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Related Concept Videos

Constraints and Statical Determinacy01:26

Constraints and Statical Determinacy

In structural engineering, the equilibrium of a system is not only determined by its equations of equilibrium but also with the help of constraints. Constraints refer to restrictions on the motion of a system. The proper combinations of constraints can minimize the total number of constraints needed to maintain a system in mechanical equilibrium. When this happens, the system is said to be statically determinate. For such systems, the unknown reaction supports can be estimated using equilibrium...
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Lagrange Multipliers: One Constraint

In constrained optimization, the objective is to maximize or minimize a quantity while satisfying a fixed condition. A standard example is a rectangular pen built against a barn wall using 100 meters of fencing. Because the wall provides one side of the enclosure, only the other three sides require fencing. The problem is to find the dimensions that produce the greatest possible area.Let L represent the length parallel to the wall and W the width perpendicular to it. The area of the pen is A =...
Lagrange Multipliers: Two Constraints01:28

Lagrange Multipliers: Two Constraints

The method of Lagrange multipliers with two constraints is used to optimize a function subject to two independent constraints. In many applications, the objective function represents a quantity to be maximized or minimized, such as cost, area, distance, or energy. The two constraints represent requirements that the solution must satisfy, such as fixed volume, limited resources, or prescribed dimensions.For a function of three variables, each constraint forms a surface in three-dimensional space.
Design Example: Maintaining Level of an Embankment01:19

Design Example: Maintaining Level of an Embankment

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Design Consideration01:22

Design Consideration

Designing a structure involves a series of considerations, primarily the material's ultimate strength, calculated through tests that measure changes under increased force until the material reaches its breaking point or limit. The ultimate load, where the material breaks, is divided by its original cross-sectional area, resulting in the ultimate normal stress or strength. The ultimate shearing stress is another significant factor taken into account.
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Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving

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Related Experiment Videos

Adaptively Secure Unitary Designs with Constant Non-Clifford Cost.

Lennart Bittel1, Lorenzo Leone1,2

  • 1Freie Universität Berlin, Dahlem Center for Complex Quantum Systems, 14195 Berlin, Germany.

Physical Review Letters
|June 12, 2026
PubMed
Summary
This summary is machine-generated.

This study introduces a new protocol for generating quantum unitary k-designs using a system-size-independent number of non-Clifford gates. This advancement makes high-order unitary designs more practical for near-term quantum computers.

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Area of Science:

  • Quantum Information Science
  • Quantum Computing Security
  • Quantum Cryptography

Background:

  • Randomness is essential for quantum information applications like cryptography and algorithms.
  • Constructing unitary k-designs that approximate Haar-random unitaries is challenging due to the high cost of non-Clifford operations.

Purpose of the Study:

  • To develop a protocol for generating unitary k-designs on n qubits with minimal non-Clifford gates.
  • To ensure the generated designs are secure against adaptive quantum adversaries.
  • To reduce the overhead of non-Clifford operations in quantum information protocols.

Main Methods:

  • A 'seed' k-design is applied to a small subsystem of size Θ(k).
  • This seed design is 'diluted' across the n-qubit system using random Clifford operators.
  • The security and efficiency of the resulting ensemble as an ϵ-approximate unitary k-design are analyzed.

Main Results:

  • The protocol generates unitary k-designs with a system-size-independent number of non-Clifford gates.
  • Full quantum security against adaptive adversaries is achieved using O[over ˜](k^{2}logϵ^{-1}) non-Clifford gates.
  • For polynomial-time adversaries, the cost reduces to O[over ˜](k+log^{1+c}ϵ^{-1}), which is shown to be optimal.

Conclusions:

  • The proposed method significantly reduces non-Clifford overhead compared to existing approaches.
  • It strengthens security guarantees to adaptive security and removes artificial assumptions between system size (n) and design order (k).
  • These findings make high-order unitary designs practically attainable in near-term fault-tolerant quantum architectures.