Disorder-driven phase transitions in weak, boundary-obstructed, and non-Hermitian topological insulators

This thesis focuses on three main areas in quantum physics. The bulk of this thesis addresses the effects of disorder on novel classes of topological insulators. Topological insulators are states of matter that display properties (most notably, protected anomalous edge states) that are robust to symmetry-preserving disorder. While the properties of "classical" tenfold way topological insulators under disorder are well-understood, there exist other topological phases whose behavior under disorder has yet to be characterized. In this portion of the thesis, we will develop real-space methods to compute weak, boundary-obstructed, and non-Hermitian topological invariants, establish their stability at weak and strong disorder, and connect these disordered topological invariants to physical signatures. The remainder of the thesis contains an eclectic mix of other work that broadly focuses on the intersection of computational complexity and quantum mechanics. The first section addresses the problem of simulating quantum mechanics on a classical computer. While exactly simulating quantum mechanics is NP hard, in this section we develop and approximate variational method to simulate quantum systems at nite temperature. The second section develops a \randomized benchmarking" method for verifying the gates of a quantum computer, a challenging task as the output of a quantum circuit is generically di cult to simulate. Finally, the third section deals with the ability of a quantum computer to simulate condensed matter systems; we study the ability of a variational quantum circuit to approximate the ground state of the mixed-spin Sherrington-Kirkpatrick spin-glass model

Tailoring fusion-based error correction for high thresholds to biased fusion failures

We introduce fault-tolerant (FT) architectures for error correction with the XZZX cluster state based on performing measurements of two-qubit Pauli operators $Z\otimes Z$ and $X\otimes X$, or fusions, on a collection of few-body entangled resource states. Our construction is tailored to be effective against noise that predominantly causes faulty $X\otimes X$ measurements during fusions. This feature offers practical advantage in linear optical quantum computing with dual-rail photonic qubits, where failed fusions only erase $X\otimes X$ measurement outcomes. By applying our construction to this platform, we find a record high FT threshold to fusion failures exceeding $25\%$ in the experimentally relevant regime of non-zero loss rate per photon, considerably simplifying hardware requirements.Comment: 7+6 pages, 4+6 figures, comments welcom

Estimating the Bias of CX Gates via Character Randomized Benchmarking

Recent work has demonstrated that high-threshold quantum error correction is possible for biased-noise qubits, provided that one can implement a controlled-sc not (CX) gate that preserves the bias. Bias-preserving CX gates have been proposed for several biased-noise qubit platforms, most notably Kerr cats. However, experimentally measuring the noise bias is challenging, as it requires accurately estimating certain low-probability Pauli errors in the presence of much larger state-preparation-and-measurement (SPAM) errors. In this paper, we introduce bias-randomized benchmarking (BRB) as a technique for measuring bias in quantum gates. BRB, like all RB protocols, is highly accurate and immune to SPAM errors. Our first protocol, CX-dihedral BRB, is a straightforward method to measure the bias of the entire CX-dihedral group. Our second protocol, interleaved-bias randomized benchmarking (IBRB), is a generalization of interleaved RB tailored to the experimental constraints of biased-noise qubits; this is a more involved procedure that directly targets the bias of the CX gate alone. Our BRB procedures occupy a middle ground between classic RB protocols that only estimate the average fidelity and tomographic RB protocols that provide more detailed characterization of noise but require more measurements as well as experimental capabilities that are not necessarily available in biased-noise qubits