Quantum state engineering, i.e., active control over the coherent dynamics of suitable quantum mechanical systems, has become a fascinating perspective of modern physics. With concepts developed in atomic and molecular physics and in the context of NMR, the field has been stimulated further by the perspectives of quantum computation and communication. For this purpose a number of individual two-state quantum systems (qubits) should be addressed and coupled in a controlled way. Several physical realizations of qubits have been considered, incl. trapped ions, NMR, and quantum optical systems. For potential applications such as logic operations, nano-electronic devices appear particularly promising because they can be embedded in electronic circuits and scaled up to large numbers of qubits.Here we review the quantum properties of low-capacitance Josephson junction devices. The relevant quantum degrees of freedom are either Cooper pair charges on small islands or fluxes in ring geometries, both in the vicinity of degeneracy points. The coherence of the superconducting state is exploited to achieve long phase coherence times. Single-and two-qubit quantum manipulations can be controlled by gate voltages or magnetic fields, by methods established for single-charge systems or the SQUID technology. Several of the interesting single-qubit properties, incl. coherent oscillations have been demonstrated in recent experiments, thus displaying in a spectacular way the laws of quantum mechanics in solid state devices. Further experiments, such as entanglement of qubit states, which are crucial steps towards a realization of logic elements, should be within reach.In addition to the manipulation of qubits the resulting quantum state has to be read out. For a Josephson charge qubit this can be accomplished by coupling it capacitively to a single-electron transistor (SET). To describe the quantum measurement process we analyze the time evolution of the density matrix of the coupled system. As long as the transport voltage is turned off, the transistor has only a weak influence on the qubit. When the voltage is switched on, the dissipative current through the SET destroys the phase coherence of the qubit within a short time. The measurement is accomplished only after a longer time, when the macroscopic signal, i.e., the current through the SET, resolves different quantum states. At still longer times the measurement process itself destroys the information about the initial state. Similar scenarios are found when the quantum state of a flux qubit is measured by a dc-SQUID, coupled to it inductively.

Decoherence in quantum bit circuits is presently a major limitation to their use for quantum computing purposes. We present experiments, inspired from NMR, that characterise decoherence in a particular superconducting quantum bit circuit, the quantronium. We introduce a general framework for the analysis of decoherence, based on the spectral densities of the noise sources coupled to the qubit. Analysis of our measurements within this framework indicates a simple model for the noise sources acting on the qubit. We discuss various methods to fight decoherence.Comment: Long paper. 65 pages, 18 Figure

Low-capacitance Josephson junctions, where Cooper pairs tunnel coherently while Coulomb blockade effects allow the control of the total charge, provide physical realizations of quantum bits (qubits), with logical states differing by one Cooper-pair charge on an island. The single-and two-bit operations required for quantum computation can be performed by applying a sequence of gate voltages. A basic design, described earlier [1], is sufficient to demonstrate the principles, but requires a high precision time control, and residual two-bit interactions introduce errors. Here we suggest a new nano-electronic design, close to ideal, where the Josephson junctions are replaced by controllable SQUIDs. This relaxes the requirements on the time control and system parameters substantially, and the two-bit coupling can be switched exactly between zero and a non-zero value for arbitrary pairs. The phase coherence time is sufficiently long to allow a series of operations.A quantum computer can perform certain tasks which no classical computer is able to do in acceptable times [2][3][4][5]. It is composed of a (large) number of coupled two-state quantum systems forming qubits; the computation is the quantum-coherent time evolution of the state of the system described by unitary transformations which are controlled by the program. Elementary steps are (i) the preparation of the initial state of the qubits, (ii) single-bit operations (gates), i.e. unitary transformation of individual qubit states, triggered by a modification of the corresponding one-qubit Hamiltonian for some period of time, (iii) two-bit gates, which require controlled inter-qubit couplings, and (iv) the measurement of the final quantum state of the system. The phase coherence time has to be long enough to allow a large number of these coherent processes. Ideally, in the idle period between the operations the Hamiltonian of the system is zero to avoid further time evolution of the states.Several physical realizations of quantum information systems have been suggested. Ions in a trap, manipulated by laser irradiation are the best studied system. However, alternatives need to be explored, in particular those which are more easily embedded in an electronic circuit as well as scaled up to large numbers of qubits. From this point of view mesoscopic and nano-electronic devices appear particularly promising [1,[6][7][8][9]. Normal-metal singleelectron devices are discussed in connection with classical digital applications and, in fact, constitute the ultimate electronic memory [10]. However, their use for quantum computation is ruled out, since, due to the large number of electron states involved, different tunneling processes are incoherent. Ultra-small quantum dots with discrete levels are candidates for qubits, but their strong coupling to the environment renders their phase coherence time short. More promising are systems built from Josephson junctions, where the coherence of the superconducting state can be exploited. Quantum extension of elements based on a sin...

