Purdue Quantum Theory Neighborhood Seminar

Discovering low-overhead quantum error-correcting codes is of significant interest for fault-tolerant quantum computation. For hardware capable of long-range connectivity, the bivariate bicycle codes offer significant overhead reduction compared to surface codes with similar performance. In this talk, I will present "ZSZ codes", a simple non-abelian generalization of the bivariate bicycle codes based on the semidirect product of cyclic groups. Numerically, ZSZ codes not only achieve competitive performance with the bivariate bicycle codes under active decoding, but also have the highest observed sustainable threshold for passive decoding of any known quantum LDPC code. Moreover, ZSZ codes admit concrete implementation on neutral-atom arrays, making them promising candidates for experimentally realizable quantum memories.

Quantum mechanics features two distinct forms of evolution-unitary dynamics and projective measurements. However, from the point of view of quantum computation, they play equivalent roles. While the standard circuit model relies on unitary evolution, quantum computational power can also emerge from quantum many-body systems through local adaptive measurements.

In this talk, I will present recent progress in understanding quantum computation as a property of entire quantum phases of matter with symmetry, rather than of individual quantum states. I will show that in one-dimensional spin chains, string order associated with symmetry-protected topological (SPT) order ensures nontrivial computational capability. I will then explain how this framework gives rise to strongly correlated and counterintuitively efficient computational regimes, and briefly discuss how the resulting optimization problems naturally connect to differential geometry. I will also showcase some recent experiments performed on IBM quantum devices that validate our theoretical predictions. Finally, I will conclude with an outlook on higher-dimensional systems and other forms of physical order.

In this talk, I study a non-invertible transformation, termed the gauging map, associated with a finite on-site symmetry G in one-dimensional quantum systems. I will characterize the conditions under which the gauging map, when restricted to the algebra of symmetric operators, realizes a quantum cellular automaton (QCA). I will then present a general constant-depth circuit construction for implementing anyon permutations in quantum double models. This construction is naturally understood through a holographic correspondence between one-dimensional QCAs on subalgebras and symmetries of two-dimensional topological orders. From a quantum-information perspective, it provides an explicit mechanism for realizing locality-preserving logical operations in topological codes. Finally, I discuss the potential non-Clifford-ness from anyon permutations in non-Abelian quantum double models, a key ingredient for fault tolerate universal quantum computation.

Two central goals in quantum science are (i) near-term simulation of many-body physics and (ii) scalable, fault-tolerant quantum computation in the long term. I will present two recent advances toward both aims. For (i), we present theoretical and experimental studies of globally controlled analog simulators, positioning them as principled platforms for quantum simulation [1]. For (ii), we develop a non-Abelian topological framework in two dimensions that provides a route to fault-tolerant universality and remains compatible with current devices [2]. Together, these results outline a practical trajectory from near-term simulation to scalable quantum computation.

For fault-tolerant quantum memory defined by periodic Pauli measurements, called Floquet codes, we prove that every correctable, undetectable spacetime error occurring during the steady stage is a product of (i) measurement operators inserted at the time of the measurement and (ii) pairs of identical Pauli operators sandwiching a measurement that commutes with the operator. We call such errors benign; they define a binary vector subspace of spacetime errors which properly generalize stabilizers of static Pauli stabilizer codes. Hence, the code distance of a Floquet code is the minimal weight of an undetectable spacetime Pauli error that is not benign. Our results apply more generally to families of dynamical codes for which every instantaneous stabilizer is inferred from measurements in a time interval of bounded length.

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Representation theoretic multiplicities like Littlewood-Richardson, Kronecker and plethysm coefficients play central roles in algebraic combinatorics, representation theory, algebraic geometry and geometric complexity theory. While no efficient formulas exist for them, the foremost quest is to understand the computational complexity of computing them or deciding vanishing/positivity. Recently, quantum computing efforts realized them as potential problems for demonstrating quantum versus classical speed-ups. In this talk we will explain the recent progress on the quantum and classical complexity of these problems and show that the realm for potential speed-ups is smaller than expected.

