# A Trotter-Suzuki approximation for Lie groups with applications to Hamiltonian simulation

## Abstract

We present a product formula to approximate the exponential of a skew-Hermitian operator that is a sum of generators of a Lie algebra. The number of terms in the product depends on the structure factors. When the generators have large norm with respect to the dimension of the Lie algebra, or when the norm of the effective operator resulting from nested commutators is less than the product of the norms, the number of terms in the product is significantly less than that obtained from well-known results. We apply our results to construct product formulas useful for the quantum simulation of some continuous-variable and bosonic physical systems, including systems whose potential is not quadratic. For many of these systems, we show that the number of terms in the product can be sublinear or even subpolynomial in the dimension of the relevant local Hilbert spaces, where such a dimension is usually determined by the energy scale of the problem. Our results emphasize the power of quantum computers for the simulation of various quantum systems.

- Authors:

- Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545 (United States)

- Publication Date:

- OSTI Identifier:
- 22596844

- Resource Type:
- Journal Article

- Journal Name:
- Journal of Mathematical Physics

- Additional Journal Information:
- Journal Volume: 57; Journal Issue: 6; Other Information: (c) 2016 Author(s); Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 0022-2488

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; APPROXIMATIONS; HAMILTONIANS; HERMITIAN OPERATORS; HILBERT SPACE; LIE GROUPS; QUANTUM COMPUTERS; QUANTUM SYSTEMS; SIMULATION; STRUCTURE FACTORS

### Citation Formats

```
Somma, Rolando D., E-mail: somma@lanl.gov.
```*A Trotter-Suzuki approximation for Lie groups with applications to Hamiltonian simulation*. United States: N. p., 2016.
Web. doi:10.1063/1.4952761.

```
Somma, Rolando D., E-mail: somma@lanl.gov.
```*A Trotter-Suzuki approximation for Lie groups with applications to Hamiltonian simulation*. United States. doi:10.1063/1.4952761.

```
Somma, Rolando D., E-mail: somma@lanl.gov. Wed .
"A Trotter-Suzuki approximation for Lie groups with applications to Hamiltonian simulation". United States. doi:10.1063/1.4952761.
```

```
@article{osti_22596844,
```

title = {A Trotter-Suzuki approximation for Lie groups with applications to Hamiltonian simulation},

author = {Somma, Rolando D., E-mail: somma@lanl.gov},

abstractNote = {We present a product formula to approximate the exponential of a skew-Hermitian operator that is a sum of generators of a Lie algebra. The number of terms in the product depends on the structure factors. When the generators have large norm with respect to the dimension of the Lie algebra, or when the norm of the effective operator resulting from nested commutators is less than the product of the norms, the number of terms in the product is significantly less than that obtained from well-known results. We apply our results to construct product formulas useful for the quantum simulation of some continuous-variable and bosonic physical systems, including systems whose potential is not quadratic. For many of these systems, we show that the number of terms in the product can be sublinear or even subpolynomial in the dimension of the relevant local Hilbert spaces, where such a dimension is usually determined by the energy scale of the problem. Our results emphasize the power of quantum computers for the simulation of various quantum systems.},

doi = {10.1063/1.4952761},

journal = {Journal of Mathematical Physics},

issn = {0022-2488},

number = 6,

volume = 57,

place = {United States},

year = {2016},

month = {6}

}