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Landau-Zener (LZ) tunneling can occur with a certain probability when crossing energy levels of a quantum two-level system are swept across the minimum energy separation. Here we present experimental evidence of quantum interference effects in solid-state LZ tunneling. We used a Cooper-pair box qubit where the LZ tunneling occurs at the charge degeneracy. By employing a weak nondemolition monitoring, we observe interference between consecutive LZ-tunneling events; we find that the average level occupancies depend on the dynamical phase. The system's unusually strong linear response is explained by interband relaxation. Our interferometer can be used as a high-resolution Mach-Zehnder-type detector for phase and charge.

Entanglement of two parts of a quantum system is a nonlocal property unaffected by local manipulations of these parts. It is described by quantities invariant under local unitary transformations. Here we present, for a system of two qubits, a set of invariants which provides a complete description of nonlocal properties. The set contains 18 real polynomials of the entries of the density matrix. We prove that one of two mixed states can be transformed into the other by singlebit operations if and only if these states have equal values of all 18 invariants. Corresponding local operations can be found efficiently. Without any of these 18 invariants the set is incomplete.Similarly, nonlocal, entangling properties of two-qubit unitary gates are invariant under single-bit operations. We present a complete set of 3 real polynomial invariants of unitary gates. Our results are useful for optimization of quantum computations since they provide an effective tool to verify if and how a given two-qubit operation can be performed using exactly one elementary two-qubit gate, implemented by a basic physical manipulation (and arbitrarily many single-bit gates).Introduction. Nonlocality is an important ingredient in quantum information processing, e.g. in quantum computation and quantum communication. Nonlocal correlations in quantum systems reflect entanglement between its parts. Genuine nonlocal properties should be described in a form invariant under local unitary operations. In this paper we discuss such locally invariant properties of (i) unitary transformations and (ii) mixed states of a two-qubit system. Two unitary transformations (logic gates), M and L, are called locally equivalent if they differ only by local operations:⊗2 are combinations of single-bit gates on two qubits [1]. A property of a two-qubit operation can be considered nonlocal only if it has the same value for locally equivalent gates. We present a complete set of local invariants of a two-qubit gate: two gates are equivalent if and only if they have equal values of all these invariants. The set contains three real polynomials of the entries of the gate's matrix. This set is minimal: the group SU (4) of two-qubit gates is 15-dimensional and local operations eliminate 2 dim[SU (2) ⊗2 ] = 12 degrees of freedom; hence any set should contain at least 15 − 12 = 3 invariants.This result can be used to optimize quantum computations. Quantum algorithms are built out of elementary quantum logic gates. Any many-qubit quantum logic circuit can be constructed out of single-bit and two-bit operations [2][3][4][5]. The ability to perform 1-bit and 2-bit

Motivated by recent experiments with Josephson-junction circuits, we analyze the influence of various noise sources on the dynamics of two-level systems at optimal operation points where the linear coupling to low-frequency fluctuations is suppressed. We study the decoherence due to nonlinear (quadratic) coupling, focusing on the experimentally relevant 1/f and Ohmic noise power spectra. For 1/f noise strong higher-order effects influence the evolution.

Recent experiments indicate a connection between the low-and high-frequency noise affecting superconducting quantum systems. We explore the possibilities that both noises can be produced by one ensemble of microscopic modes, made up, e.g., by sufficiently coherent two-level systems (TLS). This implies a relation between the noise power in different frequency domains, which depends on the distribution of the parameters of the TLSs. We show that a distribution, natural for tunneling TLSs, with a log-uniform distribution in the tunnel splitting and linear distribution in the bias, accounts for experimental observations.Recent activities and progress with quantum information systems rely on the control of decoherence processes and at the same time provide novel tools to study their mechanisms. Experiments with superconducting qubits revealed the presence of spurious quantum two-level systems [1] with strong effects on the high-frequency (∼10 GHz) qubit dynamics. Other experiments [2] suggested a connection between the strengths of the Ohmic highfrequency noise, responsible for the relaxation of the qubit (T 1 decay), and the low-frequency 1/f noise, which dominates the dephasing (T 2 decay). The noise power spectra, extrapolated from the low-and high-frequency sides, cross at ω of order T . This is also compatible with the T 2 dependence of the low-frequency part, observed earlier for the 1/f noise in Josephson devices [3,4]. Much clearer evidence for the T 2 behavior was obtained recently [5,6].In this letter we point out that a set of coherent twolevel systems (or, in fact, arbitrary quantum systems with discrete spectrum) produces both high-and lowfrequency noise with strengths that are naturally related. We show that for a realistic distribution of parameters tunnel TLSs (TTLS) produce noise with experimentally detected features: the 1/f behavior at low frequencies, the Ohmic (∝ ω) high-frequency noise, and the T 2 temperature dependence of the integrated weight of the lowfrequency noise. This implies that the 1/f and Ohmic asymptotes cross at ω ∼ T as was indeed observed in Ref.[2] at one value of T . The distribution is log-uniform in the tunnel splitting and linear in the bias. Microscopically, this distribution may describe double traps or "Andreev fluctuators" considered recently by Faoro et al. [7] in their study of the relaxation (T 1 decay) of Josephson qubits due to the high-frequency noise. Our results are obtained for environments with a large number of TLSs which are weakly coupled to the qubit. A strong coupling between a TLS and a qubit can lead to resonances [1,2].