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...

We report the first electronic structure calculation performed on a quantum computer without exponentially costly precompilation. We use a programmable array of superconducting qubits to compute the energy surface of molecular hydrogen using two distinct quantum algorithms. First, we experimentally execute the unitary coupled cluster method using the variational quantum eigensolver. Our efficient implementation predicts the correct dissociation energy to within chemical accuracy of the numerically exact result. Second, we experimentally demonstrate the canonical quantum algorithm for chemistry, which consists of Trotterization and quantum phase estimation. We compare the experimental performance of these approaches to show clear evidence that the variational quantum eigensolver is robust to certain errors. This error tolerance inspires hope that variational quantum simulations of classically intractable molecules may be viable in the near future.Comment: 13 pages, 7 figures. This revision is to correct an error in the coefficients of identity in Table

Quantum computing becomes viable when a quantum state can be preserved from environmentally-induced error. If quantum bits (qubits) are sufficiently reliable, errors are sparse and quantum error correction (QEC) 1-6 is capable of identifying and correcting them. Adding more qubits improves the preservation by guaranteeing increasingly larger clusters of errors will not cause logical failure -a key requirement for large-scale systems. Using QEC to extend the qubit lifetime remains one of the outstanding experimental challenges in quantum computing. Here, we report the protection of classical states from environmental bit-flip errors and demonstrate the suppression of these errors with increasing system size. We use a linear array of nine qubits, which is a natural precursor of the twodimensional surface code QEC scheme 7 , and track errors as they occur by repeatedly performing projective quantum non-demolition (QND) parity measurements. Relative to a single physical qubit, we reduce the failure rate in retrieving an input state by a factor of 2.7 for five qubits and a factor of 8.5 for nine qubits after eight cycles. Additionally, we tomographically verify preservation of the non-classical Greenberger-Horne-Zeilinger (GHZ) state. The successful suppression of environmentally-induced errors strongly motivates further research into the many exciting challenges associated with building a large-scale superconducting quantum computer.The ability to withstand multiple errors during computation is a critical aspect of error correction. We define n-th order fault-tolerance to mean that any combination of n errors is tolerable. Previous experiments based on nuclear magnetic resonance 8,9 , ion traps 10 , and superconducting circuits [11][12][13] have demonstrated multi-qubit states that are first-order tolerant to one type of error. Recently, experiments with ion traps and superconducting circuits have shown the simultaneous detection of multiple types of errors 14,15 . The above hallmark experiments demonstrate error correction in a single round; however, quantum information must be preserved throughout computation using multiple error correction cycles. The basics of repeating cycles have been shown in ion traps 16 and superconducting circuits 17 . Until now, it has been an open challenge to combine these elements to make the information stored in a quantum system robust against errors which intrinsically arise from the environment.The key to detecting errors in quantum information is to perform QND parity measurements. In the surface code, this is done by arranging qubits in a chequerboard pattern -with data qubits corresponding to the white, and measure qubits to the black squares (see Fig. 1) -and using these ancilla measure qubits to repetitively perform parity measurements to detect bit-flip (X) and phase-flip (Ẑ) errors 7 . A square chequerboard with (4n + 1) 2 qubits is n-th order fault tolerant, meaning at least n+1 errors must occur to cause failure in preserving a state if fidelities are above a threshold. W...

We demonstrate a planar, tunable superconducting qubit with energy relaxation times up to 44 μs. This is achieved by using a geometry designed to both minimize radiative loss and reduce coupling to materials-related defects. At these levels of coherence, we find a fine structure in the qubit energy lifetime as a function of frequency, indicating the presence of a sparse population of incoherent, weakly coupled two-level defects. We elucidate this defect physics by experimentally varying the geometry and by a model analysis. Our "Xmon" qubit combines facile fabrication, straightforward connectivity, fast control, and long coherence, opening a viable route to constructing a chip-based quantum computer.

