Joint Center for Quantum Information and Computer Science

Abstract Suppressing errors is the central challenge for useful quantum computing 1 , requiring quantum error correction (QEC) 2–6 for large-scale processing. However, the overhead in the realization of error-corrected ‘logical’ qubits, in which information is encoded across many physical qubits for redundancy 2–4 , poses substantial challenges to large-scale logical quantum computing. Here we report the realization of a programmable quantum processor based on encoded logical qubits operating with up to 280 physical qubits. Using logical-level control and a zoned architecture in reconfigurable neutral-atom arrays 7 , our system combines high two-qubit gate fidelities 8 , arbitrary connectivity 7,9 , as well as fully programmable single-qubit rotations and mid-circuit readout 10–15 . Operating this logical processor with various types of encoding, we demonstrate improvement of a two-qubit logic gate by scaling surface-code 6 distance from d = 3 to d = 7, preparation of colour-code qubits with break-even fidelities 5 , fault-tolerant creation of logical Greenberger–Horne–Zeilinger (GHZ) states and feedforward entanglement teleportation, as well as operation of 40 colour-code qubits. Finally, using 3D [[8,3,2]] code blocks 16,17 , we realize computationally complex sampling circuits 18 with up to 48 logical qubits entangled with hypercube connectivity 19 with 228 logical two-qubit gates and 48 logical CCZ gates 20 . We find that this logical encoding substantially improves algorithmic performance with error detection, outperforming physical-qubit fidelities at both cross-entropy benchmarking and quantum simulations of fast scrambling 21,22 . These results herald the advent of early error-corrected quantum computation and chart a path towards large-scale logical processors.

This review article contains the 2018 self-consistent set of values of the constants and conversion factors of physics and chemistry recommended by the Committee on Data for Science and Technology (CODATA). The CODATA values are based on a least-squares adjustment that takes into account all data available up to the end of 2018. Details of the data selection and methodology are described.

We describe a simple, efficient method for simulating Hamiltonian dynamics on a quantum computer by approximating the truncated Taylor series of the evolution operator. Our method can simulate the time evolution of a wide variety of physical systems. As in another recent algorithm, the cost of our method depends only logarithmically on the inverse of the desired precision, which is optimal. However, we simplify the algorithm and its analysis by using a method for implementing linear combinations of unitary operations together with a robust form of oblivious amplitude amplification.

Harrow, Hassidim, and Lloyd [Phys. Rev. Lett., 103 (2009), 150502] showed that for a suitably specified $N \times N$ matrix $A$ and an $N$-dimensional vector $\vec{b}$, there is a quantum algorithm that outputs a quantum state proportional to the solution of the linear system of equations $A\vec{x} = \vec{b}$. If $A$ is sparse and well-conditioned, their algorithm runs in time ${poly}(\log N, 1/\epsilon)$, where $\epsilon$ is the desired precision in the output state. We improve this to an algorithm whose running time is polynomial in $\log(1/\epsilon)$, exponentially improving the dependence on precision while keeping essentially the same dependence on other parameters. Our algorithm is based on a general technique for implementing any operator with a suitable Fourier or Chebyshev series representation. This allows us to bypass the quantum phase estimation algorithm, whose dependence on $\epsilon$ is prohibitive.

With quantum computers of significant size now on the horizon, we should understand how to best exploit their initially limited abilities. To this end, we aim to identify a practical problem that is beyond the reach of current classical computers, but that requires the fewest resources for a quantum computer. We consider quantum simulation of spin systems, which could be applied to understand condensed matter phenomena. We synthesize explicit circuits for three leading quantum simulation algorithms, using diverse techniques to tighten error bounds and optimize circuit implementations. Quantum signal processing appears to be preferred among algorithms with rigorous performance guarantees, whereas higher-order product formulas prevail if empirical error estimates suffice. Our circuits are orders of magnitude smaller than those for the simplest classically infeasible instances of factoring and quantum chemistry, bringing practical quantum computation closer to reality.

Single-qubit rotations and two-qubit CNOT operations are crucial ingredients for universal quantum computing. Although high-fidelity single-qubit operations have been achieved using the electron spin degree of freedom, realizing a robust CNOT gate has been challenging because of rapid nuclear spin dephasing and charge noise. We demonstrate an efficient resonantly driven CNOT gate for electron spins in silicon. Our platform achieves single-qubit rotations with fidelities greater than 99%, as verified by randomized benchmarking. Gate control of the exchange coupling allows a quantum CNOT gate to be implemented with resonant driving in ~200 nanoseconds. We used the CNOT gate to generate a Bell state with 78% fidelity (corrected for errors in state preparation and measurement). Our quantum dot device architecture enables multi-qubit algorithms in silicon.

