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Quantum Computation as Gravity.

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We connect geometric circuit complexity to two-dimensional conformal field theories. This reveals that gravity governs optimal quantum computation in these theories, linking quantum information to fundamental physics.

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Area of Science:

  • Quantum Field Theory
  • Quantum Information Theory
  • Quantum Gravity

Background:

  • Nielsen's geometric approach to quantum circuit complexity provides a framework for quantifying computational difficulty.
  • Two-dimensional conformal field theories (CFTs) possess rich mathematical structures with connections to gravity.

Purpose of the Study:

  • To formulate Nielsen's geometric approach to circuit complexity within the framework of two-dimensional conformal field theories.
  • To establish a connection between quantum computation and fundamental principles of gravity in CFTs.

Main Methods:

  • Interpreting series of conformal transformations in 2D CFTs as unitary circuits.
  • Utilizing energy-momentum tensor gates to construct these circuits.
  • Reformulating the complexity functional using concepts from 2D gravity and Lie group theory.

Main Results:

  • The complexity functional in 2D CFTs is shown to be equivalent to the Polyakov action of two-dimensional gravity.
  • The complexity is also shown to be equivalent to the geometric action on the coadjoint orbits of the Virasoro group.
  • A direct link is established between the geometric approach to circuit complexity and the principles of 2D gravity.

Conclusions:

  • Gravity dictates the rules for optimal quantum computation in two-dimensional conformal field theories.
  • This work bridges the gap between quantum information science and theoretical physics, specifically quantum gravity.
  • The findings suggest a deeper relationship between the geometry of spacetime and the efficiency of quantum computation.