The number of steps any classical computer requires in order to find the prime factors of an l-digit integer N increases exponentially with l, at least using algorithms known at present. Factoring large integers is therefore conjectured to be intractable classically, an observation underlying the security of widely used cryptographic codes. Quantum computers, however, could factor integers in only polynomial time, using Shor's quantum factoring algorithm. Although important for the study of quantum computers, experimental demonstration of this algorithm has proved elusive. Here we report an implementation of the simplest instance of Shor's algorithm: factorization of N = 15 (whose prime factors are 3 and 5). We use seven spin-1/2 nuclei in a molecule as quantum bits, which can be manipulated with room temperature liquid-state nuclear magnetic resonance techniques. This method of using nuclei to store quantum information is in principle scalable to systems containing many quantum bits, but such scalability is not implied by the present work. The significance of our work lies in the demonstration of experimental and theoretical techniques for precise control and modelling of complex quantum computers. In particular, we present a simple, parameter-free but predictive model of decoherence effects in our system.

Dielectric loss from two-level states is shown to be a dominant decoherence source in superconducting quantum bits. Depending on the qubit design, dielectric loss from insulating materials or the tunnel junction can lead to short coherence times. We show that a variety of microwave and qubit measurements are well modeled by loss from resonant absorption of two-level defects. Our results demonstrate that this loss can be significantly reduced by using better dielectrics and fabricating junctions of small area . With a redesigned phase qubit employing low-loss dielectrics, the energy relaxation rate has been improved by a factor of 20, opening up the possibility of multiqubit gates and algorithms.

We report a superconducting artificial atom with an observed quantum coherence time of T * 2 =95µs and energy relaxation time T1=70µs. The system consists of a single Josephson junction transmon qubit embedded in an otherwise empty copper waveguide cavity whose lowest eigenmode is dispersively coupled to the qubit transition. We attribute the factor of four increase in the coherence quality factor relative to previous reports to device modifications aimed at reducing qubit dephasing from residual cavity photons. This simple device holds great promise as a robust and easily produced artificial quantum system whose intrinsic coherence properties are sufficient to allow tests of quantum error correction. PACS numbers: 03.67.Ac, 42.50.Pq, 85.25.-j Superconducting quantum circuits are a leading candidate technology for large scale quantum computing. They have been used to show a violation of a Bell-type inequality [1]; implement a simple two-qubit gate favorable for scaling [2]; generate three-qubit entanglement [3]; perform a routine relevant to error correction [4];and very recently to demonstrate a universal set of quantum gates with fidelities greater than 95% [5]. Most of these devices employ small angle-evaporated Josephson junctions as their critical non-linear circuit components. Devices designs appear to be consistent with the basic requirements for quantum error correction (QEC) and fault tolerance [6]. However, the construction and operation of much larger systems capable of meaningful tests of such procedures will require individual qubits and junctions with a very high degree of coherence. Current estimates for threshold error rates -and the cumulative nature of errors originating from control, measurement, and decoherence -make likely the need for quantum lifetimes at least 10 3 times longer than gate and measurement times [7], corresponding to 20 to 200µs for typical systems.To this end, improvements in qubit lifetimes have continued for the past decade, spurred largely by clever methods of decoupling noise and loss mechanisms from the qubit transition and thus realizing Hamiltonians more closely resembling their idealized versions. Recently, Paik, et al. made a breakthrough advance [8] by embedding a transmon qubit [9, 10] in a superconducting waveguide cavity. Dubbed three-dimensional circuit QED (3D cQED), this system produced significantly enhanced qubit lifetimes of T 1 =25-60µs and T * 2 =10-20µs, corresponding to quality factors for dissipation and decoherence of Q 1 ≈1.8×10 6 and Q 2 ≈7×10 5 , respectively.These results lead to two important questions. First, are similar coherence properties observable using other fabrication processes, facilities, and measurement setups? Second, what is the origin of the dephasing process suppressing T * 2 well below the no-pure-dephasing limit of 2T 1 ? Is it intrinsic to the junctions or to this qubit ar-chitecture? The weight and urgency of these questions are increased by implications on scaling potential: if the results are reproducible and decoherence tim...

Demonstration of quantum entanglement, a key resource in quantum computation arising from a nonclassical correlation of states, requires complete measurement of all states in varying bases. By using simultaneous measurement and state tomography, we demonstrated entanglement between two solid-state qubits. Single qubit operations and capacitive coupling between two super-conducting phase qubits were used to generate a Bell-type state. Full two-qubit tomography yielded a density matrix showing an entangled state with fidelity up to 87%. Our results demonstrate a high degree of unitary control of the system, indicating that larger implementations are within reach.