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 [1]. 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 [2][3][4][5][6][7][8]. 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 BeH 2 . 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 [9] and a robust stochastic optimization routine [10]. We further demonstrate the flexibility of our approach by applying the technique to a problem of quantum magnetism [11]. 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.The fundamental goal of addressing molecular structure problems is to solve for the ground state energy of many-body interacting fermionic Hamiltonians. Solving this problem on a quantum computer relies on a mapping between fermionic and qubit operators [12]. This restates it as a specific instance of a local Hamiltonian problem on a set of qubits. Given a k-local Hamiltonian H, composed of terms that act on at most k qubits, the solution to the local Hamiltonian problem amounts to finding its * These authors contributed equally to this work.smallest eigenvalue E G ,To date, no efficient algorithm is known that can solve this problem in full generality. For k ≥ 2 the problem is known to be QMA-complete [13]. However, it is expected that physical systems have Hamiltonians that do not constitute hard instances of this problem, and can be solved efficiently on a quantum computer, while remaining hard to solve classically. Following Feynman's idea for quantum simulation, a quantum algorithm for the ground state problem of interacting fermions was proposed in [14] and [15]. The approach relies on a good initial state that has a large overlap with the ground state and then solves the problem using the quantum phase estimation algorithm (PEA) [16]. While PEA has been demonstrated to achieve extremely accurate energy estimates for quantum chemistry [2, 3, 5, 8], it applies stringent requirements on quantum coherence.An a...

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 square root[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.

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 Zinteractions 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.The fragile nature of quantum information means that the success of large-scale quantum computing hinges upon the successful implementation of quantum error correction (QEC) on physical qubit systems. Typically QEC protocols function through encoding of physical qubit information onto larger subspaces, which are subsequently protected against particular quantum errors [1, 2]. Amongst the many proposed QEC codes, the topological surface code [3,4] has gathered a large amount of interest by experimental implementations [5,6] due to its use of short-range nearest-neighbor interactions between physical qubits and its relatively high error thresholds.Building up a physical quantum network with the complete functionality of the surface code brings along a number of experimental challenges, some of which have yet to be explored. However, in the particular case of superconducting qubits, recent advances in coherence times [7][8][9] and in the understanding of environmental constraints [10,11] have triggered important experimental demonstrations on increasingly larger systems, including correction of bit-flip errors on linear qubit arrays [6,12], the detection of arbitrary quantum errors [13], and state preservation via encoding in cavity coherent states [14]. With gate fidelities continuing to improve [15,16], it becomes critical to demonstrate the ability to perform these operations in systems with the degree of connectivity required by the surface code. Furthermore, exploring higher-order errors in such nontrivially arranged networks of qubits are necessary for outlining the proper route towards larger numbers of interconnected qubits for QEC.In this Letter we demonstrate a plaquette of the surface code QEC protocol with an interconnected network of five superconducting transmon qubits. This network consists of four data qubits each explicitly coupled to a single syndrome qubit, through which single-shot highfidelity readout is used to measure weight-four checks of both the bit-flip and phase-flip data qubit parity. The geometrical arrangement of the network permits a systematic calibration of crosstalk noise within the plaquette, and we specifically look for errors in non-participating, or "spectator" qubits, during two-qubit gates. To make The five specific qubits used for the plaquette experiment are highlighted and labeled as data qubits (Di, i ∈ [1, 4]) and syndrome ...

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 faulttolerant 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 codewords 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.The possibility of robust quantum computation rests on the fact that quantum information can be encoded in degrees of freedom that are difficult for local noise processes to change. Quantum codes with this potential have been demonstrated in many physical systems [1][2][3][4][5][6][7][8][9][10]. To make practical use of these codes, however, it is necessary not only to encode, decode, and observe errors, but to compute with faulty and inaccurate operations in a way that does not spread errors. The well-developed theory of fault-tolerant quantum computing reveals a steep experimental path toward this goal [11,12]. Recently, the question of what constitutes a minimal experimental demonstration of fault-tolerance was considered [13]. Fault-tolerant state preparation was demonstrated soon thereafter using a quantum error detecting code with trapped atomic ions [14]. Here we go beyond that result, implementing fault-tolerant state preparation on a superconducting qubit system with supporting evidence including quantum state tomography of prepared codewords, acceptance and logical error probabilities with and without error insertion, and analysis of the measured logical observables under free evolution.We implement one of the smallest quantum codes, a four qubit code encoding two qubits [15], and characterize output states produced by fault-tolerant state preparation circuits. The circuits are fault-tolerant for only one of the two encoded qubits, which allows direct comparison of their error rates. The circuits are applied in a five-qubit transmon device with nearest-neighbor connectivity. This device is a nontrivial subset of a surface code lattice in the sense that it provides resources for detection of any single-qubit error. Although the connectivity and size places limits on the set of fault-tolerant circuits we can implement on the four-qubit code, we can use stabilizer measurements to prepare codewords in a way that is analogous to surface code state preparation.Four qubit code -The four-qubit code [15] encodes two logical qubits into four physical qubits and can detect any error that acts on one of those physical qubits. It...

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 nonmonotonic 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 a limited impact on the CR due to Kohn's theorem [W. Kohn, Phys. Rev. 123, 1242 (1961)], which does not apply in graphene as a consequence of the underlying linear band structure.