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In the study of quantum matter, measurements have traditionally been viewed as a means of learning about a system. Measurements can, nevertheless, play a more active role-generating novel quantum phenomena that may be difficult or impossible to realize in measurement-free settings. As an interesting example, I will discuss how measurements can dramatically alter universal properties of quantum systems tuned to a phase transition. I will also highlight progress to experimental realization in analog quantum simulators based on Rydberg atom arrays. Finally, I will describe how these ideas inform optimization of quantum teleportation protocols against imperfections-establishing a long-term quantum science application of 'measurement-altered quantum criticality'.

NOTE: this in-person talk will be hosted online simultaneously with the Topological Quantum Computing Online Seminar.

There is an active paradigm shift within quantum topology from semisimple to non-semisimple phenomena. The primary technology that catalyzed this shift is called the modified trace. Mathematicians have shown that non-semisimple topological quantum field theories (TQFTs) can distinguish certain topological features that their semisimple counterparts cannot.

One well-known application of quantum topology is topological quantum computation. Given this paradigm shift, it is natural to ask: what does it mean to do topological quantum computation via modified traces? In this talk, we explore this question through non-semisimple topological quantum computation accompanied with a concrete case study of the non-semisimple Ising model.

Establishing the fusion rules of anyonic quasiparticles in fractional quantum Hall fluids is essential for understanding their underlying topological order. Building on the conjecture that key topological properties are encoded in the "DNA" of candidate many-body wave functions--that is, the pattern of dominant orbital occupations restricted to a finite number of lowest Landau levels--we propose a combinatorial framework that derives these fusion rules directly from microscopic data. By extending Schrieffer's counting argument and introducing equivalences for topological excitations, our framework provides a unified route to the fusion rules for both Abelian and non-Abelian excitations. This approach elucidates the emergence of topological features from first principles in both fermionic and bosonic systems.

Topological symmetries, invertible and otherwise, play a fundamental role in the investigation of quantum field theories. Despite their ubiquitous importance across a multitude of disciplines ranging from string theory to condensed matter physics, controlled realizations of models exhibiting these symmetries in physical systems are rare. Quantum simulators based on engineered solid-state devices provide a novel alternative to conventional condensed matter systems for realizing these models.

In this work, eigenstates of impurity Hamiltonians and loop operators associated with the topological symmetries for the Ising conformal field theory in two space-time dimensions are realized on IBM's Kingston simulator. The relevant states are created on the quantum device using a hybrid quantum-classical algorithm. The latter is based on a variation of the quantum approximate optimization algorithm ansatz combined with the quantum natural gradient optimization method. Signatures of the topological symmetry are captured by measuring correlation functions of different qubit operators with results obtained from the quantum device in reasonable agreement with those obtained from classical computations. The current work demonstrates the viability of noisy quantum simulators as platforms for investigating low-dimensional quantum field theories with direct access to observables that are often difficult to probe in conventional condensed matter experiments.

This is joint work with Christopher Lamb, Robert M. Konik and Hubert Saleur.

I will present an overview of a research program dubbed "entanglement bootstrap," which aims to be a mathematical theory for defining topological phases on a lattice and classifying them. If time permits, I will describe a recent progress in making the framework invariant under constant-depth quantum circuits, as well as several classification results for topological phases in two spatial dimensions.

Jacob Lurie's classification of fully extended topological field theories applies in a broad and abstract setting across all dimensions. However, concrete examples remain scarce. In three dimensions, the Turaev-Viro theory has long been conjectured to fit within this framework. In this talk, we will establish that it necessarily does, outline a sketch of the proof, and explore the essential role of skein theory in this context.

The surface code is one of the leading approaches to building a fault-tolerant quantum computer, actively pursued by groups at Google, IBM, Microsoft, and elsewhere. A central challenge for the surface code is implementing non-Clifford -- unitary operations that map Pauli operators to non-Pauli operators. In this talk, I will introduce group surface codes, which are a natural generalization of the usual \(\mathbb{Z}_2\) surface code and can be understood as quantum double models with specific boundary conditions. I will argue that group surface codes, for suitably chosen groups, can be leveraged to perform non-Clifford gates in the \(\mathbb{Z}_2\) surface code. Time permitting I will describe three strategies for completing a universal gate set using group surface codes: (1) through magic state preparation, (2) using transversal non-Clifford gates, and (3) by sliding group surface codes. These strategies extend recent efforts in performing universal quantum computation in topological orders without the braiding of anyons.

