We show that the low-energy states of non-Abelian topological orders possess extensive magic which is long-ranged, and cannot be eliminated by a constant-depth local unitary circuit. This refines conventional notions of complexity beyond the linear circuit depth which is required to prepare any topological phase, and provides a new resource-theoretic characterization of topological orders. A central technical result is a no-go theorem establishing that stabilizer states--even up to constant-depth local unitarie--cannot approximate low-energy states of non-Abelian string-net models which satisfy the entanglement bootstrap axioms. Moreover, we show that stabilizer-realizable Abelian string-net phases have mutual braiding phases quantized by the on-site qudit dimension, and that any violation of this condition necessarily implies extensive long-range magic. Extending to higher spatial dimensions, we argue that any state obeying an entanglement area law and hosting excitations with nontrivial fusion spaces must exhibit extensive long-range magic. This applies, in particular, to ground-states and low-energy states of higher-dimensional quantum double models.
We show that the low-energy states of non-Abelian topological orders possess extensive magic which is long-ranged, and cannot be eliminated by a constant-depth local unitary circuit. This refines conventional notions of complexity beyond the linear circuit depth which is required to prepare any topological phase, and provides a new resource-theoretic characterization of topological orders. A central technical result is a no-go theorem establishing that stabilizer states--even up to constant-depth local unitarie--cannot approximate low-energy states of non-Abelian string-net models which satisfy the entanglement bootstrap axioms. Moreover, we show that stabilizer-realizable Abelian string-net phases have mutual braiding phases quantized by the on-site qudit dimension, and that any violation of this condition necessarily implies extensive long-range magic. Extending to higher spatial dimensions, we argue that any state obeying an entanglement area law and hosting excitations with nontrivial fusion spaces must exhibit extensive long-range magic. This applies, in particular, to ground-states and low-energy states of higher-dimensional quantum double models.
Nonstabilizerness, or magic, is a necessary resource for quantum advantage beyond the classically simulatable Clifford framework. Recent works have begun to chart the structure of magic in many-body states, introducing the concepts of long-range magic -- nonstabilizerness that cannot be removed by finite-depth local unitary (FDU) circuits -- and the magic hierarchy, which classifies quantum circuits by alternating layers of Clifford and FDUs. In this work, we construct explicit states that provably possess two-sided long-range magic, a stronger form of magic meaning that they cannot be prepared by a Clifford circuit and a FDU in either order, thus placing them provably outside the first level of the magic hierarchy. Our examples include the ``magical cat" state, ∣ψ⟩∝∣0n⟩+∣+n⟩, and ground states of certain nonabelian topological orders. These results provide new examples and proof techniques for circuit complexity, and in doing so, reveal the connection between long-range magic, the structure of many-body phases, and the principles of quantum error correction.
Long-range entanglement-the backbone of topologically ordered states-cannot be created in finite time using local unitary circuits, or, equivalently, adiabatic state preparation.Recently, it has come to light that single-site measurements provide a loophole, allowing for finite-time state preparation in certain cases.Here we show how this observation imposes a complexity hierarchy on long-range entangled states based on the minimal number of measurement layers required to create the state, which we call "shots."First, similar to Abelian stabilizer states, we construct single-shot protocols for creating any non-Abelian quantum double of a group with nilpotency class 2 (such as D 4 or Q 8 ).We show that after the measurement, the wave function always collapses into the desired non-Abelian topological order, conditional on recording the measurement outcome.Moreover, the clean quantum double ground state can be deterministically prepared via feedforward-gates that depend on the measurement outcomes.Second, we provide the first constructive proof that a finite number of shots can implement the Kramers-Wannier duality transformation (i.e., the gauging map) for any solvable symmetry group.As a special case, this gives an explicit protocol to prepare twisted quantum doubles for all solvable groups.Third, we argue that certain topological orders, such as nonsolvable quantum doubles or Fibonacci anyons, define nontrivial phases of matter under the equivalence class of finite-depth unitaries and measurement, which cannot be prepared by any finite number of shots.Moreover, we explore the consequences of allowing gates to have exponentially small tails, which enables, for example, the preparation of any Abelian anyon theory, including chiral ones.This hierarchy paints a new picture of the landscape of long-range entangled states, with practical implications for quantum simulators.
