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Arbitrarily long quantum computations require quantum memories that can be repeatedly measured without being corrupted. Here, we preserve the state of a quantum memory, notably with the additional use of flagged error events. All error events were extracted using fast, mid-circuit measurements and resets of the physical qubits. Among the error decoders we considered, we introduce a perfect matching decoder that was calibrated from measurements containing up to size-4 correlated events. To compare the decoders, we used a partial post-selection scheme shown to retain ten times more data than full post-selection. We observed logical errors per round of $2.2\pm0.1\times10^{-2}$ (decoded without post-selection) and $5.1\pm0.7\times10^{-4}$ (full post-selection), which was less than the physical measurement error of $7\times10^{-3}$ and therefore surpasses a pseudo-threshold for repeated logical measurements., 16 pages, 14 figures, 5 tables, for peer-review

The execution of quantum circuits on real systems has largely been limited to those which are simply time-ordered sequences of unitary operations followed by a projective measurement. As hardware platforms for quantum computing continue to mature in size and capability, it is imperative to enable quantum circuits beyond their conventional construction. Here we break into the realm of dynamic quantum circuits on a superconducting-based quantum system. Dynamic quantum circuits involve not only the evolution of the quantum state throughout the computation, but also periodic measurements of a subset of qubits mid-circuit and concurrent processing of the resulting classical information within timescales shorter than the execution times of the circuits. Using noisy quantum hardware, we explore one of the most fundamental quantum algorithms, quantum phase estimation, in its adaptive version, which exploits dynamic circuits, and compare the results to a non-adaptive implementation of the same algorithm. We demonstrate that the version of real-time quantum computing with dynamic circuits can offer a substantial and tangible advantage when noise and latency are sufficiently low in the system, opening the door to a new realm of available algorithms on real quantum systems., 7 pages, 4 figures, plus supplement

Robust quantum computation requires encoding delicate quantum information into degrees of freedom that are hard for the environment to change. Quantum encodings have been demonstrated in many physical systems by observing and correcting storage errors, but applications require not just storing information; we must accurately compute even with faulty operations. The theory of fault-tolerant quantum computing illuminates a way forward by providing a foundation and collection of techniques for limiting the spread of errors. Here we implement one of the smallest quantum codes in a five-qubit superconducting transmon device and demonstrate fault-tolerant state preparation. We characterize the resulting code words through quantum process tomography and study the free evolution of the logical observables. Our results are consistent with fault-tolerant state preparation in a protected qubit subspace.

Quantum computers can be used to address molecular structure, materials science and condensed matter physics problems, which currently stretch the limits of existing high-performance computing resources. Finding exact numerical solutions to these interacting fermion problems has exponential cost, while Monte Carlo methods are plagued by the fermionic sign problem. These limitations of classical computational methods have made even few-atom molecular structures problems of practical interest for medium-sized quantum computers. Yet, thus far experimental implementations have been restricted to molecules involving only Period I elements. Here, we demonstrate the experimental optimization of up to six-qubit Hamiltonian problems with over a hundred Pauli terms, determining the ground state energy for molecules of increasing size, up to BeH2. This is enabled by a hardware-efficient variational quantum eigensolver with trial states specifically tailored to the available interactions in our quantum processor, combined with a compact encoding of fermionic Hamiltonians and a robust stochastic optimization routine. We further demonstrate the flexibility of our approach by applying the technique to a problem of quantum magnetism. Across all studied problems, we find agreement between experiment and numerical simulations with a noisy model of the device. These results help elucidate the requirements for scaling the method to larger systems, and aim at bridging the gap between problems at the forefront of high-performance computing and their implementation on quantum hardware., 6 pages, 4 figures in main text. 18 pages, 9 figures in supplementary information

We present parity measurements on a five-qubit lattice with connectivity amenable to the surface code quantum error correction architecture. Using all-microwave controls of superconducting qubits coupled via resonators, we encode the parities of four data qubit states in either the $X$- or the $Z$-basis. Given the connectivity of the lattice, we perform full characterization of the static $Z$-interactions within the set of five qubits, as well as dynamical $Z$-interactions brought along by single- and two-qubit microwave drives. The parity measurements are significantly improved by modifying the microwave two-qubit gates to dynamically remove non-ideal $Z$ errors., Comment: 8 pages, 4 figures