We construct a new class of quantum error-correcting codes for a bosonic mode, which are advantageous for applications in quantum memories, communication, and scalable computation. These "binomial quantum codes" are formed from a finite superposition of Fock states weighted with binomial coefficients. The binomial codes can exactly correct errors that are polynomial up to a specific degree in bosonic creation and annihilation operators, including amplitude damping and displacement noise as well as boson addition and dephasing errors. For realistic continuous-time dissipative evolution, the codes can perform approximate quantum error correction to any given order in the time step between error detection measurements. We present an explicit approximate quantum error recovery operation based on projective measurements and unitary operations. The binomial codes are tailored for detecting boson loss and gain errors by means of measurements of the generalized number parity. We discuss optimization of the binomial codes and demonstrate that by relaxing the parity structure, codes with even lower unrecoverable error rates can be achieved. The binomial codes are related to existing two-mode bosonic codes, but offer the advantage of requiring only a single bosonic mode to correct amplitude damping as well as the ability to correct other errors. Our codes are similar in spirit to "cat codes" based on superpositions of the coherent states but offer several advantages such as smaller mean boson number, exact rather than approximate orthonormality of the code words, and an explicit unitary operation for repumping energy into the bosonic mode. The binomial quantum codes are realizable with current superconducting circuit technology, and they should prove useful in other quantum technologies, including bosonic quantum memories, photonic quantum communication, and optical-to-microwave up-and down-conversion.

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.

Perovskite structured materials contain myriad tunable ordered phases of electronic and magnetic origin with proven technological importance and strong promise for a variety of energy solutions. An always-contributing influence beneath these cooperative and competing interactions is the lattice, whose physics may be obscured in complex perovskites by the many coupled degrees of freedom, which makes these systems interesting. Here, we report signatures of an approach to a quantum phase transition very near the ground state of the nonmagnetic, ionic insulating, simple cubic perovskite material ScF 3 , and show that its physical properties are strongly effected as much as 100 K above the putative transition. Spatial and temporal correlations in the high-symmetry cubic phase determined using energy-and momentum-resolved inelastic x-ray scattering as well as x-ray diffraction reveal that soft mode, central peak, and thermal expansion phenomena are all strongly influenced by the transition. The class of materials with the perovskite structure and chemical formula ABX 3 contains examples of perhaps every possible type of physical behavior [1,2], much of which is difficult to understand because of the shear complexity of matter. A rich terrain of structural transitions associated with BX 6 octahedral tilting in perovskites strongly effects electronic conduction and magnetic exchange pathways, defining the framework of interactions governing a range of physical properties. The A-site tolerance appears to be an important parameter in determining the structural phase stability [1-3], but stable A-site-free perovskite structures are also thermodynamically stable. These are rare cases among oxides (X = O) because the B ions must take on rare hexavalent (+6) electronic configurations, and the only known instance is ReO 3 . In perovskites based on fluorine (X = F), however, the B ions assume the common trivalent (+3) configuration in an expanded suite of A-site-free perovskite lattices.Figure 1(a) shows a structural phase diagram of BF 3 perovskites, where B is a trivalent metal ion [4]. The 3d metal trifluorides display a reversible [5] structural cubic-torhombohedral (c-r) phase boundary. This sequence of 3d transition metal trifluoride compounds is rhombohedral at room temperature, with the exception of B = Sc, which appears at the zero-temperature terminated c-r phase boundary. Indeed, no rhombohedral phase transition has been observed for ScF 3 down to 0.4 K [6], suggesting that near this composition, the structural phase can be driven by a parameter other than temperature, implying that the ground state of this ionic insulator is very near a quantum phase transition (QPT). Cubic ScF 3 further stands out among substances in that it has the most stable structural phase of any known solid trifluoride, * Corresponding author: jason.hancock@uconn.edu retaining high cubic symmetry and a four-atom unit cell up to its high melting point, >1800 K [6,7]. Separate from the QPT reported here, further interest in this system is due t...

There are two general requirements to harness the computational power of quantum mechanics: the ability to manipulate the evolution of an isolated system and the ability to faithfully extract information from it. Quantum error correction and simulation often make a more exacting demand: the ability to perform nondestructive measurements of specific correlations within that system. We realize such measurements by employing a protocol adapted from Nigg and Girvin [Phys. Rev. Lett. 110, 243604 (2013)], enabling real-time selection of arbitrary register-wide Pauli operators. Our implementation consists of a simple circuit quantum electrodynamics module of four highly coherent 3D transmon qubits, collectively coupled to a high-Q superconducting microwave cavity. As a demonstration, we enact all seven nontrivial subset-parity measurements on our three-qubit register. For each, we fully characterize the realized measurement by analyzing the detector (observable operators) via quantum detector tomography and by analyzing the quantum backaction via conditioned process tomography. No single quantity completely encapsulates the performance of a measurement, and standard figures of merit have not yet emerged. Accordingly, we consider several new fidelity measures for both the detector and the complete measurement process. We measure all of these quantities and report high fidelities, indicating that we are measuring the desired quantities precisely and that the measurements are highly nondemolition. We further show that both results are improved significantly by an additional error-heralding measurement. The analyses we present here form a useful basis for the future characterization and validation of quantum measurements, anticipating the demands of emerging quantum technologies.