The early Gottesman, Kitaev, and Preskill (GKP) proposal for encoding a qubit in an oscillator has recently been followed by cat-and binomial-code proposals. Numerically optimized codes have also been proposed, and we introduce codes of this type here. These codes have yet to be compared using the same error model; we provide such a comparison by determining the entanglement fidelity of all codes with respect to the bosonic pure-loss channel (i.e., photon loss) after the optimal recovery operation. We then compare achievable communication rates of the combined encoding-error-recovery channel by calculating the channel's hashing bound for each code. Cat and binomial codes perform similarly, with binomial codes outperforming cat codes at small loss rates. Despite not being designed to protect against the pure-loss channel, GKP codes significantly outperform all other codes for most values of the loss rate. We show that the performance of GKP and some binomial codes increases monotonically with increasing average photon number of the codes. In order to corroborate our numerical evidence of the cat-binomial-GKP order of performance occurring at small loss rates, we analytically evaluate the quantum error-correction conditions of those codes. For GKP codes, we find an essential singularity in the entanglement fidelity in the limit of vanishing loss rate. In addition to comparing the codes, we draw parallels between binomial codes and discrete-variable systems. First, we characterize one-and two-mode binomial as well as multiqubit permutation-invariant codes in terms of spin-coherent states. Such a characterization allows us to introduce check operators and error-correction procedures for binomial codes. Second, we introduce a generalization of spin-coherent states, extending our characterization to qudit binomial codes and yielding a multiqudit code.

We investigate quantum control of an oscillator mode off-resonantly coupled to an ancillary qubit. In the strong dispersive regime, we may drive the qubit conditioned on number states of the oscillator, which together with displacement operations can achieve universal control of the oscillator. Based on our proof of universal control, we provide explicit constructions for arbitrary state preparation and arbitrary unitary operation of the oscillator. Moreover, we present an efficient procedure to prepare the number state |n using only O ( √ n) operations. We also compare our scheme with known quantum control protocols for coupled qubit-oscillator systems. This universal control scheme of the oscillator can readily be implemented using superconducting circuits.PACS numbers: 03.65. Vf, 37.10.Jk, 42.50.Lc As an important model for quantum information processing, the coupled qubit-oscillator system has been actively investigated in various platforms, including trapped ions [1], nano-photonics [2], cavity QED [3], and circuit QED [4]. Due to its convenient control, the physical qubit is usually the primary resource for quantum information processing. Meanwhile, the oscillator serves as an auxiliary system for quantum state transfer and detection [5]. In fact, the oscillator, associated with the phononic or photonic mode, may have long coherent times [1,6,7] and the large Hilbert space associated with the oscillator can be used for quantum encoding [8-10] and autonomous error correction with engineered dissipation [11]. These crucial features call for deeper investigations into quantum control theory of an oscillator.The seminal work by Law and Eberly [12] has triggered many theoretical and experimental investigations to prepare quantum states of the oscillator assisted by an ancillary qubit with Jaynes-Cummings (JC) coupling [1,[13][14][15], while the general problem of implementing arbitrary unitary operation remains an outstanding challenge. Even with recent advances, protocols for universal control require either a large number of control operations [16] or a more complicated model with an ancillary three-level system [17]. Meanwhile, development in superconducting circuits acting in the strong dispersive regime opens new possibilities for universal control of the oscillator [18].In this Letter, we provide schemes for arbitrary state preparation and universal control of the oscillator assisted by an ancillary qubit. These schemes utilize the dispersive Hamiltonian [18] along with two types of drives associated with the qubit and the oscillator, respectively.The key is the capability to drive the qubit [9,10,[18][19][20][21] and impart arbitrary phases conditioned on the number state of the oscillator.The Hamiltonian of the qubit-oscillator system iŝ Figure 1. Energy level diagram of qubit-oscillator system. In the rotating frame of the oscillator, the states {|g, n } n have the same energy. After each operation, the population (orange circles) remains in the subspace associated with |g . The displacement operatio...

Three-dimensional (3D) topological insulators in general need to be protected by certain kinds of symmetries other than the presumed U (1) charge conservation. A peculiar exception is the Hopf insulators which are 3D topological insulators characterized by an integer Hopf index. To demonstrate the existence and physical relevance of the Hopf insulators, we construct a class of tight-binding model Hamiltonians which realize all kinds of Hopf insulators with arbitrary integer Hopf index. These Hopf insulator phases have topologically protected surface states and we numerically demonstrate the robustness of these topologically protected states under general random perturbations without any symmetry other than the U (1) charge conservation that is implicit in all kinds of topological insulators.Topological phases of matter may be divided into two classes: the intrinsic ones and the symmetry protected ones 1 . Symmetry protected topological (SPT) phases are gapped quantum phases that are protected by symmetries of the Hamiltonian and cannot be smoothly connected to the trivial phases under perturbations that respect the same kind of symmetries. Intrinsic topological (IT) phases, on the other hand, do not require symmetry protection and are topologically stable under arbitrary perturbations. Unlike SPT phases, IT phases may have exotic excitations bearing fractional or even non-Abelian statistics in the bulk 2 . Fractional 3 quantum Hall states and spin liquids 4 belong to these IT phases. Remarkable examples of the SPT phases include the well known 2D and 3D topological insulators and superconductors protected by time reversal symmetry 5-7 , and the Haldane phase of the spin-1 chain protected by the SO(3) spin rotational symmetry 8 . For interacting bosonic systems with on-site symmetry G, distinct SPT phases can be systematically classified by group cohomology of G 1 , while for free fermions, the SPT phases can be systematically described by K-theory or homotopy group theory 9 , which leads to the well known periodic table for topological insulators and superconductors 10,11 .Most 3D topological insulators have to be protected by some other symmetries 10,11 , such as time reversal, particle hole or chrial symmetry, and the U (1) charge conservation symmetry 12 . A peculiar exception occurs when the Hamiltonian has just two effective bands. In this case, interesting topological phases, the so-called Hopf insulators 13 , may exist. These Hopf insulator phases have no symmetry other than the prerequisite U (1) charge conservation. To elucidate why this happens, let us consider a generic band Hamiltonian in 3D with m filled bands and n empty bands. Without symmetry constraint, the space of such Hamiltonians is topologically equivalent to the Grassmannian manifold G m,m+n and can be classified by the homotopy group of this Grassmannian 11 . Since the homotopy group π 3 (G m,m+n ) = {0} for all (m, n) = (1, 1), there exists no nontrivial topological phase in general. However, when m = n = 1, G 1,2 is topologically equival...