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The study of entanglement in many-body quantum systems has provided connections between physical properties and the computational resources required for tensor network calculations. In this talk, I will construct rigorous upper bounds on half-system entanglement entropies of states with fixed energy expectation values. These upper bounds are expressed in terms of thermal entropies of subsystems. For frustration-free systems this result shows that, when zero-temperature thermal entropies are proportional to subsystem surface areas, ground states are area-law entangled. In more general systems, and at subextensive energies, the behavior of the specific heat at low temperatures controls the scaling of entanglement with system size. For large classes of systems with conventional thermodynamic properties, I will show that the upper bounds are optimal up to subleading corrections.
Preparing low-energy states of many-body Hamiltonians is a central challenge in quantum computing, quantum complexity, and condensed matter physics. Existing approaches often get trapped in suboptimal states such as high-energy eigenstates or, more generally, low-variance states that resist further energy reduction. In this work, we explore a different perspective: instead of optimizing with respect to a single Hamiltonian, we leverage the fact that many systems admit families of Hamiltonians that share similar low-energy subspaces but differ at higher energies. We show that this redundancy can be turned into an algorithmic resource by establishing an energy-based uncertainty principle, which implies that these Hamiltonians cannot simultaneously admit low-variance states at higher energies. This suggests a simple strategy of alternating energy-lowering steps across such Hamiltonians to destabilize trapped states and enable continued descent. We investigate this approach numerically on models including the 1D AKLT chain and Heisenberg models on general graphs, and observe consistent improvements over standard methods. We also introduce a sparse variant where the uncertainty principle strengthens to yield quadratically larger variance at higher energies, leading to possibly more pronounced energy reduction. Overall, this work suggests a range of open questions at the interface of random matrix theory, local Hamiltonians and state preparation, aimed at understanding when such approaches are practical and how they can be analyzed rigorously.
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Abstract not available.
Quantum computers have the potential to perform computational tasks beyond the reach of classical machines. A prominent example is Shor’s algorithm for integer factorization and discrete logarithms, which is of both fundamental importance and practical relevance to cryptography. However, due to the high overhead of quantum error correction, optimized resource estimates for cryptographically relevant instances of Shor’s algorithm require millions of physical qubits. Here, by leveraging advances in high-rate quantum error-correcting codes, efficient logical instruction sets, and circuit design, we show that Shor's algorithm can be executed at cryptographically relevant scales with as few as 10,000 reconfigurable atomic qubits. Increasing the number of physical qubits improves time efficiency by enabling greater parallelism; under plausible assumptions, the runtime for discrete logarithms on the P-256 elliptic curve could be just a few days for a system with 26,000 physical qubits, while the runtime for factoring RSA-2048 integers is one to two orders of magnitude longer. Recent neutral-atom experiments have demonstrated universal fault-tolerant operations below the error-correction threshold, computation on arrays of hundreds of qubits, and trapping arrays with more than 6,000 highly coherent qubits. Although substantial engineering challenges remain, our theoretical analysis indicates that an appropriately designed neutral-atom architecture could support quantum computation at cryptographically relevant scales. More broadly, these results highlight the capability of neutral atoms for fault-tolerant quantum computing with wide-ranging scientific and technological applications.