PACS numbers:Quantum entanglement is the central resource behind applications in quantum information science, from quantum computers [1] and simulators of complex quantum systems [2] to metrology [3] and secure communication [1]. All of these applications require the quantum control of large networks of quantum bits (qubits) to realize gains and speedups over conventional devices. However, propagating quantum entanglement generally becomes difficult or impossible as the system grows in size, owing to the inevitable decoherence from the complexity of connections between the qubits and increased couplings to the environment. Here, we demonstrate the first step in a modular approach [4] to scaling entanglement by utilizing a hierarchy of quantum buses on a collection of three atomic ion qubits stored in two remote ion trap modules. Entanglement within a module is achieved with deterministic near-field interactions through phonons [5], and remote entanglement between modules is achieved through a probabilistic interaction through photons [6]. This minimal system allows us to address generic issues in synchronization and scalability of entanglement with multiple buses, while pointing the way toward a modular large-scale quantum computer architecture that promises less spectral crowding and less decoherence. We generate this modular entanglement faster than the observed qubit decoherence rate, thus the system can be scaled to much larger dimensions by adding more modules.Small modules of qubits have been entangled through native local interactions in many physical platforms, such as trapped atomic ions through their Coulomb interaction [5], Rydberg atoms through their electric dipoles [7,8], nitrogen-vacancy centers in diamond through their magnetic dipoles [9], and superconducting Josephson junctions through capacitive or inductive couplings [10,11]. However, each of these systems is confronted with practical limits to the number of qubits that can be reliably controlled, stemming from inhomogeneities, the complexity and density of the interactions between the qubits, or quantum decoherence. Scaling beyond these limits can be achieved by invoking a second type of interaction that can extend the entanglement to other similar qubit modules. Such an architecture should therefore exploit both the local interactions within the qubit modules, and also remote interactions between modules (an example architecture is shown in Fig 1). Optical interfaces provide ideal buses for this purpose [12,13], as optical photons can propagate over macroscopic distances with negligible loss. Several qubit systems have been entangled through remote optical buses, such as atomic ions [14], neutral atoms [15], and nitrogen-vacancy centers in diamond [16].In the experiment reported here, we juxtapose local and remote entanglement buses utilizing trapped atomic ion qubits, balancing the requirements of each interface within the same qubit system. The observed entanglement rate within and between modules is faster than the observed entangled qub...

We demonstrate a simple pulse shaping technique designed to improve the fidelity of spin-dependent force operations commonly used to implement entangling gates in trapped ion systems. This extension of the Mølmer-Sørensen gate can theoretically suppress the effects of certain frequency and timing errors to any desired order and is demonstrated through Walsh modulation of a two qubit entangling gate on trapped atomic ions. The technique is applicable to any system of qubits coupled through collective harmonic oscillator modes.

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