This talk is based on upcoming work with Vieri Mattei, Naren Manjunath, and Apoorv Tiwari.

In this talk, I will discuss the gauging of Abelian modulated symmetries -- finite counterparts of subsystem symmetries -- that act non-uniformly across space. In two dimensions, this provides a simple route to constructing symmetry-enriched topological phases (SETs), where crystalline symmetries enrich the charges carried by topological anyons. Using local Hamiltonians of spin chains as a concrete setting, I will show how the dual theories feature exact modulated 1-form symmetries accompanied by mixed anomalies.

As Moore's Law slows, quantum computers promise to accelerate computationally intensive machine learning tasks like training and data generation. However, a fundamental disconnect exists between the continuous, analog nature of modern ML models (defined by real-valued parameters and data) and the discrete, digital logic of the dominant quantum circuit model. This mismatch creates an unnatural and often unintuitive foundation for designing new quantum algorithms and frameworks. In this talk, I argue that the Hamiltonian formalism, a computational model equivalent to quantum circuits, provides a natural and powerful alternative. I will demonstrate how Hamiltonians serve as a unifying language for designing new quantum algorithms for key ML tasks, such as optimization and sampling, leading to provable speedups. Crucially, because Hamiltonians directly describe the low-level physics of quantum hardware, this approach also enables vastly more efficient algorithm compilation. This hardware-aware framework bypasses many overheads of the circuit model, paving the way for deploying large-scale quantum ML applications on near-term devices.

Last minute reschedule needed.

Newly developed experimental capabilities in implementing measurement and feedback opens up new questions about realizing non-equilibrium steady states in quantum many-body systems. For instance, can active feedback stabilize a phase of matter in the presence of noise which, in equilibrium, can only be realized at zero temperature? For certain types of noise, which leave behind classical traces, I will show that the answer is yes for a number of interesting topological states, including certain non-abelian topological orders.

The ability to coherently move nonabelian anyons is an important prerequisite for fusion-tree-based topological quantum computation. We present new results on the minimal resources needed to move and fuse nonabelian anyons in quantum double models, which are topological code with group-valued qudits on a lattice. Under geometrically local unitary circuits, we argue that moving nonabelian anyons coherently requires a linear-depth circuit and an ancilla qudit. However, the resource costs can be reduced by allowing measurement and k-local gates. With measurement, solvable anyons can be moved using finite-depth adaptive circuits. Notably, with k-local unitary circuits under mild conditions, nonabelian anyons can be moved without using ancillas, countering the conventional wisdom that an ancilla is needed. In contrast to moving, fusing anyons is argued to require an ancilla even with k-local unitary circuits, separating the resource complexity of moving anyons from that of fusing anyons.

In universal fault-tolerant quantum computing, implementing logical non-Clifford gates often demands substantial spacetime resources for many error-correcting codes, including the high-threshold surface code. In this talk, I will explore two strategies of implementing logical non-Clifford operations in the surface code with the aid of non-Abelian topological order. The first method produces a logical magic state through transformations of topological codes, including a non-Abelian code at an intermediate stage, which bypasses the need for the resource-intensive distillation procedure. In the second approach, we generalize the standard lattice surgery to hybrid lattice surgery, where operations of rough merge and rough split happen across different topological codes. Such operations are applied between Abelian and non-Abelian codes, which can provide non-Clifford operations in the standard surface code, in the form of a magic state or a non-Clifford gate teleportation.

We investigate the response of a Ramsey interferometric quantum sensor based on a frustrated, three-spin system (a Kitaev trimer) to a classical time-dependent field (signal). The system eigenspectrum is symmetric about a critical point, \(|\vec{b}| = 0\), with four of the spectral components varying approximately linearly with the magnetic field and four exhibiting a nonlinear dependence. Under the adiabatic approximation and for appropriate initial states, we show that the sensor's response to a zero-mean signal is such that below a threshold, \(|\vec{b}| < b_\mathrm{th}\), the sensor does not respond to the signal, whereas above the threshold, the sensor acts as a detector that the signal has occurred. This thresholded response is approximately omnidirectional. Moreover, when deployed in an entangled multisensor configuration, the sensor achieves sensitivity at the Heisenberg limit. Such detectors could be useful both as standalone units for signal detection above a noise threshold and in two- or three-dimensional arrays, analogous to a quantum bubble chamber, for applications such as particle track detection and long-baseline telescopy.

