware. The logical states ∣0L⟩ and ∣1L⟩ are represented by the ground and second excited states, while the dominant relaxation errors can be detected via an ancilla qubit using a microwave-activated two-qutrit SWAP gate. We demonstrate a logical qubit T1 lifetime exceeding 500μs, post-selected with repeated mid-circuit erasure detection, which is ten times longer than the T1 time of the transmon physical qubit. Coherence times beyond 300μs are achieved using dynamical decoupling. Single-qubit gate operations reach average Clifford gate infidelity on the order of 10−4. We further demonstrate dual-purposing an ancilla qubit for both erasure detection and parity checking, showing heralded generation of Bell states between erasure qubits. These results suggest that mainstream architectures of transmon qubit arrays may already be capable of implementing erasure-based QEC strategies for hardware-efficient fault-tolerant quantum computing.
Erasure qubits -- qubits designed to have an error profile that is dominated by detectable leakage errors -- are a promising way to cut down the resources needed for quantum error correction. There have been several recent experiments demonstrating erasure qubits in superconducting quantum processors, most notably the dual-rail qubit defined by the one-photon subspace of two coupled cavities. An outstanding challenge is that the ancillary transmons needed to facilitate erasure checks and two-qubit gates introduce a substantial amount of noise, limiting the benefits of working with erasure-biased qubits. Here, we show how to suppress the adverse effects of transmon-induced noise while performing erasure checks or two-qubit gates. We present control schemes for these operations that suppress erasure check errors by three orders of magnitude and reduce the logical two-qubit gate infidelities by up to three orders of magnitude.
Quantum error correction (QEC) will likely be required to realize the full potential of quantum computing, but comes with daunting hardware overheads and demands low gate errors on the physical qubits1–4. These requirements can be eased by engineering qubits with a strong error hierarchy, in which the most common noise channels are also the easiest to correct. Erasure qubits can achieve this when detectable leakage errors out of the computational subspace dominate over the residual Pauli errors5–11, resulting in higher thresholds and improved scaling with code distance5,12,13. In practice, these advantages come to fruition only if the error hierarchy is preserved as much as possible throughout all gates and operations. Here we design and realize a two-qubit entangling gate for dual-rail cavity qubits, a type of erasure qubit encoded in a pair of superconducting microwave cavities7. Our experimental demonstration confirms that the error hierarchy is largely preserved during the gate. The gate is fast (about 500 ns duration) and shows low erasure rates of approximately 0.5% per gate, remaining Pauli errors below 0.1%, and a strong bias towards dephasing errors, in which bit-flips are practically non-existent at the 10−6 level. These results enable a faster path to error-corrected systems that rapidly suppress errors as they scale; a claim we support with our detailed surface code simulations. A fast, low-error entangling gate for dual-rail cavity erasure qubits preserves a strong error hierarchy, advancing scalable quantum error correction with substantially improved fault-tolerant performance.