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.

Quantum error correction is a foundational technology for large-scale quantum computing and quantum communication, providing methods to protect fragile quantum information from noise, decoherence, and imperfect operations. This tutorial presents a concise introduction to the fundamental ideas underlying quantum error correction. We will survey several important code families and discuss how concepts from classical coding theory, algebra, and information theory have shaped current approaches to protecting quantum information.

Folded Reed-Solomon codes, introduced by Guruswami and Rudra in 2007, have been shown to achieve the information-theoretically optimal trade-off between code rate and error-correction radius. In 2022, Bergamaschi, Golowich, and Gunn initiated the study of folded codes in the quantum setting by constructing folded quantum Reed–Solomon codes and demonstrating that these codes satisfy the quantum Singleton bound. This talk focuses on the construction of quantum error-correcting codes (QECCs) using tools from algebraic geometry. In particular, we examine the role of the genus of the underlying curve and the need for distance amplification. Folded algebraic geometry codes constructed from curves of positive genus can have smaller alphabet sizes than folded Reed–Solomon codes, potentially allowing for more efficient encoding and decoding. Constructing quantum codes from folded algebraic geometry codes therefore broadens the range of available constructions for QECCs.

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.

Large-scale quantum architectures rely on error-correcting codes that encode many logical qubits, often assembled from smaller code blocks. In such codes, different logical qubits may have different local distances, and the features that govern fault tolerance (where logical information is supported, which regions are cleanable, which logical qubits remain accessible) are not captured by the standard global parameters of quantum coding theory.

In this talk, we introduce a symplectic framework for quantum error-correcting codes, with anticodes as investigative tools for the local structure of a code. Codes are modeled as symplectic spaces, and anticodes arise as maximal subspaces supported on prescribed sets of components. The framework covers several families of quantum codes, including stabilizer and subsystem codes. It extends the notion of generalized distance and produces new invariants that capture local algebraic and combinatorial features. It also gives algebraic interpretations of the cleaning lemma, complementary recovery, and quantum weight enumerators. The new results in this talk are joint work with C. Cao and B. Lackey.

Quantum low-density parity-check (QLDPC) codes promise fault tolerance with far less overhead than surface codes, but only if they can be decoded quickly and accurately. Belief propagation (BP) is the natural starting point. It is fast, local, and highly parallel, and it works extremely well for classical LDPC codes. On quantum codes, however, stabilizer degeneracy and short cycles in the Tanner graph cause plain BP to oscillate or converge to the wrong answer.

This talk surveys the growing family of decoders built around BP to overcome these obstacles. They include tuned message-passing rules, post-processing methods such as ordered-statistics decoding, guided decimation, memory-based and relay variants, local symmetry-breaking rules, redundant-check and ensemble decoders, and hybrids that combine several of these ideas. After a brief description of each approach, we compare them in a common framework. The comparison emphasizes the trade-off between logical error rate and complexity. We close with thoughts on designing practical real-time decoders.

Quantum phase transitions generate strongly entangled states, and mapping these correlations on quantum hardware provides a stringent benchmark for many-body quantum simulation. The positive partial transpose criterion, combined with quantum overlapping tomography, offers a scalable witness of pairwise entanglement from two-spin reduced density matrices, including for noisy mixed states. But on present-day processors, measurement and gate errors bias the reconstructed correlations, and obscure entanglement. Therefore, we combine matrix-free readout correction with zero-noise extrapolation (ZNE) to mitigate measurement and gate errors in transverse-field Ising and XXZ states prepared with up to 20 qubits on an IBM processor. Extrapolating the relevant correlators recovers near-critical energies with 99.53% and 99.05% accuracy relative to matrix-product-state benchmarks for the Ising and XXZ models, respectively. The mitigated results recover nearest-neighbor entanglement in the Ising model and nearest- and next-nearest-neighbor entanglement in the XXZ model near criticality, demonstrating reliable probing of quantum correlations at quantum phase transition on noisy hardware.

Topologically ordered quantum states with Abelian excitations can host defects that obey effective non-Abelian statistics, in principle allowing for quantum information processing via defect braiding. These extrinsic defects (or twists) are typically studied as static features of the lattice. However, an alternative proposal considers how an underlying topologically ordered quantum substrate can be locally perturbed to create and manipulate synthetic defects. Understanding the energy spectrum is particularly important in finite size and finitely perturbed systems, which are crucial for experimental realizations. In this work we announce a significant step in this direction by explicitly constructing, simplifying, and numerically studying the spectral properties of synthetic defects in a model system. We conclude by discussing key steps for future work to more clearly and completely study this phenomena and potential applications.

