Minimizing leakage from computational states is a challenge when using many-level systems like superconducting quantum circuits as qubits. We realize and extend the quantum-hardware-efficient,all-microwave leakage reduction unit (LRU) for transmons in a circuit QED architecture proposed by Battistel et al. This LRU effectively reduces leakage in the second- and third-excited transmon states with up to 99% efficacy in 220 ns, with minimum impact on the qubit subspace. As a first application in the context of quantum error correction, we demonstrate the ability of multiple simultaneous LRUs to reduce the error detection rate and to suppress leakage buildup within 1% in data and ancilla qubits over 50 cycles of a weight-2 parity measurement.
We realize a suite of logical operations on a distance-two logical qubit stabilized using repeated error detection cycles. Logical operations include initialization into arbitrary states,measurement in the cardinal bases of the Bloch sphere, and a universal set of single-qubit gates. For each type of operation, we observe higher performance for fault-tolerant variants over non-fault-tolerant variants, and quantify the difference through detailed characterization. In particular, we demonstrate process tomography of logical gates, using the notion of a logical Pauli transfer matrix. This integration of high-fidelity logical operations with a scalable scheme for repeated stabilization is a milestone on the road to quantum error correction with higher-distance superconducting surface codes.
Leakage outside of the qubit computational subspace, present in many leading experimental platforms, constitutes a threatening error for quantum error correction (QEC) for qubits. Wedevelop a leakage-detection scheme via Hidden Markov models (HMMs) for transmon-based implementations of the surface code. By performing realistic density-matrix simulations of the distance-3 surface code (Surface-17), we observe that leakage is sharply projected and leads to an increase in the surface-code defect probability of neighboring stabilizers. Together with the analog readout of the ancilla qubits, this increase enables the accurate detection of the time and location of leakage. We restore the logical error rate below the memory break-even point by post-selecting out leakage, discarding about 47% of the data. Leakage detection via HMMs opens the prospect for near-term QEC demonstrations, targeted leakage reduction and leakage-aware decoding and is applicable to other experimental platforms.
Gottesman, Kitaev and Preskill have formulated a way of encoding a qubit into an oscillator such that the encoded qubit is protected against small shifts in phase space. We proposeand analyze the approximate creation of these encoded states with ancilla qubits. The preparation of the code states uses the idea of phase estimation where the phase of the stabilizer checks is approximately determined. We propose a physical implementation of this protocol using the dispersive coupling between a transmon ancilla qubit and a cavity mode in circuit-QED. We provide an estimate that in a current experimental set-up one can prepare a good code state from a squeezed vacuum state using 8 rounds of adapative phase estimation, lasting in total less than 5 {\mu} sec., with 80 % (heralded) chance of success.
We perform a detailed analysis of how an amplifier-based interferometer can be used to enhance the quality of a dispersive qubit measurement, such as one performed on a superconductingtransmon qubit, using homodyne detection on an amplified microwave signal. Our modeling makes a realistic assessment of what is possible in current circuit-QED experiments; in particular, we take into account the frequency-dependence of the qubit-induced phase shift for short microwaves pulses. We compare the possible signal-to-noise ratios obtainable with (single-mode) SU(1,1) interferometers with the current coherent measurement and find a considerable reduction in measurement error probability in an experimentally-accessible range of parameters.