Estimating Truncation Errors in Quantum Bosonic Systems Using Sampling Algorithms

Основні поняття

Markov Chain Monte Carlo methods can be used to efficiently estimate the digitization errors in a wide class of bosonic quantum systems, even for large system sizes beyond the reach of exact diagonalization techniques.

Анотація

The paper introduces an MCMC-based technique to determine the digitization effects in a class of bosonic systems with Hamiltonians of the form (2), using the coordinate-basis truncation scheme. This allows estimating expectation values of various operators at finite temperature, including their digitization errors, by dialing the temperature to study different energy scales.

As a demonstration, the method is applied to the (2+1)-dimensional scalar quantum field theory regularized on a lattice. The key points are:

1. The coordinate-basis truncation scheme is introduced for single-boson and multi-boson systems. This scheme admits MCMC simulations without a sign problem.

2. The MCMC formulation and algorithm are presented, leveraging the fact that the weights in the partition function are non-negative under certain conditions. This allows efficient computations compared to exact diagonalization.

3. Numerical results are shown for a single boson and the 2D scalar QFT on a 4x4 lattice. The digitization errors are found to decay exponentially as the digitization spacing adig is reduced, matching the expected behavior.

4. Even without analytical results, the digitization errors can be determined by fitting the numerical data at finite adig values. This provides a way to estimate the resources needed for realistic quantum simulations of bosonic theories, and to cross-check the validity of quantum simulation results.

Статистика

The width of the distribution of the Fourier mode ˜ϕq is estimated by √⟨ˆ˜ϕq ˆ˜ϕ−q⟩, which is √1.0819 ≃ 1.04 for q = (0, 0) and √0.1841 ≃ 0.43 for q = (π, π).

Цитати

"To simulate bosons on a qubit- or qudit-based quantum computer, one has to regularize the theory by truncating infinite-dimensional local Hilbert spaces to finite dimensions."
"In the search for practical quantum applications, it is important to know how big the truncation errors can be."
"MCMC methods can be used only for a limited class of quantities. Although such quantities enable us to cross-check the validity of the simulation (e.g., we can check if the correct ground-state wave function is obtained), the MCMC method cannot replace the quantum simulation (e.g., it is impossible to add a small excitation to the ground state and determine the Hamiltonian time evolution)."

Ключові висновки, отримані з

Estimating truncation effects of quantum bosonic systems using sampling algorithms

by Masanori Han... о arxiv.org 04-03-2024

https://arxiv.org/pdf/2212.08546.pdf

Глибші Запити

How can the MCMC-based digitization error estimation be extended to other types of bosonic systems beyond the class considered in this work

The MCMC-based digitization error estimation technique can be extended to other types of bosonic systems by adapting the coordinate-basis truncation scheme to suit the specific characteristics of the new systems. This extension would involve defining the appropriate coordinate variables and conjugate momenta for the new system, as well as formulating the Hamiltonian in terms of these variables. By following a similar approach to the one outlined in the context provided, researchers can apply the MCMC method to estimate the truncation effects in these different bosonic systems. Additionally, the Fourier transform and Trotterization steps may need to be adjusted based on the specific properties of the new system to ensure accurate estimation of the digitization errors.

What are the limitations of the MCMC approach in terms of the types of observables that can be reliably estimated, and how can these limitations be overcome

The limitations of the MCMC approach in estimating observables primarily stem from the complexity and computational cost associated with certain types of observables. Observables that involve highly entangled states or non-local correlations may pose challenges for accurate estimation using MCMC methods. To overcome these limitations, researchers can explore advanced MCMC techniques such as cluster algorithms or parallel tempering to improve the sampling efficiency and reduce the autocorrelation time. Additionally, optimizing the proposal distribution and updating strategies can enhance the exploration of the configuration space, leading to more reliable estimates of observables. Furthermore, combining MCMC with other numerical methods like tensor network approaches can provide a complementary way to estimate observables that are challenging for MCMC alone.

Given the ability to estimate digitization errors using classical MCMC, how can this information be leveraged to optimize the design of quantum algorithms and hardware for simulating bosonic quantum systems

The information obtained from estimating digitization errors using classical MCMC can be instrumental in optimizing the design of quantum algorithms and hardware for simulating bosonic quantum systems. By quantifying the truncation effects, researchers can determine the resources needed for realistic quantum simulations and validate the results obtained from quantum devices. This information can guide the development of error mitigation strategies and resource-efficient quantum algorithms tailored to specific bosonic systems. Moreover, understanding the digitization errors can aid in benchmarking quantum hardware performance and identifying areas for improvement in quantum simulation protocols. By leveraging the insights gained from classical MCMC-based error estimation, researchers can enhance the accuracy and efficiency of quantum simulations of bosonic systems.