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Efficient computation of the Zassenhaus formula
Fernando Casas, Ander Murua, Mladen Nadinic
TL;DR
Computing high-order Zassenhaus terms is costly because common expansions can use dependent commutators that require rewriting in a free-Lie-algebra basis. This paper develops a recurrence that produces independent commutators directly and derives sharper Banach-algebra convergence bounds. It reports a computation of C20 with 48,528 independent terms and a considerably enlarged convergence domain.
Problem
Many procedures for computing high-degree commutator expansions do not yield linearly independent terms, making conversion to a free-Lie-algebra basis time- and memory-intensive as term counts grow exponentially with degree.
Source: §1 · p. 2Method
A recursive procedure computes Zassenhaus terms directly as linear combinations of independent commutators and bounds the terms to analyze convergence in Banach algebras.
Source: §1 · p. 4Results
C20 contains 48,528 independent terms and took less than 20 seconds of CPU time and 35 MBytes of memory to construct; the authors report a considerably enlarged convergence domain.
Source: §2.3 · p. 9, §3 · p. 12Takeaways & Limitations
Direct output in independent commutators avoids a separate basis-rewriting process and its associated computing-time and memory costs.
Source: §1 · p. 4The convergence analysis is formulated for X and Y defined in a Banach algebra, a complete normed linear space with a submultiplicative norm.
Source: §3 · p. 10
Abstract
from arXiv · showhide
A new recursive procedure to compute the Zassenhaus formula up to high order is presented, providing each exponent in the factorization directly as a linear combination of independent commutators and thus containing the minimum number of terms. The recursion can be easily implemented in a symbolic algebra package and requires much less computational effort, both in time and memory resources, than previous algorithms. In addition, by bounding appropriately each term in the recursion, it is possible to get a larger convergence domain of the Zassenhaus formula when it is formulated in a Banach algebra.
1 Introduction
The Zassenhaus formula factors an exponential of a sum of non-commuting operators into a product of exponentials, but computing its terms efficiently is difficult because many representations use dependent commutators and require costly rewriting. This paper introduces a recurrence that outputs independent commutators directly and sharpens convergence bounds in Banach algebras.
1 Introduction
- The Zassenhaus exponents are homogeneous Lie polynomials: rational linear combinations of nested commutators in X and Y.The factorization disentangles the exponential of a sum of non-commuting operators into a product of exponentials.Source: §1 · pp. 1–2
- Common computations yield commutators that are not linearly independent, so rewriting them in a free-Lie-algebra basis costs time and memory as the number of terms grows exponentially with degree.The introduction notes a recent algorithm that computes BCH expressions in Hall and Lyndon bases.Source: §1 · p. 2
- The new recurrence expresses each Zassenhaus term directly using the minimum number of independent commutators, avoiding basis rewriting and saving computing time and memory.It can be implemented in a symbolic algebra package.Source: §1 · p. 4
- The paper also derives sharper bounds for Zassenhaus terms, enlarging the formula’s convergence domain when formulated in a Banach algebra.The reported domain contains x+y < 1.054 and is considerably larger than the earlier Suzuki and Bayen bounds.Source: §1 · p. 4
2 Constructing the Zassenhaus terms
A recursive procedure computes Zassenhaus exponents directly as homogeneous Lie polynomials in independent commutators. The paper examines their linear independence and reports computational results through degree 20.
2.1 A new recurrence
- The recurrence computes Zassenhaus terms of arbitrarily large degree as homogeneous Lie polynomials and can be implemented in a symbolic algebra package.Source: §2.1 · p. 7
2.2 Linear independence
- By construction, the computed exponents use only independent commutators and cannot be further simplified using Jacobi or commutator antisymmetry.The corresponding representation of the Zassenhaus terms is linearly independent in the free Lie algebra.Source: §2.2 · pp. 7–8
2.3 Computational aspects
- On a 2.4 GHz Core 2 Duo with 2 GBytes of RAM, constructing terms through degree 20 took less than 20 seconds and 35 MBytes.C20 has 48,528 independent terms, and the expressions through C8 match the classical Hall basis.Source: §2.3 · p. 9
- Table 1 compares CPU time and memory for the new recurrence and W-S, which expresses exponents as words rather than directly as independent commutators.Source: §2.3 · p. 9, Table 1 · p. 10
- For C16, the new formulation uses 3,711 terms versus 54,146 terms as combinations of words, translating directly into memory requirements.Source: §2.3 · p. 10
3 Convergence of the Zassenhaus formula
The section establishes a larger convergence domain for the Zassenhaus product in a Banach algebra by bounding terms in the recursion. Numerically, this domain exceeds prior sum-norm bounds and includes points where one norm is zero and the other is arbitrarily large.
3 Convergence of the Zassenhaus formula
- The recursion establishes e^X e^Y e^C2 ... e^Cn → e^(X+Y) on a domain larger than the prior sum-norm region.Source: §3 · p. 10
- The proof bounds recursive terms via δ_n, whose bounds depend on x=||X|| and y=||Y||, then imposes a sufficient convergence condition.Source: §3 · pp. 11–12
- The computed domain contains x+y<1.054, beyond Suzuki’s x+y<0.1931 and Bayen’s x+y<0.5967 bounds.Source: Figure 1 · p. 12, §3 · p. 12