In recent years, surface codes have become a leading method for quantum error correction in theoretical large-scale computational and communications architecture designs. Their comparatively high fault-tolerant thresholds and their natural two-dimensional nearest-neighbour (2DNN) structure make them an obvious choice for large scale designs in experimentally realistic systems. While fundamentally based on the toric code of Kitaev, there are many variants, two of which are the planar-and defect-based codes. Planar codes require fewer qubits to implement (for the same strength of error correction), but are restricted to encoding a single qubit of information. Interactions between encoded qubits are achieved via transversal operations, thus destroying the inherent 2DNN nature of the code. In this paper we introduce a new technique enabling the coupling of two planar codes without transversal operations, maintaining the 2DNN of the encoded computer. Our lattice surgery technique comprises splitting and merging planar code surfaces, and enables us to perform universal quantum computation (including magic state injection) while removing the need for braided logic in a strictly 2DNN design, and hence reduces the overall qubit resources for logic operations. Those resources are 2 further reduced by the use of a rotated lattice for the planar encoding. We show how lattice surgery allows us to distribute encoded GHZ states in a more direct (and overhead friendly) manner, and how a demonstration of an encoded cnot between two distance-3 logical states is possible with 53 physical qubits, half of that required in any other known construction in 2D.

We develop a layered quantum-computer architecture, which is a systematic framework for tackling the individual challenges of developing a quantum computer while constructing a cohesive device design. We discuss many of the prominent techniques for implementing circuit-model quantum computing and introduce several new methods, with an emphasis on employing surface-code quantum error correction. In doing so, we propose a new quantum-computer architecture based on optical control of quantum dots. The time scales of physical-hardware operations and logical, error-corrected quantum gates differ by several orders of magnitude. By dividing functionality into layers, we can design and analyze subsystems independently, demonstrating the value of our layered architectural approach. Using this concrete hardware platform, we provide resource analysis for executing fault-tolerant quantum algorithms for integer factoring and quantum simulation, finding that the quantum-dot architecture we study could solve such problems on the time scale of days.

We propose an approach to implement quantum repeaters for long-distance quantum communication. Our protocol generates a backbone of encoded Bell pairs and uses the procedure of classical error correction during simultaneous entanglement connection. We illustrate that the repeater protocol with simple Calderbank-ShorSteane encoding can significantly extend the communication distance, while still maintaining a fast key generation rate.

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Quantum networks will support long-distance quantum key distribution (QKD) and distributed quantum computation, and are an active area of both experimental and theoretical research. Here, we present an analysis of topologically complex networks of quantum repeaters composed of heterogeneous links. Quantum networks have fundamental behavioral differences from classical networks; the delicacy of quantum states makes a practical path selection algorithm imperative, but classical notions of resource utilization are not directly applicable, rendering known path selection mechanisms inadequate. To adapt Dijkstra's algorithm for quantum repeater networks that generate entangled Bell pairs, we quantify the key differences and define a link cost metric, seconds per Bell pair of a particular fidelity, where a single Bell pair is the resource consumed to perform one quantum teleportation. Simulations that include both the physical interactions and the extensive classical messaging confirm that Dijkstra's algorithm works well in a quantum context. Simulating about three hundred heterogeneous paths, comparing our path cost and the total work along the path gives a coefficient of determination of 0.88 or better.

Abstract-We present a new control algorithm and system design for a network of quantum repeaters, and outline the endto-end protocol architecture. Such a network will create longdistance quantum states, supporting quantum key distribution as well as distributed quantum computation. Quantum repeaters improve the reduction of quantum-communication throughput with distance from exponential to polynomial. Because a quantum state cannot be copied, a quantum repeater is not a signal amplifier. Rather, it executes algorithms for quantum teleportation in conjunction with a specialized type of quantum error correction called purification to raise the fidelity of the quantum states. We introduce our banded purification scheme, which is especially effective when the fidelity of coupled qubits is low, improving the prospects for experimental realization of such systems. The resulting throughput is calculated via detailed simulations of a long line composed of shorter hops. Our algorithmic improvements increase throughput by a factor of up to fifty compared to earlier approaches, for a broad range of physical characteristics.

We present a detailed analysis of the impact on quantum modular exponentiation of architectural features and possible concurrent gate execution. Various arithmetic algorithms are evaluated for execution time, potential concurrency, and space tradeoffs. We find that to exponentiate an n-bit number, for storage space 100n (twenty times the minimum 5n), we can execute modular exponentiation two hundred to seven hundred times faster than optimized versions of the basic algorithms, depending on architecture, for n = 128. Addition on a neighboronly architecture is limited to O(n) time while non-neighbor architectures can reach O(log n), demonstrating that physical characteristics of a computing device have an important impact on both real-world running time and asymptotic behavior. Our results will help guide experimental implementations of quantum algorithms and devices.

Quantum communication typically involves a linear chain of repeater stations, each capable of reliable local quantum computation and connected to their nearest neighbors by unreliable communication links. The communication rate of existing protocols is low as two-way classical communication is used. By using a surface code across the repeater chain and generating Bell pairs between neighboring stations with probability of heralded success greater than 0.65 and fidelity greater than 0.96, we show that two-way communication can be avoided and quantum information can be sent over arbitrary distances with arbitrarily low error at a rate limited only by the local gate speed. This is achieved by using the unreliable Bell pairs to measure nonlocal stabilizers and feeding heralded failure information into post-transmission error correction. Our scheme also applies when the probability of heralded success is arbitrarily low.