Shor's algorithm, which given appropriate hardware can factorise an integer N in a time polynomial in its binary length L, has arguably spurred the race to build a practical quantum computer. Several different quantum circuits implementing Shor's algorithm have been designed, but each tacitly assumes that arbitrary pairs of qubits within the computer can be interacted. While some quantum computer architectures possess this property, many promising proposals are best suited to realising a single line of qubits with nearest neighbour interactions only. In light of this, we present a circuit implementing Shor's factorisation algorithm designed for such a linear nearest neighbour architecture. Despite the interaction restrictions, the circuit requires just 2L+4 qubits and to leading order requires 8L^4 2-qubit gates arranged in a circuit of depth 32L^3 --- identical to leading order to that possible using an architecture that can interact arbitrary pairs of qubits.

COMMUNICATION (1 of 8)Diamond materials are central to an increasing range of advanced technological demonstrations, from high power electronics to nanoscale quantum bioimaging with unprecedented sensitivity. [1] However, the full exploitation of diamond for these applications is often limited by the uncontrolled nature of the diamond material surface, which suffers from Fermi-level pinning and hosts a significant density of electromagnetic noise sources. [2] These issues occur despite the oxide-free and air-stable nature of the diamond crystal surface, which should be an ideal candidate for functionalization and chemical engineering. In this work, a family of previously unidentified and near-ubiquitous primal surface defects, which are assigned to differently reconstructed surface vacancies, is revealed. The density of these defects is quantified with X-ray absorption spectroscopy, their energy structures are elucidated by ab initio calculations, and their effect on near-surface quantum Many advanced applications of diamond materials are now being limited by unknown surface defects, including in the fields of high power/ frequency electronics and quantum computing and quantum sensing. Of acute interest to diamond researchers worldwide is the loss of quantum coherence in near-surface nitrogen-vacancy (NV) centers and the generation of associated magnetic noise at the diamond surface. Here for the first time is presented the observation of a family of primal diamond surface defects, which is suggested as the leading cause of band-bending and Fermi-pinning phenomena in diamond devices. A combination of density functional theory and synchrotron-based X-ray absorption spectroscopy is used to show that these defects introduce low-lying electronic trap states. The effect of these states is modeled on band-bending into the diamond bulk and it is shown that the properties of the important NV defect centers are affected by these defects. Due to the paramount importance of near-surface NV center properties in a growing number of fields, the density of these defects is further quantified at the surface of a variety of differently-treated device surfaces, consistent with best-practice processing techniques in the literature. The identification and characterization of these defects has wide-ranging implications for diamond devices across many fields.

Band bending is a central concept in solid-state physics that arises from local variations in charge distribution especially near semiconductor interfaces and surfaces [1][2][3]. Its precision measurement is vital in a variety of contexts from the optimisation of field effect transistors [4][5][6] to the engineering of qubit devices with enhanced stability and coherence [7][8][9]. Existing methods are surface sensitive and are unable to probe band bending at depth from surface or bulk charges related to crystal defects [1, 10-12]. Here we propose an in-situ method for probing band bending in a semiconductor device by imaging an array of atomic-sized quantum sensing defects to report on the local electric field. We implement the concept using the nitrogen-vacancy centre in diamond [13,14], and map the electric field at different depths under various surface terminations. We then fabricate a two-terminal device based on the conductive two-dimensional hole gas formed at a hydrogen-terminated diamond surface [15], and observe an unexpected spatial modulation of the electric field attributed to a complex interplay between charge injection and photo-ionisation effects. Our method opens the way to three-dimensional mapping of band bending in diamond and other semiconductors hosting suitable quantum sensors, combined with simultaneous imaging of charge transport in complex operating devices [16].The emergence of semiconductor-based quantum sensing technologies in the last decade has opened new opportunities in a range of disciplines across physics, materials science and biology [17]. While most existing applications involve sensors that are external to the target sample to be measured [18,19], in-situ quantum sensors can also be an extremely valuable resource to study the sample itself by enabling three-dimensional (3D) mapping [20]. For semiconductor materials this is especially advantageous as it allows information to be gained on * These authors contributed equally to this work. †