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
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We show how to design block-encoding circuits and analyze their optimality in terms of encoding subnormalization for finite-difference stencils.

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We study semi-Clifford and generalized semi-Clifford structures in the Clifford hierarchy as a function of hierarchy level $k$ and qubit number $n$. We give a complete picture on when a cell $(k,n)$ is semi-Clifford and an almost complete picture on generalized semi-Clifford property. We also propose a structure beyond generalized semi-Clifford.

Quantum error suppression and mitigation are crucial for enabling accurate quantum computations before full fault tolerance and remain valuable as error correction matures. In this tutorial, we will begin with accessible intuition for how errors affect quantum circuits and introduce the main ideas behind error suppression and characterization-based mitigation. We will then examine state-of-the-art methods and show how they enable applications across quantum simulation, quantum chemistry, and other domains, focusing on a recent high-accuracy simulation of prethermal Floquet dynamics in a regime beyond the reach of leading classical simulations. Finally, we will discuss the interplay between error mitigation and error correction, focusing on how syndrome information gathered during error-corrected computations can improve the efficiency of logical error mitigation. We will highlight recent work revealing how, in a sense, error mitigation is built into error-corrected circuits.

In this talk, I will discuss the physical mechanisms that give rise to errors in quantum systems, including decoherence and relaxation, and present methods to characterize and mitigate them. Specifically, I will introduce coherent control techniques that can identify and quantify noise sources in qubits, as well as optimized dynamical decoupling pulse sequences capable of suppressing these noise sources while remaining robust against experimental imperfections.

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.

Using photons for fault tolerant quantum computation is challenging due to their extremely weak interactions with each other. One approach for overcoming this limitation is measurement based quantum computation, which utilizes highly entangled states, known as cluster states, to perform computation. However, the generation of large cluster states remains difficult due to the complex entanglement structure of these states, as well as experimental losses. In this talk, I will present a deterministic method for generating cluster states by coupling a single matter qubit to a photonic cavity, in which cavity quantum electrodynamics mediates the entanglement between a group of photons. I will describe our approach of utilizing multiple matter qubits to parallelize cluster-state generation and emphasize the advantages of this approach compared to the use of a single matter qubit. Most notably, for certain subgroups of two- and three-dimensional cluster states, our approach offers exponential speedup in the successful generation of these states due to the added parallelization and reduction of photon losses. These improvements may enable the generation of previously infeasible cluster states, paving the way toward the use of multiple logical qubits and deep circuits for fault-tolerant measurement-based quantum computation.

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.

Large-scale fault-tolerant quantum computing is likely to require increasing specialization and distribution. Storage, computation, and communication place different demands on hardware, connectivity, and quantum error correction, creating significant opportunities for architectural and algorithmic co-design. In this talk, I will discuss our work exploring these opportunities across multiple layers of the quantum computing stack.

First, I will present our circuit-level study of transversal fault-tolerant distributed operations. We show how transversal operations between qLDPC code blocks can enable efficient distributed computation, and find that non-local CNOTs can achieve up to an order of magnitude lower logical error rates than logical teleportation under matched conditions. I will also introduce the Transversal Multiple Code Block Simulator (TMCBS), our scalable framework for studying distributed operations across different codes and noise models.

I will then present a heterogeneous distributed architecture for fault-tolerant quantum simulation. By separating an extensible storage system from a centralized magic core and co-designing the architecture around the structure of fermionic simulation workloads, we efficiently execute highly non-local Pauli-string operations. I will compare this design with homogeneous distributed architectures and discuss the resulting trade-offs between communication, connectivity, parallelism, and resource cost. 

This talk is based on doi.org/10.1038/s41467-026-75693-3 and https://doi.org/10.48550/arXiv.2608.09215.

I will present some of our published results related to computation of return probability and quantum chaos in a paradigmatic model of quantum gravity known as the Sachdev-Ye-Kitaev model. We use self-mitigation (an improvement over the CNOT-only mitigation) to obtain results consistent with exact diagonalization for circuit depth of more than 300 CNOT gates. This work is based on https://arxiv.org/abs/2311.17991, published as Phys. Rev. D, 109, 105002 (2024).