We construct a local decoder for the 2D toric code using ideas from the hierarchical classical cellular automata of Tsirelson and Gács. Our decoder is a circuit of strictly local quantum operations preserving a logical state for exponential time in the presence of circuit-level noise without the need for non-local classical computation or communication. Our construction is not translation invariant in spacetime but can be made time-translation invariant in 3D with stacks of 2D toric codes. This solves the open problem of constructing a local topological quantum memory below four dimensions.

This talk is based on joint work [arXiv:2412.19803] with Shankar Balasubramanian and Ethan Lake.

In this talk, I will explain the concept of SPT-Sewing, which is a systematic method for constructing gapped domain walls of topologically ordered systems by gauging a lower-dimensional symmetry-protected topological (SPT) order. Based on our construction, we propose a correspondence between 1d SPT phases with a non-invertible \(G \times \mathrm{Rep}(G) \times G\) symmetry and invertible domain walls in the quantum double associated with the group \(G\). We prove this correspondence when \(G\) is Abelian and provide evidence for the general case by studying the quantum double model for \(G = S_3\). We also use our method to construct anchoring domain walls, which are novel exotic domain walls in the 3d toric code that transform point-like excitations to semi-loop-like excitations anchored on these domain walls.

I will report on arXiv:2501.0528 where we consider the following decision problem: on input \(G\in \mathcal D\) and \(H\in \mathcal T\), decide if there is a surjective homomorphism from \(G\) onto \(H\).

We prove that the problem is NP-complete when \(\mathcal D\) is the class of all finitely presented groups and \(\mathcal T\) is the class of direct products \(\mathbb Z^d\times Q\) where \(Q\) is a finite group, the class of virtually cyclic groups, or the class of a single fixed dihedral group that is not nilpotent.

Work of Kuperberg and Samperton previously showed the problem is NP-complete for target a single non-abelian simple group arXiv:1707.03811.

Our techniques involve reducing epimorphism to decision problems about equations over groups, which I will explain in the talk.

This is joint work with Jerry Shen (UTS) and Armin Weiß (Stuttgart).

The problem of classifying modular categories is motivated by applications to topological quantum computation as algebraic models for topological phases of matter. These categories have also applications in different areas of mathematics like topological quantum field theory, von Neumann algebras, representation theory, and others.

In this talk, we will give an overview of the current status of the classification program for modular categories. We will also present a construction of non-group-theoretical modular categories of certain dimensions. If time allows, we will present some open questions and the relation of this result and the classification by rank.

From physics lore, topological phases in the presence of a symmetry have the structure of a G-crossed braided tensor category, where the objects are the defects of the symmetry. In this talk we will show how to arrive at this structure using operator algebras, starting from lattice systems in the presence of a symmetry. Along the way, we will give a general recipe to create a symmetry defect. Finally we will discuss the novel consequences of this approach. In particular, it gives us a way to calculate the symmetry fractionalization class in the bulk using local unitaries.

(2+1)D topological orders possess emergent symmetries that consists of the braided tensor autoequivalences of the modular tensor category. In this talk we discuss cases where symmetries neither permute anyons nor are associated to any symmetry fractionalization but are still non-trivial, which we refer to as soft symmetries. We point out that one can construct topological defects corresponding to such exotic symmetry actions by decorating with a certain class of gauged SPT states that cannot be distinguished by their torus partition function. This gives a physical interpretation to work by Davydov on soft braided tensor autoequivalences. This has a number of important implications for the classification of gapped boundaries, non-invertible spontaneous symmetry breaking, and the general classification of symmetry-enriched topological phases of matter. We also demonstrate analogous phenomena in higher dimensions, such as (3+1)D gauge theory with gauge group given by the quaternion group Q8.

In this talk, we will use the SymTFT picture to give a precise definition of Kramer-Wannier type dualities on the 1D lattice with categorical symmetry. We will describe some topological invariants of dualities and present a recent result which gives a complete classification of G-dualities up to symmetric finite depth circuits, where G is a finite group.