TrotterSuzukidekompositioner

The Trotter-Suzuki decomposition is a family of numerical methods for approximating the exponential of a sum of non-commuting operators, a common problem in simulating quantum dynamics and statistical mechanics. The idea is to factor exp((A+B)t) into a product of exponentials of A and B separately, so that each exponential can be computed efficiently. The original Trotter formula, attributed to H. F. Trotter, states that exp((A+B)t) ≈ (exp(A t/n) exp(B t/n))^n as n becomes large, with the leading error term involving the commutator [A,B].

Suzuki extended and generalized the approach to higher orders by using symmetric factorization. A widely used second-order (symmetric) decomposition is exp((A+B)t) ≈ (exp(A t/2n) exp(B t/n) exp(A t/2n))^n, which typically yields a global error proportional to t^3/n^2. Suzuki and collaborators developed systematic recursive constructions to obtain even higher-order decompositions, improving accuracy at the cost of more factors in the product. These higher-order schemes are designed to reduce the step size required for a given accuracy.

Applications are broad in computational physics and chemistry. They are used to simulate real-time quantum evolution, compute thermal properties via exp(-βH) with H = T+V, and enable efficient quantum Monte Carlo and tensor-network methods. The methods are particularly valuable when A and B represent local or easily exponentiable terms in lattice Hamiltonians or other structured systems.

Limitations include the need for tractable exponentials of A and B and the growth of computational cost with higher-order schemes. The accuracy depends on the size of commutators like [A,B] and may require small time steps or problem-specific optimizations.