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Whether solid, liquid, or gas, a substance's state depends on the order and arrangement of its particles (atoms, molecules, or ions). Particles in the solid pack closely together, generally in a pattern. The particles vibrate about their fixed positions but do not move or squeeze past their neighbors. In liquids, although the particles are closely spaced, they are randomly arranged. The position of the particles are not fixed—that is, they are free to move past their neighbors to...
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The internal energy of a substance—the total kinetic energy of all its molecules and the potential energy of their associated forces—depends on the strength of the intermolecular forces in the condensed phases and the pressure exerted on the substance. The internal energy of a substance is the highest in the gaseous state, the lowest in the solid state, and intermediate in the liquid state. Phase transitions are caused by changes in physical conditions, such as temperature and...
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The physical form of a substance changes on changing its temperature. For example, raising the temperature of a liquid causes the liquid to vaporize (convert into vapor). The process is called vaporization—a surface phenomenon. Vaporization occurs when the thermal motion of the molecules overcome the intermolecular forces, and the molecules (at the surface) escape into the gaseous state. When a liquid vaporizes in a closed container, gas molecules cannot escape. As these gas phase...
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Heating a crystalline solid increases the average energy of its atoms, molecules, or ions, and the solid gets hotter. At some point, the added energy becomes large enough to partially overcome the forces holding the molecules or ions of the solid in their fixed positions, and the solid begins the process of transitioning to the liquid state or melting. At this point, the temperature of the solid stops rising, despite the continual input of heat, and it remains constant until all of the solid is...
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Quantum entanglement

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Area of Science:

  • Quantum physics
  • Computational complexity
  • Quantum information science

Background:

  • Entanglement is a key property of quantum systems.
  • Its role in computational complexity for quantum system simulation is not fully understood.
  • Current algorithms like tensor networks show explicit dependence on entanglement.

Purpose of the Study:

  • To quantitatively link quantum entanglement to the inherent complexity of simulating quantum systems.
  • To characterize entanglement and complexity as a function of system parameters.
  • To investigate the simulation of k-regular graph states on n qubits.

Main Methods:

  • Analysis of single-qubit measurements on k-regular graph states.
  • Quantitative assessment of entanglement in these states.
  • Characterization of computational complexity related to simulation.
  • Proof of a duality for simulation complexity.

Main Results:

  • A sharp transition in entanglement and simulation complexity occurs at k=3 (from easy/low entanglement to hard/high entanglement).
  • A reverse transition occurs at k=n-3 (from hard/high entanglement back to easy/low entanglement).
  • A duality is proven between low and high regularity simulation complexity for graph states.

Conclusions:

  • Entanglement directly correlates with algorithm-independent computational complexity in simulating quantum systems.
  • The k-regular graph states exhibit a clear entanglement-complexity phase transition.
  • The proven duality offers new insights into the simulation of quantum states.