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\b[A <quantum computing> apparatus needs a scalable <quantum register> of controllable <qubits>, reproducible initialization, a <universal quantum gate set>, sufficiently long <quantum coherence>, and selective readout.] A <qubit> is a two-dimensional state space; several <qubits> occupy a <tensor product> space and can develop <entanglement>. <Quantum state preparation> must establish a known low-entropy input and supply fresh <ancillas> during <quantum error correction>. A <universal quantum gate set> may use arbitrary single-qubit <rotations> with an entangling two-qubit gate such as <controlled-NOT gate>. <Quantum coherence> and control fidelity must suffice for the intended circuit, and measurement must recover chosen output <probabilities> without uncontrolled disturbance of other <qubits>. These are the five <DiVincenzo criteria>.<The Physical Implementation of Quantum Computation by David P. DiVincenzo>[Section III, requirements 1–5]{ref}
An atomic implementation can encode <qubits> in two long-lived internal levels, initialize them by <optical pumping>, drive coherent transitions with <laser> or microwave pulses, couple selected <qubits> through a shared interaction, and distinguish states by state-dependent fluorescence. For trapped ions, a shared vibrational mode can mediate entangling gates, so motional control and cooling are part of the computational apparatus. The motion is a resource only while uncontrolled heating does not erase the required <quantum coherence>.
The computational resource is coherent control of amplitudes and <entanglement>, with <interference> making useful outcomes likely. An $n$-qubit <pure state> has $2^n$ basis amplitudes, but a single measurement does \b[not] reveal those amplitudes or all possible answers. Algorithms exploit structure rather than simply reading out parallel classical computations. The <Shor algorithm> gives factoring with polynomial resource scaling in the input bit length, compared with the best known classical algorithms' superpolynomial scaling; an exponential classical lower bound has not been proved.<Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer by Peter W. Shor>[Abstract, polynomial running time in the number of input digits]{ref} <Grover's algorithm> uses order $\sqrt M$ oracle queries for unstructured search among $M$ possibilities, a quadratic query improvement.<A fast quantum mechanical algorithm for database search by Lov Grover>[Abstract and algorithmic query scaling]{ref} <Quantum simulation> is a further motivation because interacting quantum systems can require exponentially many classical amplitudes for a direct generic representation.
The main challenges are <quantum decoherence>, fluctuating control fields, imperfect gates, leakage out of the <qubit> subspace, readout errors, crosstalk, and the difficulty of scaling connectivity and calibration. Faster control can introduce extra errors, while stronger isolation can make preparation and readout harder. <Quantum error correction> encodes logical information redundantly in several physical <qubits> and extracts error syndromes without measuring the unknown logical state itself. <Fault-tolerant quantum computation> prevents the correction apparatus and gates from spreading a few faults into uncorrectable logical errors. It requires error rates below an architecture- and noise-dependent threshold and substantial resources; there is no universal numerical threshold for all implementations. \b[Advantages are task-dependent, while reliable scaling requires both coherent hardware and error control.]
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