This is the talk that I gave at Wolfram Tech Conference 2025. It presents an in-depth exploration of the Zassenhaus, Baker–Campbell–Hausdorff and Magnus formulas, which decompose exponentials of operator sums in noncommutative settings. We detail their derivations, applications and recent algorithmic formulations for high-degree expansions. Special attention is given to symbolic implementations in Wolfram Language, showcasing efficient computation using custom-built functions and algebraic structures. Applications to operator theory, Lie algebras and symbolic matrix analysis will be discussed.I believe new tools for performing non-commutative algebraic computations can open up new horizons in quantum computing, from simulating error correction to studying open quantum systems and noise. CITATION: Mads Bahrami, Expanding Exponentials in Noncommutative Algebras: The Zassenhaus and Baker–Campbell–Hausdorff Formulas, Wolfram Technology Conference 2025. https://www.wolfram.com/events/technology-conference/2025/​​Baker-Campbell-Hausdorff (BCH) series:

1


2


m
…
=
m
∑
j=1

j
+
∞
∑
n=2

n

​Zassenhaus series:
m
∑
j=1

m

=
m
∏
j=1

j

∏
k=2

k

​Magnus series:
Y(t)
t
=A(t)Y(t)⇒Y(t)=
∞
∑
n=1
Ω
n
(t)
e

Initialization


Adjoint action

The adjoint action of
A
on another operator
B
is define as the commutation
[A,B]
. A direct interesting result is the exponential adjoint action
A

B
-A

=
ad
A

B
with
ad
A
[B]=[A,B]
In[]:=
ClearAll[A,B]​​adjointAction[A,B,1]
Out[]=
A**B-B**A
The
j
-th iterated adjoint of
A
is defined as the
n
-fold nested commutator of
A
with
B
, denoted by
(j)
ad
A
[B]=[A,[A,…[A,B]…]]
:
In[]:=
adjointAction[A,B,2]==Commutator[A,Commutator[A,B]]
Out[]=
True
In[]:=
adjointAction[A,B,3]==Commutator[A,Commutator[A,Commutator[A,B]]]
Out[]=
True
We shall study some examples from quantum optics. To do so, let us first define a function to reduce noncommutative polynomials of annihilation and creation operators.

Examples

Bosonic commutation relation


a
,
†

a
=1
Consider
λ
†
a
a

a
-λ
†
a
a

. Compute 10 terms in the exponential adjoint:
In[]:=
NCPolynomialReductionBosonicSumNCPolynomialReductionBosonicadjointActionλ
†
a
**a,a,j,{λ}j!,{j,0,10},{λ}
Out[]=
a1-λ+
2
λ
2
-
3
λ
6
+
4
λ
24
-
5
λ
120
+
6
λ
720
-
7
λ
5040
+
8
λ
40320
-
9
λ
362880
+
10
λ
3628800
Verify the summation of 10 terms is the same as Taylor expansion of
-λ

In[]:=
NormalSeries
-λ

,{λ,0,10}==CoefficientRules%,a,
†
a
[[1,-1]]
Out[]=
True
So we managed to verify
λ
†
a
a

a
-λ
†
a
a

=
-λ

a
Consider
1
2

*
ζ
2
a
-ζ
2
†
a


a
-
1
2

*
ζ
2
a
-ζ
2
†
a


with
ζ=r
ϕ

. Compute 10 terms in the exponential adjoint:
In[]:=
NCPolynomialReductionBosonicSumNCPolynomialReductionBosonicadjointAction

-ϕ

a**a-
ϕ

†
a
**
†
a
2
,a,jj!,{,ϕ},{j,0,10},{,ϕ}
Out[]=
a1+
2

2
+
4

24
+
6

720
+
8

40320
+
10

3628800
+
ϕ

+
1
6
ϕ

3

+
1
120
ϕ

5

+
ϕ

7

5040
+
ϕ

9

362880
†
a
Verify
1
2

*
ζ
2
a
-ζ
2
†
a


a
-
1
2

*
ζ
2
a
-ζ
2
†
a


=ua+v
†
a
with
u=cosh(r)
and
v=-
ϕ

sinh(r)
:
In[]:=
NormalSeriesCosh[],
ϕ

Sinh[],{,0,10}==CoefficientRules%,a,
†
a
[[All,-1]]
Out[]=
True

Baker-Campbell-Hausdorff (BCH) series

The BCH series expresses the product of two or more exponentials of non-commuting operators as a single exponential:

1


2


m
…
=
m
∑
j=1

j
+
∞
∑
n=2

n

For two operators,
X+Y

, one gets the Dynkin’s formula:

m
=
m
∑
n=1
n-1
(-1)
n
∑
p
1
+
q
1
>0,…,
p
n
+
q
n
>0,
n
∑
j=1
(
p
j
+
q
j
)=m-1
p
1
ad
X
q
1
ad
Y
…
p
n
ad
X
q
n
ad
Y
p
1
!
q
1
!…
p
n
!
q
n
!
(X+Y)
.
Show a few terms of
A

