Reliable quantum information processing in the face of errors is a major fundamental and technological challenge. Quantum error correction protects quantum states by encoding a logical quantum bit (qubit) in multiple physical qubits. To be compatible with universal fault-tolerant computations, it is essential that states remain encoded at all times and that errors are actively corrected. Here we demonstrate such active error correction on a continuously protected logical qubit using a diamond quantum processor. We encode the logical qubit in three long-lived nuclear spins, repeatedly detect phase errors by non-destructive measurements, and apply corrections by real-time feedback. The actively error-corrected qubit is robust against errors and encoded quantum superposition states are preserved beyond the natural dephasing time of the best physical qubit in the encoding. These results establish a powerful platform to investigate error correction under different types of noise and mark an important step towards fault-tolerant quantum information processing.

Conditional-phase (CZ) gates in transmons can be realized by flux pulsing computational states towards resonance with noncomputational ones. We present a 40 ns CZ gate based on a bipolar flux pulse suppressing leakage (0.1%) by interference and approaching the speed limit set by exchange coupling. This pulse harnesses a built-in echo to enhance fidelity (99.1%) and is robust to long-timescale distortion in the flux-control line, ensuring repeatability. Numerical simulations matching experiment show that fidelity is limited by high-frequency dephasing and leakage by short-timescale distortion.

Variational quantum eigensolvers offer a small-scale testbed to demonstrate the performance of error mitigation techniques with low experimental overhead. We present successful error mitigation by applying the recently proposed symmetry verification technique to the experimental estimation of the ground-state energy and ground state of the hydrogen molecule. A finely adjustable exchange interaction between two qubits in a circuit QED processor efficiently prepares variational ansatz states in the single-excitation subspace respecting the parity symmetry of the qubit-mapped Hamiltonian. Symmetry verification improves the energy and state estimates by mitigating the effects of qubit relaxation and residual qubit excitation, which violate the symmetry. A full-density-matrix simulation matching the experiment dissects the contribution of these mechanisms from other calibrated error sources. Enforcing positivity of the measured density matrix via scalable convex optimization correlates the energy and state estimate improvements when using symmetry verification, with interesting implications for determining system properties beyond the ground-state energy.Noisy intermediate-scale quantum (NISQ) devices [1], despite lacking layers of quantum error correction (QEC), may already be able to demonstrate quantum advantage over classical computers for select problems [2,3]. In particular, the hybrid quantum-classical variational quantum eigensolver (VQE) [4,5] may have sufficiently low experimental requirements to allow estimation of ground-state energies of quantum systems that are difficult to simulate purely classically [6][7][8][9]. To date, VQEs have been used to study small examples of the electronic structure problem, such as H 2 [10][11][12][13][14][15], HeH+ [4,16], LiH [13][14][15], and BeH 2 [14], as well as exciton systems [17], strongly correlated magnetic models [15], and the Schwinger model [18]. Although these experimental efforts have achieved impressive coherent control of up to 20 qubits, the error in the resulting estimations has remained relatively high due to performance limitations in the NISQ hardware. Consequently, much focus has recently been placed on developing error mitigation techiques that offer order-of-magnitude accuracy improvement without the costly overhead of full QEC. This may be achieved by using known properties of the target state, e.g., by checking known symmetries in a manner inspired by QEC stabilizer measurements [19,20], or by expanding around the experimentally-obtained state via a linear (or higher-order) response framework [21]. The former, termed symmetry verification (SV), is of particular interest because it is comparatively low-cost in terms of required hardware and additional measurements. Other mitigation techniques require understanding the underlying error models of the quantum device, allowing for an extrapolation of the calculation to the zero-error limit [22][23][24], or the summing of multiple calculations to probabilistically cancel errors [23,25,26].In this Rapid ...