In this thesis, we analyse and improve the best known ecient quantum algorithm for solving the Short Principal Ideal Problem (SPIP) in cyclotomic elds, which, given an ideal, determines whether the ideal is principal and outputs a suciently short generator.SPIP is fundamental in cryptography and computational number theory.The standard solution to SPIP can unfold in a two-part process involving the Principal Ideal Problem (PIP) and the Short Generator Problem (SGP).This thesis consists of the following two results.For PIP, we calculate the upper bound on quantum space complexity of the quantum algorithm proposed by Biasse and Song at SODA 2016 (BS) for solving PIP.PIP is the problem that, given an ideal, determines whether the ideal is principal and outputs a generator of the ideal if it is principal.The best known classical algorithm solves PIP in subexponential time, while the BS quantum algorithm nds a generator in polynomial time.Whether the cryptographic systems relying on the hardness of PIP have become insecure depends not only on the existence of polynomial time algorithms solving it but also on whether the algorithms should be implementable.Quantum computers nowadays have only hundreds of qubits, which means that quantum algorithms using too many qubits are not feasible for now.By estimating the qubits used in the BS algorithm, we want to estimate the threat at the moment.
Quantum geometry is universally bounded from below by Wilson-loop windings. In this work, we define an isolated set of bands to be Wilson-loop-ideal, if their quantum metric saturates the Wilson-loop lower bound. The definition naturally incorporates the known Chern-ideal and Euler-ideal bands, and allows us to define other types of ideal bands, such as Kane-Mele
Z2-ideal and inversion-fragile-ideal bands. In particular, we find that in the case of zero total Chern number, an isolated WL-ideal set of two bands with non-singular nonabelian Berry curvature and nontrivial normal Wilson-loop winding always admits a Chern-ideal gauge, without the need of a global good quantum number (such as spin). This enables the direct construction of new topologically ordered states, such as fractional topological insulator wavefunctions. We further propose a general framework of constructing monotonic flows that achieve Wilson-loop-ideal states starting from non-ideal bands through band mixing, where Wilson-loop-ideal states are not energy eigenstates but have smooth projectors similar to isolated bands. We apply the constructed flows to the realistic model of
3.89∘ twisted bilayer MoTe
2, a moiré Rashba model and another moiré time-reversal-breaking models, and numerically find Chern-ideal,
Z2-ideal and inversion-fragile states, respectively, with relative error in the integrated quantum metric below
5×10−3. Our exact-diagonalization calculations on the numerically ideal states demonstrate the potential of our general definition of Wilson-loop-ideal bands and general procedure of constructing Wilson-loop-ideal states for future study of novel correlated physics.
We generalize the standard quantum model (Caves 1982 Phys.Rev. D 26 1817) of the noise properties of ideal linear amplifiers to include the possibility of non-ideal behavior.We find that under many conditions the non-ideal behavior can be described simply by assuming that the internal noise source that describes the quantum noise of the amplifier is not in its ground state.We validate this model by showing that it reproduces the known predictions of two models of non-ideal amplifiers: a laser amplifier with incomplete inversion and a cascade of alternating ideal amplifiers and ideal attenuators.We also describe implications of this model for practical devices.