B

=
A+B+
∑
n=2

n

In[]:=
ClearAll[A,B];​​TableForm[Table[{n,TraditionalForm@BakerCampbellHausdorffTerms[{A,B},n,"CommutatorForm"->True]},{n,2,4}],TableHeadings->{None,{"n","

n
"}},TableAlignments->Left]
Out[]//TableForm=
n

n
2
[A,B]
2
3
-
[A,[A,B]]
12
-
[A,[B,A]]
6
-
[B,[A,B]]
6
-
[B,[B,A]]
12
4
[A,[A,[A,B]]]
16
+
[A,[A,[B,A]]]
16
+
[A,[B,[A,B]]]
24
+
[A,[B,[B,A]]]
16
+
[B,[A,[A,B]]]
16
+
[B,[A,[B,A]]]
12
+
[B,[B,[A,B]]]
16
+
[B,[B,[B,A]]]
16
Most quantum operators belong to small Lie algebras (Heisenberg–Weyl, SU(2), SU(1,1)), where the commutator chains truncate or close, allowing the BCH formula to yield clean, closed-form exponential identities.

Normal ordering and Weyl relations

We will explore some examples involving photonic operators, where the single-mode photon field satisfies the Bosonic commutation relation


a
,
†

a
=1
.

Annihilation and creation operator:

Consider the expression:
μ
†

a

ν

a

=
μ
†

a
+ν

a
+
∑
n

n

Compute the order-2 term in BCH series:
In[]:=
BakerCampbellHausdorffTermsμa,ν
†
a
,2
Out[]=
1
2
(aμ)**(ν
†
a
)-
1
2
(ν
†
a
)**(aμ)
Reduce/simplify it:
In[]:=
NCPolynomialReductionBosonic[%,{μ,ν}]
Out[]=
μν
2
Note the result is a c-number only.
Compute the order-3 term of BCH series:
In[]:=
BakerCampbellHausdorffTermsμa,ν
†
a
,3
Out[]=
1
12
(aμ)**(aμ)**(ν
†
a
)-
1
6
(aμ)**(ν
†
a
)**(aμ)+
1
12
(aμ)**(ν
†
a
)**(ν
†
a
)+
1
12
(ν
†
a
)**(aμ)**(aμ)-
1
6
(ν
†
a
)**(aμ)**(ν
†
a
)+
1
12
(ν
†
a
)**(ν
†
a
)**(aμ)
Reduce it:
In[]:=
NCPolynomialReductionBosonic[%,{μ,ν}]
Out[]=
0
Likewise, the other terms disappear as well. Thus one finds:
μ
†

a

ν

a

=
μ
†

a
+ν

a
+μν/2


Canonical position and momentum

Consider canonical position and momentum operators: ​

q
=
1
2
(

a
+
†

a
)
and

p
=
1

2
(

a
-
†

a
)
Define canonical position and momentum operators:
In[]:=
q=(a+
†
a
)
2
;p=(a-
†
a
)
2
;
Verify


q
,

p
=
In[]:=
NCPolynomialReductionBosonic@Commutator[q,p]
Out[]=

Now let’s try to compute
α

q

β

p

.
Compute the second term in the BCH series:
In[]:=
NCPolynomialReductionBosonic[BakerCampbellHausdorffTerms[{αq,βp},2],{α,β}]
Out[]=
-
1
2
αβ
Show that the higher-order terms are zero:
In[]:=
Table[NCPolynomialReductionBosonic[BakerCampbellHausdorffTerms[{αq,βp},n],{α,β}],{n,3,6}]
Out[]=
{0,0,0,0}
Thus it means
α

q

β

p

=
-αβ/2

α

q
+β

p)


Displacement operator

Displacement operator is defined as
(α)=
α
†
a
-
*
α
a

. Show that
(α)(β)=
α
*
β
-
*
α
β
2

(α+β)
In[]:=
NCPolynomialReductionBosonicBakerCampbellHausdorffTermsα
†
a
-Conjugate[α]a,β
†
a
-Conjugate[β]a,2,{α,β}
Out[]=
-
1
2
βConjugate[α]+
1
2
αConjugate[β]
Note that the result is a c-number.
Show that the higher-order terms are zero:

SU(2) spin rotations

Compute the second term of the BCH series:

Numerical example

Note that in the previous examples, all operators were symbolic. One can use the same thing with numerical operators, too.

Zassenhaus series

Displacement operator:

Compute the reduced form of the order-2 term in Zassenhaus series:
Note the result is a c-number, which implies higher order terms are zero:
Compute the second term in Zassenhaus series:
Compute a few terms (3rd and more) and show they are zero:

SU(2) Trotter error

SU(2) Gauss/Wei–Norman decomposition

Putting them together, one finds:
Define noncommutative variables:
Putting above results together and simplify it, one finds:
So the same for the final result we obtained above:
Equate coefficients:
Solve them:
Plug the result back into the solution:

Magnus series

Show three terms of Magnus series (without time integration):

Examples

For each term, find the coefficient for Pauli matrices (keep in mind that we will have only linear terms):
Now let's repeat the same process, but this time assigning explicit values to Pauli matrics and doing the commutation/dot calculations:
As expected, the coefficients in front of Pauli matrices are the same the ones we obtained from non - commutative algebraic calculations.
Calculate the unitary operator:

CITE THIS NOTEBOOK

Exponentials in Noncommutative Algebras Zassenhaus, BCH and Magnus​
by Mohammad Bahrami​
Wolfram Community, STAFF PICKS, December 2, 2025
​https://community.wolfram.com/groups/-/m/t/3584499