A quantum computer can solve hard problems -such as prime factoring 1,2 , database searching 3,4 , and quantum simulation 5 -at the cost of needing to protect fragile quantum states from error. Quantum error correction 6 provides this protection, by distributing a logical state among many physical qubits via quantum entanglement. Superconductivity is an appealing platform, as it allows for constructing large quantum circuits, and is compatible with microfabrication. For superconducting qubits the surface code 7 is a natural choice for error correction, as it uses only nearest-neighbour coupling and rapidly-cycled entangling gates. The gate fidelity requirements are modest: The per-step fidelity threshold is only about 99%. Here, we demonstrate a universal set of logic gates in a superconducting multi-qubit processor, achieving an average single-qubit gate fidelity of 99.92% and a two-qubit gate fidelity up to 99.4%. This places Josephson quantum computing at the fault-tolerant threshold for surface code error correction. Our quantum processor is a first step towards the surface code, using five qubits arranged in a linear array with nearest-neighbour coupling. As a further demonstration, we construct a five-qubit GreenbergerHorne-Zeilinger (GHZ) state 8,9 using the complete circuit and full set of gates. The results demonstrate that Josephson quantum computing is a high-fidelity technology, with a clear path to scaling up to large-scale, fault-tolerant quantum circuits.The high fidelity performance we demonstrate here is achieved through a combination of highly coherent qubits, a straightforward interconnection architecture, and a novel implementation of the two-qubit controlled-phase (CZ) entangling gate. The CZ gate uses a fast but adiabatic frequency tuning of the qubits 10 , which is easily adjusted yet minimises decoherence and leakage from the computational basis [Martinis, J., et al., in preparation]. We note that previous demonstrations of two-qubit gates achieving better than 99% fidelity used fixed-frequency qubits: Systems based on nuclear magnetic resonance and ion traps have shown two-qubit gates with fidelities of 99.5% 11 and 99.3% 12 . Here, the tuneable nature of the qubits and their entangling gates provides, remarkably, both high fidelity and fast control.Superconducting integrated circuits give flexibility in building quantum systems due to the macroscopic nature of the electron condensate. As shown in Fig. 1, we have designed a processor consisting of five Xmon qubits with nearestneighbour coupling, arranged in a linear array. The crossshaped qubit 14 offers a nodal approach to connectivity while maintaining a high level of coherence (see Supplementary Information for decoherence times). Here, the four legs of the cross allow for a natural segmentation of the design into coupling, control and readout. We chose a modest inter-qubit capacitive coupling strength of g/2π = 30 MHz and use alternating qubit idle frequencies of 5.5 and 4.7 GHz, enabling a CZ gate in 40 ns when two qubits are brough...

The complete gauge-invariant set of the one-loop QED corrections to the parity-nonconserving 6s-7s amplitude in 133 Cs is evaluated to all orders in αZ using a local version of the Dirac-Hartree-Fock potential. The calculations are peformed in both length and velocity gauges for the absorbed photon. The total binding QED correction is found to be-0.27(3)%, which differs from previous evaluations of this effect. The weak charge of 133 Cs, derived using two most accurate values of the vector transition polarizability β, is QW = −72.57(46) for β = 26.957(51)a 3 B and QW = −73.09(54) for β = 27.15(11)a 3 B. The first value deviates by 1.1σ from the prediction of the Standard Model, while the second one is in perfect agreement with it. PACS numbers: 11.30.Er,31.30.Jv,32.80.Ys Investigations of parity noncoservation (PNC) effects in atomic systems play a prominent role in tests of the Standard Model (SM) and impose constraints on physics beyond it [1, 2]. The 6s-7s PNC amplitude in 133 Cs [3] remains one of the most attractive subject for such investigations. The measurement of this amplitude to a 0.3% accuracy [4, 5] has stimulated a reanalysis of related theoretical contributions. First, it was found [6, 7, 8, 9] that the role of the Breit interaction had been underestimated in previous evaluations of this effect [10, 11]. Then, it was pointed out [12] that the QED corrections may be comparable with the Breit corrections. The numerical evaluation of the vacuum-polarization (VP) correction [13] led to a 0.4% increase of the 6s-7s PNC amplitude in 133 Cs, which resulted in a 2.2σ deviation of the weak charge of 133 Cs from the SM prediction. This has triggered a great interest to calculations of the one-loop QED corrections to the PNC amplitude. While the VP contribution can easily be evaluated to a high accuracy within the Uehling approximation, the calculation of the self-energy (SE) contribution is a much more demanding problem (here and below we imply that the SE term embraces all one-loop vertex diagrams as well). To zeroth order in αZ, it was derived in Refs. [14, 15]. This correction, whose relative value equals to −α/(2π), is commonly included in the definition of the nuclear weak charge. The αZ-dependent part of the SE correction to the PNC matrix element between s and p states was evaluated in Refs. [16, 17]. These calculations , which are exact to first order in αZ and partially include higher-order binding effects, yield the correction of-0.9(1)% [16, 18] and-0.85% [17]. This restored the agreement with SM. Despite of the close agreement of the results obtained in Refs. [17, 18], the status of the QED correction to PNC in 133 Cs cannot be considered as resolved until a complete αZ-dependence calculation of the SE correction to the 6s-7s transition amplitude is accomplished. The reasons for that are the following. First, in case of cesium (Z = 55) the parameter αZ ≈ 0.4 is not small and, therefore, the higher-order corrections can be significant. Second, because the calculations [16, 17, 18] are performed f...

We apply the method of compressed sensing (CS) quantum process tomography (QPT) to characterize quantum gates based on superconducting Xmon and phase qubits. Using experimental data for a two-qubit controlled-Z gate, we obtain an estimate for the process matrix χ with reasonably high fidelity compared to full QPT, but using a significantly reduced set of initial states and measurement configurations. We show that the CS method still works when the amount of used data is so small that the standard QPT would have an underdetermined system of equations. We also apply the CS method to the analysis of the three-qubit Toffoli gate with numerically added noise, and similarly show that the method works well for a substantially reduced set of data. For the CS calculations we use two different bases in which the process matrix χ is approximately sparse, and show that the resulting estimates of the process matrices match each other with reasonably high fidelity. For both two-qubit and three-qubit gates, we characterize the quantum process by not only its process matrix and fidelity, but also by the corresponding standard deviation, defined via variation of the state fidelity for different initial states.

One of the key challenges in quantum information is coherently manipulating the quantum state. However, it is an outstanding question whether control can be realized with low error. Only gates from the Clifford group -- containing $\pi$, $\pi/2$, and Hadamard gates -- have been characterized with high accuracy. Here, we show how the Platonic solids enable implementing and characterizing larger gate sets. We find that all gates can be implemented with low error. The results fundamentally imply arbitrary manipulation of the quantum state can be realized with high precision, providing new practical possibilities for designing efficient quantum algorithms.Comment: 8 pages, 4 figures, including supplementary materia

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