This tutorial will introduce the fundamental concepts of quantum computing, primarily from a programming and algorithms perspective. Topics include quantum bits (qubits), superposition, entanglement, quantum gates and circuits, and some important quantum algorithms. Knowledge of quantum mechanics is not required, but familiarity with basic linear algebra (vectors and matrices) is helpful. Emphasis will be placed on topics that will be further explored during the rest of the workshop.

The ultimate goal of quantum error correction is to create logical qubits with very low error rates (e.g., 1e-12) and assemble them into large-scale quantum computers capable of performing many (e.g., billions) of logical gates on many (e.g., thousands) of logical qubits. However, it is necessarily difficult to directly assess the performance of such high-quality logical qubits using standard Monte Carlo sampling because logical failure events become very rare. Building on existing approaches to this problem, we develop three complementary techniques to characterize the rare-event regime for general quantum low-density parity-check (qLDPC) codes under circuit noise. (I) We propose a well-motivated, low-parameter ansatz for the failure spectrum (the fraction of fault sets of each size that fail) that empirically fits all the QEC systems we studied and predicts logical error rates at all physical error rates. (II) We find min-weight logical operators of syndrome measurement circuits and exactly compute the number of min-weight failing configurations. (III) We generalize the splitting method to qLDPC codes using multi-seeded Metropolis sampling to improve convergence for systems with many inequivalent logical operators. We apply these tools to distance-6, -12, and -18 bivariate bicycle codes under circuit noise, observing strong low-error-rate performance with the recently proposed Relay decoder but also considerable scope for further improvement.

Fault-tolerant quantum computation requires error-correcting codes that not only protect quantum information but also support useful logical operations with practical physical realizations. Existing gate-design methods often rely on transversality, CSS structure, code automorphisms, or additional geometric and homological structure, and systematic constructions become substantially more difficult beyond the Clifford group. Here we develop a general framework in which the stabilizer syndromes of commuting Pauli rotations directly organize logical-gate design. We derive an exact codespace-preservation condition valid at arbitrary rotation angles, providing a common stabilizer- level description for CSS and non-CSS codes and for both Clifford and non-Clifford operations. A particularly useful equal-syndrome specialization reduces Clifford gate design to stabilizer factorization, yields a general fragment-weight bound of $\lceild/2\rceil$ for distance-$d$ codes, and systematically extends to logical rotations higher in the Clifford hierarchy. Applied to bivariate-bicycle and non-CSS Mirror qLDPC codes, the framework uncovers large families of Clifford gates together with native controlled-phase and $\pi/8$ $T$-type operations. We also develop analytic constructions using Clifford algebra bases, which turn the syndrome constraints into explicit algebraic relations and produce addressed CNOT, SWAP, controlled, and CCZ-type gates on recursive hypercube codes. An independent construction on middle quantum Reed–Muller codes yields disjoint $C^rZ$ layers that saturate the equal-syndrome weight bound. These results establish stabilizer syndromes as a constructive link between the stabilizer structure of a quantum code and the logical computations it can natively
support.

In addition to complex quantum phenomena described by Schrodinger’s wave equation, modern quantum systems inevitably couple and interact with both classical and quantum environments. This leads to rich, novel quantum physics. This talk will describe a new, infinite family of quantum error correcting codes and multilinear methods to simulate open quantum systems at scale. Our error correction portion will focus on the benefits of confined anyons, with respect to decoder complexity, in error correcting codes. The open systems simulator will focus on applications to predicting future quantum device’s algorithmic performance and dissipative steady states.

Many quantum algorithms ultimately require useful bitstrings–not only accurate expectation values–from a finite set of noisy measurements. This talk presents dominance-aware (DA) clustering, a sample-level quantum error mitigation framework for discovering and refining high-probability solution regions directly from measured bitstrings. We discuss its finite-sample recovery guarantees, practical variants, and experiments on noisy QAOA instances.