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Trotterization is Substantially Efficient for Low-Energy States.

Kaoru Mizuta1,2,3, Tomotaka Kuwahara4,5,6

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Trotterization simulations show reduced error and cost for low-energy quantum states. This study proves optimal bounds, revealing a genuine advantage for these states in quantum computing and tensor network simulations.

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Area of Science:

  • Quantum Computing
  • Quantum Many-Body Dynamics
  • Condensed Matter Physics

Background:

  • Trotterization is a key method for simulating quantum dynamics.
  • Low-energy initial states can reduce Trotterization error and cost.
  • Previous studies showed diminishing improvements with higher Trotter orders, questioning the low-energy advantage.

Purpose of the Study:

  • To resolve the mystery of the genuine advantage of low-energy initial states in Trotterization.
  • To prove the optimal error bound and computational cost for Trotterization of low-energy states.
  • To establish theoretical best scaling for Trotterization accuracy and efficiency.

Main Methods:

  • Proving optimal error bounds for Trotterization with low-energy initial states.
  • Analyzing generic local Hamiltonians with positive-semidefinite terms.
  • Deriving computational cost scaling with initial state energy and system size.

Main Results:

  • Trotter error scales linearly with initial state energy (Δ) and polylogarithmically with system size (N).
  • Computational cost is substantially reduced for low-energy states (Δ∈o(Ng)).
  • Results partially extend to weakly correlated initial states with low-energy expectation values.

Conclusions:

  • This work establishes the theoretically optimal scaling for Trotterization error and cost concerning initial state energy.
  • The findings demonstrate a genuine advantage of using low-energy states for Trotterization.
  • The results facilitate faster and more accurate simulations of low-energy states, crucial for condensed matter physics and quantum chemistry.