Purdue Quantum Theory 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.

Understanding the multiplicities of decompositions of tensor products and compositions of irreducible representations of the symmetric and general linear groups is a central problem. While we have been concerned with classical computation and are searching for computational hardness results, recently quantum algorithms were developed which put the problems in QMA (vanishing) and #BQP (counting). Even more recently quantum algorithms that run in polynomial time were developed for certain cases. We will discuss the classical hardness results, the quantum complexity and ultimately show that most of the quantum easy cases are also poly-time computable on a classical computer thereby refuting some hopes for superpolynomial quantum speedups.

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