A major challenge in quantum computing is to solve general problems with limited physical hardware. Here, we implement digitized adiabatic quantum computing, combining the generality of the adiabatic algorithm with the universality of the digital approach, using a superconducting circuit with nine qubits. We probe the adiabatic evolutions, and quantify the success of the algorithm for random spin problems. We find that the system can approximate the solutions to both frustrated Ising problems and problems with more complex interactions, with a performance that is comparable. The presented approach is compatible with small-scale systems as well as future error-corrected quantum computers.Quantum mechanics can help solve complex problems in physics [1], chemistry [2], and machine learning [3], provided they can be programmed in a physical device. In adiabatic quantum computing [4][5][6], the system is slowly evolved from the ground state of a simple initial Hamiltonian to a final Hamiltonian that encodes a computational problem. The appeal of this analog method lies in its combination of simplicity and generality; in principle, any problem can be encoded. In practice, applications are restricted by limited connectivity, available interactions, and noise. A complementary approach is digital quantum computing, where logic gates combine to form quantum circuit algorithms [7]. The digital approach allows for constructing arbitrary interactions and is compatible with error correction [8, 9], but requires devising tailor-made algorithms. Here, we combine the advantages of both approaches by implementing digitized adiabatic quantum computing in a superconducting system. We tomographically probe the system during the digitized evolution, explore the scaling of errors with system size, and measure the influence of local fields. We conclude by having the full system find the solution to random Ising problems with frustration, and problems with more complex interactions. This digital quantum simulation [10][11][12][13] consists of up to nine qubits and up to 10 3 quantum logic gates. This demonstration of digitized quantum adiabatic computing in the solid state opens a path to solving complex problems, and we hope it will motivate further research into the efficient synthesis of adiabatic algorithms, on small-scale systems with noise as well as future large-scale quantum computers with error correction.A key challenge in adiabatic quantum computing is to construct a device that is capable of encoding problem Hamiltonians that are non-stoquastic [14]. Such Hamiltonians would allow for universal adiabatic quantum computing [15, 16] as well as improving the performance for difficult instances * Present address: IBM T. J. Watson Research Center, Yorktown Heights, NY 10598, USA of classical optimization problems [17]. Additionally, simulating interacting fermions for physics and chemistry requires non-stoquastic Hamiltonians [1, 18]. In general, nonstoquastic Hamiltonians are more difficult to study classically, as Monte Carlo ...

We introduce a superconducting qubit architecture that combines high-coherence qubits and tunable qubit-qubit coupling. With the ability to set the coupling to zero, we demonstrate that this architecture is protected from the frequency crowding problems that arise from fixed coupling. More importantly, the coupling can be tuned dynamically with nanosecond resolution, making this architecture a versatile platform with applications ranging from quantum logic gates to quantum simulation. We illustrate the advantages of dynamic coupling by implementing a novel adiabatic controlled-Z gate, at a speed approaching that of single-qubit gates. Integrating coherence and scalable control, our "gmon" architecture is a promising path towards large-scale quantum computation and simulation.The fundamental challenge for quantum computation and simulation is to construct a large-scale network of highly connected coherent qubits [1, 2]. Superconducting qubits use macroscopic circuits to process quantum information and are a promising candidate towards this end [3]. Over the last several years, materials research and circuit optimization have led to significant progress in qubit coherence [4][5][6]. Superconducting qubits can now perform hundreds of operations within their coherence times, allowing for research into complex algorithms such as error correction [7,8].It is desirable to combine these high-coherence qubits with tunable inter-qubit coupling; the resulting architecture would allow for both coherent local operations and dynamically varying qubit interactions. For quantum simulation, this would provide a unique opportunity to investigate dynamic processes in non-equilibrium condensed matter phenomena [9][10][11][12][13]. For quantum computation, such an architecture would provide isolation for single-qubit gates while at the same time enabling fast two-qubit gates that minimize errors from decoherence. Despite previous successful demonstrations of tunable coupling [14][15][16][17][18][19][20][21][22][23], these applications have yet to be realized due to the challenge of incorporating tunable coupling with high coherence devices.Here, we introduce a planar qubit architecture that combines high coherence with tunable inter-qubit coupling g. This "gmon" device is based on the Xmon transmon design [5], but now gives nanosecond control of the coupling strength with a measured on/off coupling ratio exceeding 1000. We find that our device retains the high coherence inherent in the Xmon design, with the coupler providing unique advantages in constructing single-and two-qubit quantum logic gates. With the coupling turned off, we demonstrate that our architecture is protected from the frequency crowding problems that arise from fixed coupling. Our single-qubit gate fidelity is nearly independent of the qubit-qubit detuning, even when operating the qubits on resonance. By dynamically tuning the coupling, we implement a novel adiabatic controlled-Z gate at a speed approaching that of single-qubit gates.A two-qubit unit cell with tun...