We study fermionic non-local magic in higher-dimensional free-fermion systems through the lens of the entanglement spectrum. For translationally invariant Gaussian states in strip geometries, dimensional reduction maps the problem to independent lower-dimensional momentum sectors, leading in two dimensions to a multiplicative logarithmic violation of the boundary law analogous to Gioev-Klich-Widom scaling. We further show that the second Rényi fermionic non-local magic is bounded by the capacity of entanglement and introduce an entanglement-temperature deformation that probes the low-energy structure of the entanglement Hamiltonian. Applied to the matter-Majorana sector of the Kitaev model, this response is exponentially suppressed in the Abelian phase and algebraic in the non-Abelian phase, where its ratio with the capacity approaches a universal constant. Out of equilibrium, we extend the quasi-particle description to fermionic non-local magic in higher-dimensional quenches and show that random Gaussian circuits exhibit diffusive spreading in two spatial dimensions.
As a necessary resource for quantum computational advantage, quantum magic (nonstabilizerness) is of fundamental importance in the study of quantum computation and physics. We develop a systematic theory of \emph{long-range magic (LRM)} -- nonstabilizerness that cannot be erased by shallow unitary circuits -- and demonstrate its broad relevance. By bridging LRM with fault-tolerant logical gate theory, we show the emergence of LRM families from quantum error-correcting codes and devise a simple yet powerful method for testing transversal logical gates. Further, we introduce and characterize \emph{LRM phases} in which all ground states exhibit LRM, and identify certain non-Abelian topological orders as representative examples. Then, adopting a complexity theory perspective, we demonstrate the classicality of non-LRM systems in e.g.~preparation and learning settings, and present a ``no low-energy trivial magic'' (NLTM) conjecture with key motivation in the quantum PCP context, for which our LRM results suggest a promising route. Additionally, we demonstrate how to diagnose LRM with correlation functions. The concepts and results admit substantive extensions to approximate (robust) and nongeometric scenarios. Our LRM theory illuminates profound new connections among quantum resources, computational advantage, error correction and fault tolerance, many-body physics, and complexity theory.
This thesis concerns itself with two fundamental tasks in the theory of quantum information: the preparation of ground and low-energy states of local Hamiltonians on a quantum computer and the secure transmission of classical information over a quantum channel. The first part of this thesis addresses the former task and is dedicated in its entirety to quantum many-body physics. We will be focusing on topologically ordered lattice models in two spatial dimensions. It is known that the presence of long-range entanglement in the ground states of these models necessitates local quantum circuits of depth linear in the diameter of the system for their preparation. We study the consequences of this hardness result for the preparation of low-energy states of these models. In particular we show that preparing such states requires local quantum circuits of polynomial depth in the inverse energy density of the states and comment on the relevance of these results to quantum computation in the near term. Our technical contribution is to give a circuit depth lower bound for low-energy states independent of the dimension of the ground space. We then experimentally demonstrate how currently existing ion trap hardware has reached a level of maturity where noisy adaptive circuits of constant depth outperform noiseless, non-adaptive circuits of the same depth for the task of approximating a ground state of the the toric code. In the last part of this thesis, we switch gears to quantum Shannon theory and study private communication over a classical-quantum wiretap channel. We study the problem of non-additivity of the private information for this model. Surprisingly, we find that the private information is non-additive when either of the outputs of the channel is allowed to be quantum, while the input is classical. However, there exists a single-letter formula when the input state is quantum and the outputs are classical due to the private information of the channel becoming additive.
We present a versatile and mathematically rigorous technique for bounding recovery thresholds in topological codes subject to noise. Our method captures the effect of applying an arbitrary (possibly non-Pauli) local noise channel to the code state of a broad class of two-dimensional codes, including surface codes, non-Abelian quantum doubles, and string-net codes. In each case, we prove that for noise strengths up to some explicit constant value, any initially encoded logical information can be recovered to high precision, and that the noise-corrupted state exhibits key hallmarks of mixed-state topological order: long-range entanglement and emergent higher-form symmetries. We also describe how these methods can be adapted to higher dimensions and correlated noise models.