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Reducing the Sampling Complexity of Energy Estimation in Quantum Many-Body Systems Using Empirical Variance

Alexander Gresch1,2, Uğur Tepe1, Martin Kliesch2

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This study introduces an adaptive quantum energy estimation method that improves sampling complexity guarantees. The novel approach, using empirical Bernstein stopping, enhances accuracy by up to one order of magnitude compared to basic methods.

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Area of Science:

  • Quantum computing
  • Quantum chemistry
  • Computational physics

Background:

  • Accurate energy estimation is vital for quantum algorithms, especially in quantum chemistry.
  • Current methods using Pauli decomposition and Hoeffding's inequality offer limited sampling complexity guarantees.
  • Optimizing sampling complexity is crucial for efficient quantum state preparation.

Purpose of the Study:

  • To develop a more efficient adaptive estimator for quantum state preparation energy.
  • To improve upon existing single-shot estimators by leveraging the state's actual variance.
  • To provide rigorous tail bounds for the new estimation method.

Main Methods:

  • Constructed an adaptive estimator based on the empirical Bernstein stopping (EBS) algorithm.
  • Utilized grouping schemes and the state's empirical variance for estimation.
  • Provided rigorous tail bounds leveraging empirical variance for theoretical guarantees.

Main Results:

  • The adaptive EBS estimator consistently improved upon elementary readout guarantees.
  • Demonstrated up to a 1-order-of-magnitude improvement in numerical benchmarks.
  • Successfully estimated ground-state energies of various Hamiltonians with enhanced accuracy.

Conclusions:

  • The developed adaptive estimator offers significant improvements in sampling complexity for quantum energy estimation.
  • Leveraging empirical variance with EBS provides a robust and more efficient approach.
  • This method has strong implications for advancing quantum algorithms in fields like quantum chemistry.