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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

Quantum computing holds the promise of a novel form of information processing in which data resides in quantum states. By exploiting superpositions and entanglement among these states, quantum logic operations are expected to far surpass conventional digital logic in certain classes of problems. To realize this prospect, we employ superconducting circuits at millikelvin temperatures. The nonlinearity of a Josephson tunnel junction enables a two-level quantum system, or qubit, with excitation energy in the microwave regime. Key recent advances in such devices are the demonstration of robust and simple multi-qubit gates, the integration of superconducting qubits into arrays of four or more, and gate fidelities reaching 99%. [1,2,3]

Electrons floating on the surface of superfluid helium have been suggested as promising mobile spin quantum bits (qubits). Transferring electrons extremely efficiently in a narrow channel structure with underlying gates has been demonstrated, showing no transfer error while clocking $10^9$ pixels in a 3-phase charge coupled device (CCD). While on average, one electron per channel was clocked, it is desirable to reliably obtain a single electron per channel. We have designed an electron turnstile consisting of a narrow (0.8$\mu$m) channel and narrow underlying gates (0.5$\mu$m) operating across seventy-eight parallel channels. Initially, we find that more than one electron can be held above the small gates. Underlying gates in the turnstile region allow us to repeatedly split these electron packets. Results show a plateau in the electron signal as a function of the applied gate voltages, indicating quantization of the number of electrons per pixel, simultaneously across the seventy-eight parallel channels.

Unprecedented transport efficiency is demonstrated for electrons on the surface of micron-scale superfluid helium-filled channels by co-opting silicon processing technology to construct the equivalent of a charge-coupled device. Strong fringing fields lead to undetectably rare transfer failures after over a billion cycles in two dimensions. This extremely efficient transport is measured in 120 channels simultaneously with packets of up to 20 electrons, and down to singly occupied pixels. These results point the way towards the large scale transport of either computational qubits or electron spin qubits used for communications in a hybrid qubit system.

We report a study of the cyclotron resonance (CR) transitions to and from the unusual $n=0$ Landau level (LL) in monolayer graphene. Unexpectedly, we find the CR transition energy exhibits large (up to 10%) and non-monotonic shifts as a function of the LL filling factor, with the energy being largest at half-filling of the $n=0$ level. The magnitude of these shifts, and their magnetic field dependence, suggests that an interaction-enhanced energy gap opens in the $n=0$ level at high magnetic fields. Such interaction effects normally have limited impact on the CR due to Kohn's theorem [W. Kohn, Phys. Rev. {\bf 123}, 1242 (1961)], which does not apply in graphene as a consequence of the underlying linear band structure., 4 pages, 4 figures. Version 2, edited for publication. Includes a number of edits for clarity; also added a paragraph contrasting our work w/ previous CR expts. in 2D Si and GaAs

We present the first measurements of cyclotron resonance of electrons and holes in bilayer graphene. In magnetic fields up to B = 18 T we observe four distinct intraband transitions in both the conduction and valence bands. The transition energies are roughly linear in B between the lowest Landau levels, whereas they follow \sqrt{B} for the higher transitions. This highly unusual behavior represents a change from a parabolic to a linear energy dispersion. The density of states derived from our data generally agrees with the existing lowest order tight binding calculation for bilayer graphene. However in comparing data to theory, a single set of fitting parameters fails to describe the experimental results., Comment: to appear in Phys. Rev. Lett. Updated version with two added references and minor text editing

Electrons floating on the surface of superfluid helium have been suggested as promising mobile spin qubits. Three micron wide channels fabricated with standard silicon processing are filled with superfluid helium by capillary action. Photoemitted electrons are held by voltages applied to underlying gates. The gates are connected as a 3-phase charge-coupled device (CCD). Starting with approximately one electron per channel, no detectable transfer errors occur while clocking $10^9$ pixels. One channel with its associated gates is perpendicular to the other 120, providing a CCD which can transfer electrons between the others. This perpendicular channel has not only shown efficient electron transport but also serves as a way to measure the uniformity of the electron occupancy in the 120 parallel channels.