Exactly Solvable Lattice Hamiltonians and Gravitational Anomalies

We construct infinitely many new exactly solvable local commuting projector lattice Hamiltonian models for general bosonic beyond group cohomology invertible topological phases of order two and four in any spacetime dimensions, whose boundaries are characterized by gravitational anomalies. Examples include the beyond group cohomology invertible phase without symmetry in (4+1)D that has an anomalous boundary $\mathbb{Z}_2$ topological order with fermionic particle and fermionic loop excitations that have mutual $\pi$ statistics. We argue that this construction gives a new non-trivial quantum cellular automaton (QCA) in (4+1)D of order two. We also present an explicit construction of gapped symmetric boundary state for the bosonic beyond group cohomology invertible phase with unitary $\mathbb{Z}_2$ symmetry in (4+1)D. We discuss new quantum phase transitions protected by different invertible phases across the transitions.Comment: 60 pages, 14 figures, 3 tables; v2: typos corrected, references adde

Analysis and Design of High-Speed A/D Converters in SiGe Technology

Mixed-signal systems play a key role in modern communications and electronics. The quality of A/D and D/A conversions deeply affects what we see and what we hear in the real world video and radio. This dissertation deals with high-speed ADCs: a 5-bit 500-MSPS ADC and an 8-bit 2-GSPS ADC. These units can be applied in flat panel display, image enhancement and in high-speed data link. To achieve the state-of-the-art performance, we employed a 0.13-μm/2.5-V 210-GHz (unity-gain frequency) BiCMOS SiGe process for all the implementations. The circuit building blocks, such as the Track-and-Hold circuit (T/H) and the comparator, required by an ADC not only benefit from SiGe's superior ultra-high frequency properties but also by its power drive capability. The T/H described here achieved a dynamic performance of 8-bit accuracy at 2-GHz Nyquist rate with an input full scale range of 1 Vp-p. The T/H consumed 13 mW of power. The unique 4-in/2-out comparator was made of fully differential emitter couple pairs in order to operate at such a high frequency. Cascaded cross-coupled amplifier core was employed to reduce Miller effect and to avoid collector-emitter breakdown of the HBTs. We utilized the comparator interpolation technique between the preamplifer stages and the latches to reduce the total power dissipated by the comparator array. In addition, we developed an innovative D/A conversion and analog subtraction approach necessary for two-step conversion by using a bipolar pre-distortion technique. This innovation enabled us to decrease the design complexity in the subranging process of a two-step ADC. The 5-bit interpolating ADC operated at 2-GSPS achieved a differential nonlinearity (DNL) of 0.114 LSB and an integral nonlinearity (INL) of 0.076 LSB. The effective number of bits (ENOBs) are 4.3 bits at low frequency and 4.1 bits near Nyquist rate. The power dissipation was reduced more than half to 66.14 mW, with comparator interpolation. The 8-bit two-step interpolating ADC operated at 500-MSPS. It achieved a DNL of 0.33 LSB and an INL of 0.40 LSB with a power consumption of 172 mW. The ENOBs are 7.5 bits at low frequency and 6.9 bits near Nyquist rate

Phase and Amplitude Responses of Narrow-Band Optical Filter Measured by Microwave Network Analyzer

The phase and amplitude responses of a narrow-band optical filter are measured simultaneously using a microwave network analyzer. The measurement is based on an interferometric arrangement to split light into two paths and then combine them. In one of the two paths, a Mach-Zehnder modulator generates two tones without carrier and the narrow-band optical filter just passes through one of the tones. The temperature and environmental variations are removed by separated phase and amplitude averaging. The amplitude and phase responses of the optical filter are measured to the resolution and accuracy of the network analyzer

Higher-group symmetry in finite gauge theory and stabilizer codes

A large class of gapped phases of matter can be described by topological finite group gauge theories. In this paper, we derive the $d$-group global symmetry and its 't Hooft anomaly for topological finite group gauge theories in $(d+1)$ space-time dimensions, including non-Abelian gauge groups and Dijkgraaf-Witten twists. We focus on the 1-form symmetry generated by invertible (Abelian) magnetic defects and the higher-form symmetries generated by invertible topological defects decorated with lower dimensional gauged symmetry-protected topological (SPT) phases. We show that due to a generalization of the Witten effect and charge-flux attachment, the 1-form symmetry generated by the magnetic defects mixes with other symmetries into a higher group. We describe such higher-group symmetry in various lattice model examples. We discuss several applications, including the classification of fermionic SPT phases in (3+1)D for general fermionic symmetry groups, where we also derive a simpler formula for the $[O_5] \in H^5(BG, U(1))$ obstruction than has appeared in previous work. We also show how the $d$-group symmetry is related to fault-tolerant non-Pauli logical gates and a refined Clifford hierarchy in stabilizer codes. We construct new logical gates in stabilizer codes using the $d$-group symmetry, such as the control-Z gate in (3+1)D $\mathbb{Z}_2$ toric code.Comment: 41 pages, 6 figure

Retraction and Generalized Extension of Computing with Words

Fuzzy automata, whose input alphabet is a set of numbers or symbols, are a formal model of computing with values. Motivated by Zadeh's paradigm of computing with words rather than numbers, Ying proposed a kind of fuzzy automata, whose input alphabet consists of all fuzzy subsets of a set of symbols, as a formal model of computing with all words. In this paper, we introduce a somewhat general formal model of computing with (some special) words. The new features of the model are that the input alphabet only comprises some (not necessarily all) fuzzy subsets of a set of symbols and the fuzzy transition function can be specified arbitrarily. By employing the methodology of fuzzy control, we establish a retraction principle from computing with words to computing with values for handling crisp inputs and a generalized extension principle from computing with words to computing with all words for handling fuzzy inputs. These principles show that computing with values and computing with all words can be respectively implemented by computing with words. Some algebraic properties of retractions and generalized extensions are addressed as well.Comment: 13 double column pages; 3 figures; to be published in the IEEE Transactions on Fuzzy System