The ubiquitous noise in quantum system hinders the advancement of quantum information processing and has driven the emergence of different hardware-efficient quantum error correctionprotocols. Among them, qubits with structured noise, especially with biased noise, are one of the most promising platform to achieve fault-tolerance due to the high error thresholds of quantum error correction codes tailored for them. Nevertheless, their quantum operations are challenging and the demonstration of their performance beyond the fault-tolerant threshold remain incomplete. Here, we leverage Schrödinger cat states in a scalable planar superconducting nonlinear oscillator to thoroughly characterize the high-fidelity single-qubit quantum operations with systematic quantum tomography and benchmarking tools, demonstrating the state-of-the-art performance of operations crossing the fault-tolerant threshold of the XZZX surface code. These results thus embody a transformative milestone in the exploration of quantum systems with structured error channels. Notably, our framework is extensible to other types of structured-noise systems, paving the way for systematic characterization and validation of novel quantum platforms with structured noise.
The Kerr-cat qubit is a bosonic qubit in which multi-photon Schrodinger cat states are stabilized by applying a two-photon drive to an oscillator with a Kerr nonlinearity. The suppressedbit-flip rate with increasing cat size makes this qubit a promising candidate to implement quantum error correction codes tailored for noise-biased qubits. However, achieving strong light-matter interactions necessary for stabilizing and controlling this qubit has traditionally required strong microwave drives that heat the qubit and degrade its performance. In contrast, increasing the coupling to the drive port removes the need for strong drives at the expense of large Purcell decay. By integrating an effective band-block filter on-chip, we overcome this trade-off and realize a Kerr-cat qubit in a scalable 2D superconducting circuit with high coherence. This filter provides 30 dB of isolation at the qubit frequency with negligible attenuation at the frequencies required for stabilization and readout. We experimentally demonstrate quantum non-demolition readout fidelity of 99.6% for a cat with 8 photons. Also, to have high-fidelity universal control over this qubit, we combine fast Rabi oscillations with a new demonstration of the X(90) gate through phase modulation of the stabilization drive. Finally, the lifetime in this architecture is examined as a function of the cat size of up to 10 photons in the oscillator achieving a bit-flip time higher than 1 ms and only a linear decrease in the phase-flip time, in good agreement with the theoretical analysis of the circuit. Our qubit shows promise as a building block for fault-tolerant quantum processors with a small footprint.
The storage and processing of quantum information are susceptible to external noise, resulting in computational errors that are inherently continuous A powerful method to suppress theseeffects is to use quantum error correction. Typically, quantum error correction is executed in discrete rounds where errors are digitized and detected by projective multi-qubit parity measurements. These stabilizer measurements are traditionally realized with entangling gates and projective measurement on ancillary qubits to complete a round of error correction. However, their gate structure makes them vulnerable to errors occurring at specific times in the code and errors on the ancilla qubits. Here we use direct parity measurements to implement a continuous quantum bit-flip correction code in a resource-efficient manner, eliminating entangling gates, ancilla qubits, and their associated errors. The continuous measurements are monitored by an FPGA controller that actively corrects errors as they are detected. Using this method, we achieve an average bit-flip detection efficiency of up to 91%. Furthermore, we use the protocol to increase the relaxation time of the protected logical qubit by a factor of 2.7 over the relaxation times of the bare comprising qubits. Our results showcase resource-efficient stabilizer measurements in a multi-qubit architecture and demonstrate how continuous error correction codes can address challenges in realizing a fault-tolerant system.
The quantum Zeno effect is the suppression of Hamiltonian evolution by repeated observation, resulting in the pinning of the state to an eigenstate of the measurement observable. Usingmeasurement only, control of the state can be achieved if the observable is slowly varied such that the state tracks the now time-dependent eigenstate. We demonstrate this using a circuit-QED readout technique that couples to a dynamically controllable observable of a qubit. Continuous monitoring of the measurement record allows us to detect an escape from the eigenstate, thus serving as a built-in form of error detection. We show this by post-selecting on realizations with arbitrarily high fidelity with respect to the target state. Our dynamical measurement operator technique offers a new tool for numerous forms of quantum feedback protocols, including adaptive measurements and rapid state purification.
The state of a continuously monitored qubit evolves stochastically, exhibiting competition between coherent Hamiltonian dynamics and diffusive partial collapse dynamics that followthe measurement record. We couple these distinct types of dynamics together by linearly feeding the collected record for dispersive energy measurements directly back into a coherent Rabi drive amplitude. Such feedback turns the competition cooperative, and effectively stabilizes the qubit state near a target state. We derive the conditions for obtaining such dispersive state stabilization and verify the stabilization conditions numerically. We include common experimental nonidealities, such as energy decay, environmental dephasing, detector efficiency, and feedback delay, and show that the feedback delay has the most significant negative effect on the feedback protocol. Setting the measurement collapse timescale to be long compared to the feedback delay yields the best stabilization.
Using circuit QED, we consider the measurement of a superconducting transmon qubit via a coupled microwave resonator. For ideally dispersive coupling, ringing up the resonator producescoherent states with frequencies matched to transmon energy states. Realistic coupling is not ideally dispersive, however, so transmon-resonator energy levels hybridize into joint eigenstate ladders of the Jaynes-Cummings type. Previous work has shown that ringing up the resonator approximately respects this ladder structure to produce a coherent state in the eigenbasis (a dressed coherent state). We numerically investigate the validity of this coherent state approximation to find two primary deviations. First, resonator ring-up leaks small stray populations into eigenstate ladders corresponding to different transmon states. Second, within an eigenstate ladder the transmon nonlinearity shears the coherent state as it evolves. We then show that the next natural approximation for this sheared state in the eigenbasis is a dressed squeezed state, and derive simple evolution equations for such states using a hybrid phase-Fock-space description.
In modern circuit QED architectures, superconducting transmon qubits are measured via the state-dependent phase and amplitude shift of a microwave field leaking from a coupled resonator.Determining this shift requires integrating the field quadratures for a nonzero duration, which can permit unwanted concurrent evolution. Here we investigate such dynamical degradation of the measurement fidelity caused by a detuned neighboring qubit. We find that in realistic parameter regimes, where the qubit ensemble-dephasing rate is slower than the qubit-qubit detuning, the joint qubit-qubit eigenstates are better discriminated by measurement than the bare states. Furthermore, we show that when the resonator leaks much more slowly than the qubit-qubit detuning, the measurement tracks the joint eigenstates nearly adiabatically. However, the measurement process also causes rare quantum jumps between the eigenstates. The rate of these jumps becomes significant if the resonator decay is comparable to or faster than the qubit-qubit detuning, thus significantly degrading the measurement fidelity in a manner reminiscent of energy relaxation processes.
We describe a method to perform any generalized purity-preserving measurement of a qubit with techniques tailored to superconducting systems. We start with considering two methods forrealizing a two-outcome partial projection: using a thresholded continuous measurement in the circuit QED setup and using an indirect ancilla qubit measurement. Then we decompose an arbitrary purity-preserving two-outcome measurement into single qubit unitary rotations and a partial projection. Finally, we systematically reduce any multiple-outcome measurement to a sequence of two-outcome measurements and unitary operations.