(c) The process map Φ α β γ as measured by the quantum process tomography. The 64 real parameters Φ α β γ thus fully specify Φ. Here, α, β, and γ run over O, X, Y, and Z, where σ ^ 0 is the identity matrix and σ ^ X, σ ^ Y, and σ ^ Z are the corresponding Pauli matrices. The process map Φ, which describes the evolution of the system in each cycle of the protocol, is given by Φ ( ρ ^ DE ) = ∑ α, β, γ Φ α β γ ρ γ DE σ ^ α σ ^ β, where ρ ^ DE = ∑ γ ρ γ DE σ ^ γ is the density matrix that describes the input DE state and Φ ( ρ ^ DE ) describes the DE-plus-one-photon state after applying the process on the input DE qubit. In both cases, the DE spin is initialized to the | − X DE ⟩ state with a measured fidelity of F = 0.86 ± 0.02 and then one or two applications of the process. The fidelity of the measured density matrix to the ideal one is F = 0.77 ± 0.04 in (a) and F = 0.68 ± 0.07 in (b). In (a) and (b), the colored bars represent measured density-matrix elements, while the empty bars represent the ideal maximally entangled two- and three-qubit states. (b) The measured DE-plus-two-photons density matrix. (a) The measured DE-plus-one-photon density matrix. We project the polarization of the photons using liquid-crystal variable retarders (LCVRs) and polarizing beam splitters (PBSs) while looking for up to three photon correlations. The upward-pointing arrows are 12-ps-long π-area laser pulses, while the downward-pointing pink exponential decays represent the emitted photons. The blue downward-pointing broad arrow is an optical-depletion 7-ns-long pulse, which depletes the QD of carriers. (e) The experimental setup, pulse sequence, photon emissions, and detection for realizing and characterizing the cluster state. (d) The cluster-state repeating protocol containing Hadamard gates acting on the QD confined spin followed by a two-qubit controlled- not ( cnot) gate, which entangles the spin polarization with the polarization of the emitted photons.

A coherent superposition of the DE (BIE) eigenstates precesses with a period marked by T DE ( T BIE). The short-range electron-hole exchange interaction removes the degeneracy between the two eigenstates of both the DE and the BIE. Here, + X ( − X) describes a horizontally (vertically) rectilinearly polarized optical transition. (c) The BIE-DE*-DE energy levels and the polarization-selection rules for the optical transitions. The DE* then decays to its ground level by emitting an acoustic-phonon (curly downward-pointing arrow). The BIE then decays to an excited DE* by emitting a single-photon (downward pink arrow). Each laser pulse (red upward-pointing arrow) excites the DE and photogenerates a BIE. (b) The energy levels of the DE-DE*-BIE system and the transitions between these levels.

(a) A schematic description of the QD-based device: DBR, distributed Bragg reflector.

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