We show how to construct a large class of quantum error-correcting codes, known as CalderbankSteane-Shor codes, from highly entangled cluster states. This becomes a primitive in a protocol that foliates a series of such cluster states into a much larger cluster state, implementing foliated quantum error correction. We exemplify this construction with several familiar quantum error-correction codes and propose a generic method for decoding foliated codes. We numerically evaluate the error-correction performance of a family of finite-rate Calderbank-Steane-Shor codes known as turbo codes, finding that they perform well over moderate depth foliations. Foliated codes have applications for quantum repeaters and fault-tolerant measurement-based quantum computation. DOI: 10.1103/PhysRevLett.117.070501 Quantum error correction is critical to building practical quantum-information processors (QIPs). In an influential series of papers, Raussendorf and co-workers described a measurement-based approach to fault-tolerant quantum processing using highly entangled cluster states, defined on a 3D lattice [1-4]. Raussendorf's 3D cluster state can be visualized as a foliation of Kitaev's surface code [5,6], i.e., a sequence of 2D surface-code "sheets," stacked together to form a 3D lattice. This is evident in Ref. [1], where it is shown that measuring the "bulk" qubits of a 3D cluster state leaves the two logical surface-code qubits encoded in the boundary faces in an entangled Bell pair.Raussendorf's 3D cluster gained prominence for its high fault-tolerant computational error thresholds ≲1%. It has applications in various QIP tasks, including long-range entanglement sharing, in which surface-code cluster states are created at regularly spaced local nodes, which are linked by medium-range optical channels into a 3D cluster state [7]. It is capable of fault-tolerant, measurement-based quantum computation, using an elegant geometric construction that braids defects in the interior of the 3D cluster state to produce robust Clifford gates. Universality is afforded by magic state injection and distillation [3,4,8].The robustness of Refs. [3,4] is inherited from the underlying surface code, which has a high error-correction threshold ∼11% [6,9-11]. The surface code has large distance and zero rate (in regards to the asymptotic ratio of the number of logical and physical qubits), reflecting the trade-off between distance and rate in two spatial dimensions [12]. It is natural to ask how to adapt the foliated structure of Refs. [1,2] to use other underlying codes that could achieve a higher encoding rate.Another motivation for our work is recent fault-tolerant schemes that produce a universal gate set by code deformation and code switching [13][14][15]. Extending code foliation to codes that circumvent magic state distillation [8] may produce cluster states with a lower resource overhead for fault-tolerant measurement-based QIPs.In this Letter we show that all Calderbank-Steane-Shor (CSS) codes can be clusterized, meaning that they can...

We present spectroscopic observations of ultracompact dwarf (UCD) galaxies in the Fornax and Virgo clusters made to measure and compare their stellar populations. The spectra were obtained on the Gemini‐North (Virgo) and Gemini‐South (Fornax) telescopes using the respective Gemini Multi‐Object Spectrographs. We estimated the ages, metallicities and abundances of the objects from measurements of Lick line‐strength indices in the spectra; we also estimated the ages and metallicities independently using a direct spectral fitting technique. Both methods revealed that the UCDs are old (mean age 10.8 ± 0.7 Gyr) and (generally) metal rich (mean [Fe/H] = −0.8 ± 0.1). The α‐element abundances of the objects measured from the Lick indices are super‐solar. We used these measurements to test the hypothesis that UCDs are formed by the tidal disruption of present‐day nucleated dwarf elliptical galaxies. The data are not consistent with this hypothesis because both the ages and abundances are significantly higher than those of observed dwarf galaxy nuclei (this does not exclude disruption of an earlier generation of dwarf galaxies). They are more consistent with the properties of globular star clusters, although at higher mean metallicity. The UCDs display a very wide range of metallicity (−1.7 < [Fe/H] < 0.0), spanning the full range of both globular clusters and dwarf galaxy nuclei. We confirm previous reports that most UCDs have high metalliticities for their luminosities, lying significantly above the canonical metallicitiy–luminosity relation followed by early‐type galaxies. In contrast to previous work, we find that there is no significant difference in either the mean ages or the mean metallicities of the Virgo and Fornax UCD populations.

Foliated quantum codes are a resource for fault-tolerant measurement-based quantum error correction for quantum repeaters and for quantum computation. They represent a general approach to integrating a range of possible quantum error correcting codes into larger fault-tolerant networks.Here we present an efficient heuristic decoding scheme for foliated quantum codes, based on message passing between primal and dual code 'sheets'. We test this decoder on two different families of sparse quantum error correcting code: turbo codes and bicycle codes, and show reasonably high numerical performance thresholds. We also present a construction schedule for building such code states.Quantum information processing (QIP) requires that the computational process must be performed with high fidelity. In a noisy environment this will require quantum error correction (QEC) [1,2]. Depending on the computational model, this noise manifests in different ways. A conceptually and technologically important step in the project to build quantum computers was the observation by Raussendorf et al. [3,4] that highly-entangled cluster states, are universal resource states with which to perform a quantum computation. In cluster-state-based computation, the computation is driven forward by a series of measurements.Subsequently, Raussendorf et al. [5][6][7][8] proposed a method of fault-tolerant quantum computation utilising cluster states. In this scheme a 3D cluster state lattice is constructed, which can be viewed as a foliation of Kitaev's surface code [9,10]. Alternating sheets within this foliated structure correspond to primal or dual surface codes. Measurements on the bulk qubits generate correlations between corresponding logical qubits on the boundary of the lattice. In these schemes, errors are partially revealed through parity checks operators, which can be determined from the outcomes of single qubit measurements.Raussendorf's 3D measurement-based computation scheme has proved important for the practical development of quantum computers [11][12][13][14], due to its high faulttolerant computational error thresholds 1%[10]. Furthermore, the robustness to erasure errors [15-17] makes these schemes attractive for various architectures, including optical networks [18]. This high threshold is a result of the underlying surface code, which itself has a high computational error threshold ∼ 10% [9,10,15,16].Fault-tolerant, measurement-based quantum computation is achieved, in part, by 'braiding' defects within the foliated cluster, to generate subgroup of the Clifford group. By virtue of their topologically protected nature, braiding operations can be made extremely ro- * stace@physics.uq.edu.au bust, and so can be used to distill magic states. Together, these resources allows for universal quantum computation [7,8,19].In an earlier paper, we showed that all Calderbank-Steane-Shor (CSS) codes can be clusterised using a larger cluster state resource [20]. These cluster state codes can be foliated as a generalisation of Raussendorf's 3D latti...