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Related Concept Videos

Spin–Spin Coupling Constant: Overview01:08

Spin–Spin Coupling Constant: Overview

In bromoethane, the three methyl protons are coupled to the two methylene protons that are three bonds away. In accordance with the n+1 rule, the signal from the methyl protons is split into three peaks with 1:2:1 relative intensities. The methylene protons appear as a quartet, with the relative intensities of 1:3:3:1.
Qualitatively, any spin plus-half nucleus polarizes the spins of its electrons to the minus-half state. Consequently, the paired electron in the hydrogen–carbon bond must have a...
Spin–Spin Coupling: One-Bond Coupling01:17

Spin–Spin Coupling: One-Bond Coupling

Coupling interactions are strongest between NMR-active nuclei bonded to each other, where spin information can be transmitted directly through the pair of bonding electrons. While nuclei polarize their electrons to the opposite spins, the bonding electron pair has opposite spins. Configurations with antiparallel nuclear spins are expected to be lower in energy. When coupling makes antiparallel states more favorable, J is considered to have a positive value. The one-bond coupling constant, 1J,...
Linear Approximations01:23

Linear Approximations

For a differentiable function of two variables, linear approximation estimates values near a known point by replacing the curved surface with its tangent plane. Consider the function\begin{equation*}f(x,y)=x^2+3y^2\end{equation*}near the point (2, 1). The exact value at this point is f(2, 1) = 22 + 3(1)2 = 4 + 3 = 7.The linear approximation of f(x, y)) near (a, b) is\begin{equation*}L(x,y)=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)\end{equation*}First, compute the partial derivatives: fx(x, y) = 2x and...
Linearization and Approximation01:26

Linearization and Approximation

Linearization is a mathematical technique used to approximate complex, nonlinear functions with simpler linear models in the vicinity of a chosen reference point. The method is based on the idea that, although a function may be difficult to evaluate exactly, its behavior near a specific input value can often be closely approximated by the tangent line at that point. This approach is particularly useful when small deviations from a known value are involved.Consider the square root function, for...
Spin–Spin Coupling: Two-Bond Coupling (Geminal Coupling)01:20

Spin–Spin Coupling: Two-Bond Coupling (Geminal Coupling)

Two NMR-active nuclei bonded to a central atom can be involved in geminal or two-bond coupling. Geminal coupling is commonly seen between diastereotopic protons in chiral molecules and unsymmetrical alkenes, among others.
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Vectors in Space: Problem Solving01:26

Vectors in Space: Problem Solving

A chandelier suspended by multiple cables can be analyzed using principles of three-dimensional static equilibrium. In this setup, a chandelier weighing 1000 N is positioned at the origin of a three-dimensional coordinate system, while three ceiling anchor points are fixed at known locations above it. Each cable connects the chandelier to one anchor point and transmits a tensile force along its length.To find out the forces in the cables, the spatial direction of each cable must first be...

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Related Experiment Videos

Spin-Boson Mapping of the Quantum Approximate Optimization Algorithm.

Sami Boulebnane1, Abid Khan1, Minzhao Liu1

  • 1JPMorganChase, Global Technology Applied Research, New York, New York 10001, USA.

Physical Review Letters
|July 7, 2026
PubMed
Summary
This summary is machine-generated.

High-depth quantum approximate optimization algorithms (QAOA) for the Sherrington-Kirkpatrick model are now more accessible. This research maps QAOA to a spin-boson system, enabling study of previously unreachable depths.

Related Experiment Videos

Area of Science:

  • Quantum computing
  • Condensed matter physics
  • Optimization algorithms

Background:

  • The quantum approximate optimization algorithm (QAOA) shows improved performance with circuit depth.
  • Evaluating high-depth QAOA is computationally expensive, limiting research.
  • The Sherrington-Kirkpatrick (SK) model is a standard benchmark for optimization problems.

Purpose of the Study:

  • To overcome the computational limitations of studying high-depth QAOA.
  • To develop a new method for analyzing QAOA performance in the high-depth regime.
  • To investigate the behavior of QAOA for the SK model at large circuit depths.

Main Methods:

  • Proving the convergence of depth-p QAOA states to a spin-boson system in the infinite-size limit.
  • Simulating the equivalent spin-boson system using matrix product states.
  • Optimizing QAOA parameters through the spin-boson mapping.

Main Results:

  • The depth-p QAOA state for the SK model converges to a spin coupled to p bosonic modes in the infinite-size limit.
  • QAOA achieves a (1-ε) approximation to the optimal SK model energy with circuit depth O(n/ε^{1.13}) in the average case.
  • QAOA performance reaches ϵ≲2.2% at p=160, significantly beyond previously accessible depths (p≤20).

Conclusions:

  • The spin-boson mapping provides an efficient many-body approach to study and optimize high-depth QAOA.
  • This method opens new avenues for exploring regimes of QAOA previously limited by computational cost.
  • The findings advance the understanding of QAOA scalability and its application to complex optimization problems.