We run a selection of algorithms on two state-of-the-art 5-qubit quantum computers that are based on different technology platforms. One is a publicly accessible superconducting transmon device (www. RESEARCH: ibm.com/ibm-q) with limited connectivity, and the other is a fully connected trapped-ion system. Even though the two systems have different native quantum interactions, both can be programed in a way that is blind to the underlying hardware, thus allowing a comparison of identical quantum algorithms between different physical systems. We show that quantum algorithms and circuits that use more connectivity clearly benefit from a better-connected system of qubits. Although the quantum systems here are not yet large enough to eclipse classical computers, this experiment exposes critical factors of scaling quantum computers, such as qubit connectivity and gate expressivity. In addition, the results suggest that codesigning particular quantum applications with the hardware itself will be paramount in successfully using quantum computers in the future.

The calculation of excited state energies of electronic structure Hamiltonians has many important applications, such as the calculation of optical spectra and reaction rates. While low-depth quantum algorithms, such as the variational quantum eigenvalue solver (VQE), have been used to determine ground state energies, methods for calculating excited states currently involve the implementation of high-depth controlled-unitaries or a large number of additional samples. Here we show how overlap estimation can be used to deflate eigenstates once they are found, enabling the calculation of excited state energies and their degeneracies. We propose an implementation that requires the same number of qubits as VQE and at most twice the circuit depth. Our method is robust to control errors, is compatible with error-mitigation strategies and can be implemented on near-term quantum computers.

Product formulas offer a powerful, simple approach to quantum simulation. A new theory quantifying their errors puts these algorithms on a rigorous foundation, showcasing their superiority over other methods.

The emerging field of quantum machine learning has the potential to substantially aid in the problems and scope of artificial intelligence. This is only enhanced by recent successes in the field of classical machine learning. In this work we propose an approach for the systematic treatment of machine learning, from the perspective of quantum information. Our approach is general and covers all three main branches of machine learning: supervised, unsupervised, and reinforcement learning. While quantum improvements in supervised and unsupervised learning have been reported, reinforcement learning has received much less attention. Within our approach, we tackle the problem of quantum enhancements in reinforcement learning as well, and propose a systematic scheme for providing improvements. As an example, we show that quadratic improvements in learning efficiency, and exponential improvements in performance over limited time periods, can be obtained for a broad class of learning problems.

Modern quantum machine learning (QML) methods involve variationally optimizing a parameterized quantum circuit on a training data set, and subsequently making predictions on a testing data set (i.e., generalizing). In this work, we provide a comprehensive study of generalization performance in QML after training on a limited number N of training data points. We show that the generalization error of a quantum machine learning model with T trainable gates scales at worst as [Formula: see text]. When only K ≪ T gates have undergone substantial change in the optimization process, we prove that the generalization error improves to [Formula: see text]. Our results imply that the compiling of unitaries into a polynomial number of native gates, a crucial application for the quantum computing industry that typically uses exponential-size training data, can be sped up significantly. We also show that classification of quantum states across a phase transition with a quantum convolutional neural network requires only a very small training data set. Other potential applications include learning quantum error correcting codes or quantum dynamical simulation. Our work injects new hope into the field of QML, as good generalization is guaranteed from few training data.

Silicon spin qubits satisfy the necessary criteria for quantum information processing. However, a demonstration of high-fidelity state preparation and readout combined with high-fidelity single- and two-qubit gates, all of which must be present for quantum error correction, has been lacking. We use a two-qubit Si/SiGe quantum processor to demonstrate state preparation and readout with fidelity greater than 97%, combined with both single- and two-qubit control fidelities exceeding 99%. The operation of the quantum processor is quantitatively characterized using gate set tomography and randomized benchmarking. Our results highlight the potential of silicon spin qubits to become a dominant technology in the development of intermediate-scale quantum processors.

Various fundamental phenomena of strongly correlated quantum systems such as high-T(c) superconductivity, the fractional quantum-Hall effect and quark confinement are still awaiting a universally accepted explanation. The main obstacle is the computational complexity of solving even the most simplified theoretical models which are designed to capture the relevant quantum correlations of the many-body system of interest. In his seminal 1982 paper (Feynman 1982 Int. J. Theor. Phys. 21 467), Richard Feynman suggested that such models might be solved by 'simulation' with a new type of computer whose constituent parts are effectively governed by a desired quantum many-body dynamics. Measurements on this engineered machine, now known as a 'quantum simulator,' would reveal some unknown or difficult to compute properties of a model of interest. We argue that a useful quantum simulator must satisfy four conditions: relevance, controllability, reliability and efficiency. We review the current state of the art of digital and analog quantum simulators. Whereas so far the majority of the focus, both theoretically and experimentally, has been on controllability of relevant models, we emphasize here the need for a careful analysis of reliability and efficiency in the presence of imperfections. We discuss how disorder and noise can impact these conditions, and illustrate our concerns with novel numerical simulations of a paradigmatic example: a disordered quantum spin chain governed by the Ising model in a transverse magnetic field. We find that disorder can decrease the reliability of an analog quantum simulator of this model, although large errors in local observables are introduced only for strong levels of disorder. We conclude that the answer to the question 'Can we trust quantum simulators?' is … to some extent.

