There are several known schemes for entangling trapped ion quantum bits for large-scale quantum computation. Most are based on an interaction between the ions and external optical fields, coupling internal qubit states of trapped ions to their Coulomb-coupled motion. In this paper, we examine the sensitivity of these motional gate schemes to phase fluctuations introduced through noisy external control fields, and suggest techniques for suppressing the resulting phase decoherence.

Spontaneous symmetry breaking can lead to the formation of time crystals, as well as spatial crystals. Here we propose a space-time crystal of trapped ions and a method to realize it experimentally by confining ions in a ring-shaped trapping potential with a static magnetic field. The ions spontaneously form a spatial ring crystal due to Coulomb repulsion. This ion crystal can rotate persistently at the lowest quantum energy state in magnetic fields with fractional fluxes. The persistent rotation of trapped ions produces the temporal order, leading to the formation of a space-time crystal. We show that these space-time crystals are robust for direct experimental observation. We also study the effects of finite temperatures on the persistent rotation. The proposed space-time crystals of trapped ions provide a new dimension for exploring many-body physics and emerging properties of matter.

Logical qubit encoding and quantum error correction (QEC) have been experimentally demonstrated in various physical systems with multiple physical qubits, however, logical operations are challenging due to the necessary nonlocal operations. Alternatively, logical qubits with bosonic-mode-encoding are of particular interest because their QEC protection is hardware efficient, but gate operations on QEC protected logical qubits remain elusive. Here, we experimentally demonstrate full control on a single logical qubit with a binomial bosonic code, including encoding, decoding, repetitive QEC, and high-fidelity (97.0% process fidelity on average) universal quantum gate set on the logical qubit. The protected logical qubit has shown 2.8 times longer lifetime than the uncorrected one. A Ramsey experiment on a protected logical qubit is demonstrated for the first time with two times longer coherence than the unprotected one. Our experiment represents an important step towards fault-tolerant quantum computation based on bosonic encoding.

We propose a method to generate massive entanglement in a spinor Bose-Einstein condensate from an initial product state through adiabatic sweep of magnetic field across a quantum phase transition induced by competition between the spin-dependent collision interaction and the quadratic Zeeman effect. The generated many-body entanglement is characterized by the experimentally measurable entanglement depth in the proximity of the Dicke state. We show that the scheme is robust to practical noise and experimental imperfection and under realistic conditions it is possible to generate genuine entanglement for hundreds of atoms.Generation of massive entanglement, besides its interest for foundational research of quantum theory, is of great importance for applications in quantum information processing and precision measurements. Entanglement is a valuable resource that can be used to enhance the performance of quantum computation, the security of quantum communication, and the precision of quantum measurements. For these applications, it is desirable to get as many particles as possible into entangled states. However, entanglement is typically fragile and many-particle entangled states can be easily destroyed by decoherence due to inevitable coupling to the environment. As an experimental record, so far fourteen qubits carried by trapped ions have been successfully prepared into genuine entangled states [1]. Pushing up this number represents a challenging goal in the experimental frontier.The Bose Einstein condensate of ultracold atoms is in a pure quantum mechanical state with strong collision interaction. In a spinor condensate [2-4], the spin-dependent collision interaction can be used to produce spin squeezing [5,6], which is an indicator of many-particle entanglement [8]. Spin squeezing has been demonstrated in condensates in recent experiments through spin-dependent collision dynamics [6,7]. A squeezed state is typically sensitive to noise and generation of substantial squeezing requires accurate control of experimental systems, which is typically challenging. In quantum information theory, the Dicke states are known to be relatively robust to noise and they have important applications for quantum metrology [9] and implementation of quantum information protocols [10]. For instance, the three-particle Dicke state, the so-called W state, has been proven to be the most robust entangled state under the particle loss [11]. Because of their applications and nice noise properties, Dicke states represent an important class of many-body states that are pursed in physical implementation. For a few particles, Dicke states have been generated in several experimental systems [12] .In this paper, we propose a robust method to generate massive entanglement in the proximity of many-particle Dicke states through control of adiabatic passage across a quantum phase transition in a spinor condensate. Using conservation of the magnetic quantum number, we show that sweep of the magnetic field across the polarferromagnetic phase transiti...

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Dynamical quantum phase transitions are closely related to equilibrium quantum phase transitions for ground states. Here, we report an experimental observation of a dynamical quantum phase transition in a spinor condensate with correspondence in an excited state phase diagram, instead of the ground state one. We observe that the quench dynamics exhibits a non-analytical change with respect to a parameter in the final Hamiltonian in the absence of a corresponding phase transition for the ground state there. We make a connection between this singular point and a phase transition point for the highest energy level in a subspace with zero spin magnetization of a Hamiltonian. We further show the existence of dynamical phase transitions for finite magnetization corresponding to the phase transition of the highest energy level in the subspace with the same magnetization. Our results open a door for using dynamical phase transitions as a tool to probe physics at higher energy eigenlevels of many-body Hamiltonians.

We prove that all inseparable Gaussian states of two modes can be distilled into maximally entangled pure states by local operations. Using this result we show that a bipartite Gaussian state of arbitrarily many modes can be distilled if and only if its partial transpose is not positive.

The realization of high fidelity quantum gates in a multi-qubit system, with a typical target set at 99.9%, is a critical requirement for the implementation of fault-tolerant quantum computation. To reach this level of fidelity, one needs to carefully analyze the noises and imperfections in the experimental system and optimize the gate operations to mitigate their effects. Here, we consider one of the leading experimental systems for the fault-tolerant quantum computation, ions in an anharmonic linear Paul trap, and optimize entangling quantum gates using segmented laser pulses with the assistance of all the collective transverse phonon modes of the ion crystal. We present detailed analyses of the effects of various kinds of intrinsic experimental noises as well as errors from imperfect experimental controls. Through explicit calculations, we find the requirements on these relevant noise levels and control precisions to achieve the targeted high fidelity of 99.9% for the entangling quantum gates in a multi-ion crystal.