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Training Variational Quantum Algorithms Is NP-Hard.

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Variational quantum algorithms face NP-hard optimization challenges, even for simple systems. This intrinsic difficulty, stemming from numerous local minima, hinders convergence to optimal solutions for quantum computing tasks.

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Area of Science:

  • Quantum Computing
  • Computational Complexity Theory
  • Quantum Chemistry

Background:

  • Variational quantum algorithms (VQAs) like VQE and QAOA are prominent for near-term quantum devices.
  • These algorithms train parametrized quantum circuits using classical optimization.

Purpose of the Study:

  • To analyze the computational complexity of the classical optimization problems inherent in VQAs.
  • To determine if the hardness of VQAs originates from the quantum problem or the classical optimization itself.

Main Methods:

  • Theoretical analysis of the classical optimization landscape associated with VQAs.
  • Investigation of worst-case instances for polynomial-time classical algorithms.
  • Examination of specific cases like logarithmic qubits and free fermions.

Main Results:

  • The classical optimization problems for VQAs are proven to be NP-hard.
  • This hardness is robust, implying significant approximation errors for polynomial-time algorithms (assuming P≠NP).
  • Optimization remains NP-hard even for classically tractable quantum systems, indicating intrinsic classical difficulty.

Conclusions:

  • The classical optimization component of VQAs is intrinsically hard, not just a reflection of the underlying quantum problem.
  • Training landscapes often contain numerous persistent local minima, impeding gradient-based optimization.
  • VQAs may generally converge to suboptimal solutions due to these challenging optimization landscapes.