Quantum error correction enables the preservation of logical qubits with a lower logical error rate than the physical error rate, with performance depending on the decoding method.Traditional error decoding approaches, relying on the binarization (`hardening‘) of readout data, often ignore valuable information embedded in the analog (`soft‘) readout signal. We present experimental results showcasing the advantages of incorporating soft information into the decoding process of a distance-three (d=3) bit-flip surface code with transmons. To this end, we use the 3×3 data-qubit array to encode each of the 16 computational states that make up the logical state $\ket{0_{\mathrm{L}}}$, and protect them against bit-flip errors by performing repeated Z-basis stabilizer measurements. To infer the logical fidelity for the $\ket{0_{\mathrm{L}}}$ state, we average across the 16 computational states and employ two decoding strategies: minimum weight perfect matching and a recurrent neural network. Our results show a reduction of up to 6.8% in the extracted logical error rate with the use of soft information. Decoding with soft information is widely applicable, independent of the physical qubit platform, and could reduce the readout duration, further minimizing logical error rates.
During the last 30 years, stimulated by the quest to build superconducting quantum processors, a theory of quantum electrical circuits has emerged and this theory goes under the nameof circuit quantum electrodynamics or circuit-QED. The goal of the theory is to provide a quantum description of the most relevant degrees of freedom. The central objects to be derived and studied are the Lagrangian and the Hamiltonian governing these degrees of freedom. Central concepts in classical network theory such as impedance and scattering matrices can be used to obtain the Hamiltonian and Lagrangian description for the lossless (linear) part of the circuits. Methods of analysis, both classical and quantum, can also be developed for nonreciprocal circuits. These lecture notes aim at giving a pedagogical overview of this subject for theoretically-oriented Master or PhD students in physics and electrical engineering, as well as Master and PhD students who work on experimental superconducting quantum devices and wish to learn more theory.
We propose and analyze two types of microwave-activated gates between a fluxonium and a transmon qubit, namely a cross-resonance (CR) and a CPHASE gate. The large frequency differencebetween a transmon and a fluxonium makes the realization of a two-qubit gate challenging. For a medium-frequency fluxonium qubit, the transmon-fluxonium system allows for a cross-resonance effect mediated by the higher levels of the fluxonium over a wide range of transmon frequencies. This allows one to realize the cross-resonance gate by driving the fluxonium at the transmon frequency, mitigating typical problems of the cross-resonance gate in transmon-transmon chips related to frequency targeting and residual ZZ coupling. However, when the fundamental frequency of the fluxonium enters the low-frequency regime below 100 MHz, the cross-resonance effect decreases leading to long gate times. For this range of parameters, a fast microwave CPHASE gate can be implemented using the higher levels of the fluxonium. In both cases, we perform numerical simulations of the gate showing that a gate fidelity above 99% can be obtained with gate times between 100 and 300 ns. Next to a detailed gate analysis, we perform a study of chip yield for a surface code lattice of fluxonia and transmons interacting via the proposed cross-resonance gate. We find a much better yield as compared to a transmon-only architecture with the cross-resonance gate as native two-qubit gate.
Leakage outside of the qubit computational subspace poses a threatening challenge to quantum error correction (QEC). We propose a scheme using two leakage-reduction units (LRUs) thatmitigate these issues for a transmon-based surface code, without requiring an overhead in terms of hardware or QEC-cycle time as in previous proposals. For data qubits we consider a microwave drive to transfer leakage to the readout resonator, where it quickly decays, ensuring that this negligibly affects the coherence within the computational subspace for realistic system parameters. For ancilla qubits we apply a |1⟩↔|2⟩ π pulse conditioned on the measurement outcome. Using density-matrix simulations of the distance-3 surface code we show that the average leakage lifetime is reduced to almost 1 QEC cycle, even when the LRUs are implemented with limited fidelity. Furthermore, we show that this leads to a significant reduction of the logical error rate. This LRU scheme opens the prospect for near-term scalable QEC demonstrations.
We analyze whether circuit-QED Hamiltonians are stoquastic focusing on systems of coupled flux qubits: we show that scalable sign-problem free path integral Monte Carlo simulationscan typically be performed for such systems. Despite this, we corroborate the recent finding [1] that an effective, non-stoquastic qubit Hamiltonian can emerge in a system of capacitively coupled flux qubits. We find that if the capacitive coupling is sufficiently small, this non-stoquasticity of the effective qubit Hamiltonian can be avoided if we perform a canonical transformation prior to projecting onto an effective qubit Hamiltonian. Our results shed light on the power of circuit-QED Hamiltonians for the use of quantum adiabatic computation and the subtlety of finding a representation which cures the sign problem in these systems
One of the most direct preparations of a Gottesman-Kitaev-Preskill qubit in an oscillator uses a tunable photon-pressure (also called optomechanical) coupling of the form gq^a†a,enabling to imprint the modular value of the position q^ of one oscillator onto the state of an ancilla oscillator. We analyze the practical feasibility of executing such modular quadrature measurements in a parametric circuit-QED realization of this coupling. We provide estimates for the expected GKP squeezing induced by the protocol and discuss the effect of photon loss and other errors on the resulting squeezing.