We describe the fabrication and measurement of microwave coplanar waveguide resonators with internal quality factors above 10 7 at high microwave powers and over 10 6 at low powers, with the best low power results approaching 2 × 10 6 , corresponding to ∼ 1 photon in the resonator. These quality factors are achieved by controllably producing very smooth and clean interfaces between the resonators' aluminum metallization and the underlying single crystal sapphire substrate. Additionally, we describe a method for analyzing the resonator microwave response, with which we can directly determine the internal quality factor and frequency of a resonator embedded in an imperfect measurement circuit.High quality factor microwave resonators provide critical elements for superconducting electromagnetic radiation detectors 1 , quantum memories 2,3 , and quantum computer architectures 4 . Good performance and stability can be achieved for such applications using aluminum resonators patterned on sapphire substrates. Aluminum is a favored material due to its robust oxide and reasonable transition temperature, and sapphire provides an excellent substrate due to its very low microwave loss tangent 5 δ ∼ 10 −8 and its chemical inertness. However, the quality factors measured in such resonators is lower than expected; recent simulations 6 and experiments 7 suggest that the unexplained loss arises mostly from imperfections at the metal-substrate interface. Using an experimental apparatus with minimal stray magnetic fields and infrared light at the sample 8 , here we show that careful substrate preparation and cleaning yields aluminumon-sapphire resonators with significantly higher internal quality factors Q i . We also introduce a new method for evaluating the resonator microwave response.The aluminum for the resonators was deposited on cplane sapphire substrates in one of three deposition systems: A high vacuum DC sputter system (base pressure P base = 6 × 10 −8 Torr), a high vacuum electron-beam evaporator (P base = 5 × 10 −8 Torr) or an ultra-high vacuum (UHV) molecular beam epitaxy (MBE) system (P base = 6 × 10 −10 Torr) with electron-beam deposition. The sapphire substrates were first sonicated in a bath of acetone then isopropanol followed by a deionized water rinse. For the sputter-deposited and e-beam evaporated samples, we further cleaned the substrates prior to Al de-

Progress in superconducting qubit experiments with greater numbers of qubits or advanced techniques such as feedback requires faster and more accurate state measurement. We have designed a multiplexed measurement system with a bandpass filter that allows fast measurement without increasing environmental damping of the qubits. We use this to demonstrate simultaneous measurement of four qubits on a single superconducting integrated circuit, the fastest of which can be measured to 99.8% accuracy in 140 ns. This accuracy and speed is suitable for advanced multi-qubit experiments including surface code error correction.With recent results showing high fidelity one and two qubit logic gates [1, 2], superconducting qubits have become a leading candidate for experiments in large scale engineered quantum systems. Realization of complex experiments in quantum information such as error correction [3, 4], quantum simulation [5], cluster state quantum computing [6,7], and measurement feedback [8,9] will require state measurements to be interleaved with coherent manipulations. For example, error correction protocols like the surface code repeatedly measure parity operators to detect and correct errors. This requires the measurement process, like the gates, to be much faster than the qubit coherence time. In particular, the measurements must be switched on and off quickly so that the measurement channel does not continuously collapse the qubit state during coherent manipulations. Additionally, an ideal detector suitable for a large system multiplexes to many qubits without introducing correlated qubit errors.Accurate measurement of superconducting qubits is a major challenge because the measurement apparatus introduces damping which lowers the qubit's energy relaxation time T 1 . Transmon qubits [10] are measured dispersively; a probe signal applied to an auxiliary linear resonator coupled to the qubit acquires a phase shift that depends on the qubit's quantum state [11]. Coupling to the environment through the resonator leads to qubit damping via the Purcell effect [12,13]. This places a limit on measurement speed as the resonator coupling to the environment, characterized by a leakage rate κ r , must be large enough to get photons into and out of the resonators quickly, but weak enough to prevent environmental damping from lowering T 1 . Introducing a filter between the qubit and environment eases this constraint by suppressing damping at the qubit frequency while maintaining strong coupling between the resonator and environment. Increased T 1 was demonstrated previously with a notch filter placed in series with the resonator, but measurement speed was not studied [13].Recent experiments demonstrating quantum jumps, state heralding, dressed dephasing, single quantum tra- jectories, and joint qubit readout have focused on a single channel of quantum information [14][15][16][17][18]. Furthermore these experiments used either long measurement times or qubits with coherence strongly limited by the measurement system. To make pro...