We report the 2018 self-consistent values of constants and conversion factors of physics and chemistry recommended by the Committee on Data of the International Science Council. The recommended values can also be found at physics.nist.gov/constants. The values are based on a least-squares adjustment that takes into account all theoretical and experimental data available through 31 December 2018. A discussion of the major improvements as well as inconsistencies within the data is given. The former include a decrease in the uncertainty of the dimensionless fine-structure constant and a nearly two orders of magnitude improvement of particle masses expressed in units of kg due to the transition to the revised International System of Units (SI) with an exact value for the Planck constant. Further, because the elementary charge, Boltzmann constant, and Avogadro constant also have exact values in the revised SI, many other constants are either exact or have significantly reduced uncertainties. Inconsistencies remain for the gravitational constant and the muon magnetic-moment anomaly. The proton charge radius puzzle has been partially resolved by improved measurements of hydrogen energy levels.

Limited quantum memory is one of the most important constraints for near-term quantum devices. Understanding whether a small quantum computer can simulate a larger quantum system, or execute an algorithm requiring more qubits than available, is both of theoretical and practical importance. In this Letter, we introduce cluster parameters K and d of a quantum circuit. The tensor network of such a circuit can be decomposed into clusters of size at most d with at most K qubits of inter-cluster quantum communication. We propose a cluster simulation scheme that can simulate any (K,d)-clustered quantum circuit on a d-qubit machine in time roughly 2^{O(K)}, with further speedups possible when taking more fine-grained circuit structure into account. We show how our scheme can be used to simulate clustered quantum systems-such as large molecules-that can be partitioned into multiple significantly smaller clusters with weak interactions among them. By using a suitable clustered ansatz, we also experimentally demonstrate that a quantum variational eigensolver can still achieve the desired performance for estimating the energy of the BeH_{2} molecule while running on a physical quantum device with half the number of required qubits.

Quantum computers and simulators may offer significant advantages over their classical counterparts, providing insights into quantum many-body systems and possibly improving performance for solving exponentially hard problems, such as optimization and satisfiability. Here, we report the implementation of a low-depth Quantum Approximate Optimization Algorithm (QAOA) using an analog quantum simulator. We estimate the ground-state energy of the Transverse Field Ising Model with long-range interactions with tunable range, and we optimize the corresponding combinatorial classical problem by sampling the QAOA output with high-fidelity, single-shot, individual qubit measurements. We execute the algorithm with both an exhaustive search and closed-loop optimization of the variational parameters, approximating the ground-state energy with up to 40 trapped-ion qubits. We benchmark the experiment with bootstrapping heuristic methods scaling polynomially with the system size. We observe, in agreement with numerics, that the QAOA performance does not degrade significantly as we scale up the system size and that the runtime is approximately independent from the number of qubits. We finally give a comprehensive analysis of the errors occurring in our system, a crucial step in the path forward toward the application of the QAOA to more general problem instances.

Abstract The first generation of quantum computers are on the horizon, fabricated from quantum hardware platforms that may soon be able to tackle certain tasks that cannot be performed or modelled with conventional computers. These quantum devices will not likely be universal or fully programmable, but special-purpose processors whose hardware will be tightly co-designed with particular target applications. Trapped atomic ions are a leading platform for first-generation quantum computers, but they are also fundamentally scalable to more powerful general purpose devices in future generations. This is because trapped ion qubits are atomic clock standards that can be made identical to a part in 10 15 , and their quantum circuit connectivity can be reconfigured through the use of external fields, without modifying the arrangement or architecture of the qubits themselves. In this forward-looking overview, we show how a modular quantum computer with thousands or more qubits can be engineered from ion crystals, and how the linkage between ion trap qubits might be tailored to a variety of applications and quantum-computing protocols.

New calculations show that a single stationary state of an isolated quantum system encodes physical properties of the entire system at all temperatures, which provides new, fundamental insight into the quantum nature of thermalization.

The great promise of quantum computers comes with the dual challenges of building them and finding their useful applications. We argue that these two challenges should be considered together, by codesigning full-stack quantum computer systems along with their applications in order to hasten their development and potential for scientific discovery. In this context, we identify scientific and community needs, opportunities, a sampling of a few use case studies, and significant challenges for the development of quantum computers for science over the next 2-10 years.

The Grover quantum search algorithm is a hallmark application of a quantum computer with a well-known speedup over classical searches of an unsorted database. Here, we report results for a complete three-qubit Grover search algorithm using the scalable quantum computing technology of trapped atomic ions, with better-than-classical performance. Two methods of state marking are used for the oracles: a phase-flip method employed by other experimental demonstrations, and a Boolean method requiring an ancilla qubit that is directly equivalent to the state marking scheme required to perform a classical search. We also report the deterministic implementation of a Toffoli-4 gate, which is used along with Toffoli-3 gates to construct the algorithms; these gates have process fidelities of 70.5% and 89.6%, respectively.