The potential impact of future quantum networks hinges on high-quality quantum entanglement shared between network nodes. Unavoidable real-world imperfections necessitate means to improve remote entanglement by local quantum operations. Here we realize entanglement distillation on a quantum network primitive of distant electron-nuclear two-qubit nodes. We demonstrate the heralded generation of two copies of a remote entangled state through single-photon-mediated entangling of the electrons and robust storage in the nuclear spins. After applying local two-qubit gates, single-shot measurements herald the distillation of an entangled state with increased fidelity that is available for further use. In addition, this distillation protocol significantly speeds up entanglement generation compared to previous two-photon-mediated schemes. The key combination of generating, storing and processing entangled states demonstrated here opens the door to exploring and utilizing multi-particle entanglement on an extended quantum network.Future quantum networks connecting nodes of longlived stationary qubits through photonic channels may enable secure communication, quantum computation and simulation, and enhanced metrology [1][2][3][4][5][6][7][8][9]. The power of these applications fundamentally derives from quantum entanglement shared between the network nodes. The key experimental challenge is therefore to establish high-quality remote entanglement in the presence of unavoidable errors such as decoherence, photon loss and imperfect quantum control. Remarkably, by only using classical communication and local quantum operations, a high-fidelity remote entangled state can be distilled from several lower-fidelity copies [10,11] (Fig. 1A). Success of this intrinsically probabilistic distillation can be nondestructively heralded by measurement outcomes such that the distilled state is available for further use, a critical requirement for scalable networks. Owing to these unique features, entanglement distillation, also known as purification, has become a central building block of quantum network proposals [6-9, 12, 13]. GENERATION AND DISTILLATION OF REMOTE ENTANGLED STATESTo run entanglement distillation on a quantum network, several copies of a raw entangled state must first be shared between the nodes. This can be achieved using a network primitive of two nodes with two qubits * These authors contributed equally.† Present address: Max-Planck-Institute for Quantum Optics, Hans-Kopfermann-Str. 1, 85748 Garching, Germany ‡ To whom correspondence should be addressed; E-mail: r.hanson@tudelft.nl each: a communication qubit with an optical interface for generating remote entanglement and a memory qubit for storage (Fig. 1B). First the communication qubits run the entangling protocol, which due to photon loss is intrinsically probabilistic. After photon detection heralds the generation of a raw entangled state on the communication qubits, this state is swapped onto the memory qubits. The communication qubits are then used to generate a...

A scalable quantum computer could be built by networking together many simple processor cells, thus avoiding the need to create a single complex structure. The difficulty is that realistic quantum links are very error prone. A solution is for cells to repeatedly communicate with each other and so purify any imperfections; however prior studies suggest that the cells themselves must then have prohibitively low internal error rates. Here we describe a method by which even error-prone cells can perform purification: groups of cells generate shared resource states, which then enable stabilization of topologically encoded data. Given a realistically noisy network (≥10% error rate) we find that our protocol can succeed provided that intra-cell error rates for initialisation, state manipulation and measurement are below 0.82%. This level of fidelity is already achievable in several laboratory systems.

Exquisite quantum control has now been achieved in small ion traps, in nitrogen-vacancy centers and in superconducting qubit clusters. We can regard such a system as a universal cell with diverse technological uses from communication to large-scale computing, provided that the cell is able to network with others and overcome any noise in the interlinks. Here, we show that loss-tolerant entanglement purification makes quantum computing feasible with the noisy and lossy links that are realistic today: With a modestly complex cell design, and using a surface code protocol with a network noise threshold of 13.3%, we find that interlinks that attempt entanglement at a rate of 2 MHz but suffer 98% photon loss can result in kilohertz computer clock speeds (i.e., rate of high-fidelity stabilizer measurements). Improved links would dramatically increase the clock speed. Our simulations employ local gates of a fidelity already achieved in ion trap devices. DOI: 10.1103/PhysRevX.4.041041 Subject Areas: Quantum PhysicsWithin the past year, there have been remarkable advances in the fidelity with which small quantum devices can be controlled. The two most mature systems are ion traps and superconducting qubits. In ion trap devices, single-qubit fidelities [1] have reached 99.9999%, with combined preparation and measurement of 99.93%. Moreover, two-qubit operations [2] have been reported with fidelities up to 99.9%. Meanwhile, a superconducting qubit device (SQD) containing five qubits [3] has been demonstrated with all qubit manipulations above 99.3%. At the same time, there has been rapid progress in the study of nitrogen-vacancy (NV) centers in diamond-single electron spin manipulation is possible with 99% fidelity [4], and it is possible to manipulate nuclei that are relatively far from the center, so that each NV center may be thought of as a group of several qubits interacting with an optically active core [5].These prototype systems are small; none of them contain as many as 20 qubits. But importantly in each case, it is possible to bridge between small systems using photonic channels, albeit with lower entanglement fidelities and in a probabilistic way that may require many attempts. In the ion trap community, there are well-established methods for entangling ions in separate traps, and recent progress associated with projects such as the MUSIQC initiative [6] have led to successful entanglement at a rate of hertz [7]. This can be improved by orders of magnitude by hardware advances and by loss-adapted protocols, as we describe in this paper. In SQDs, a well-established means of interfacing qubits is to exploit microwave photons in cavities [8]. This suffices for short-range bridging, and, moreover, remote entanglement of two superconducting qubits separated by more than a meter of coaxial cable has recently been demonstrated [9]. In the case of NV center research, successful optical linking of qubits can occur either within the same sample [4] or over meters of separation [10], and teleportation [11] with fidelity ...

We introduce fusion-based quantum computing (FBQC) -a model of universal quantum computation in which entangling measurements, called fusions, are performed on the qubits of small constant-sized entangled resource states. We introduce a stabilizer formalism for analyzing fault tolerance and computation in these schemes. This framework naturally captures the error structure that arises in certain physical systems for quantum computing, such as photonics. FBQC can offer significant architectural simplifications, enabling hardware made up of many identical modules, requiring an extremely low depth of operations on each physical qubit and reducing classical processing requirements. We present two pedagogical examples of fault-tolerant schemes constructed in this framework and numerically evaluate their threshold under a hardware agnostic fusion error model including both erasure and Pauli error. We also study an error model of linear optical quantum computing with probabilistic fusion and photon loss. In FBQC the non-determinism of fusion is directly dealt with by the quantum error correction protocol, along with other errors. We find that tailoring the fault-tolerance framework to the physical system allows the scheme to have a higher threshold than schemes reported in literature. We present a ballistic scheme which can tolerate a 10.4% probability of suffering photon loss in each fusion.

The constituent parts of a quantum computer are inherently vulnerable to errors. To this end, we have developed quantum error-correcting codes to protect quantum information from noise. However, discovering codes that are capable of a universal set of computational operations with the minimal cost in quantum resources remains an important and ongoing challenge. One proposal of significant recent interest is the gauge color code. Notably, this code may offer a reduced resource cost over other well-studied fault-tolerant architectures by using a new method, known as gauge fixing, for performing the non-Clifford operations that are essential for universal quantum computation. Here we examine the gauge color code when it is subject to noise. Specifically, we make use of single-shot error correction to develop a simple decoding algorithm for the gauge color code, and we numerically analyse its performance. Remarkably, we find threshold error rates comparable to those of other leading proposals. Our results thus provide the first steps of a comparative study between the gauge color code and other promising computational architectures.