Quantum Computing
(Fall 2013)
Instructor: Shengyu Zhang.
Lecture 3 Hidden Subgroup Problem 1: Group Representations
Studies of Hidden Subgroup Problem for non-Abelian groups often need knowledge of group
representations, which is the subject of this lecture.
1. Linear representation of finite groups
Suppose that 𝑉 is a vector space over ℂ. The general linear group 𝐺𝐿(𝑉) is the group of
invertible linear operators from 𝑉 to itself. Here we only consider the case 𝑉 = ℂ𝑛 . The
group 𝐺𝐿(𝑉) can also be identified with the group of invertible matrices in ℂ𝑛×𝑛 .
For a finite group 𝐺, a linear representation is a homomorphism 𝜌: 𝐺 → 𝐺𝐿(𝑉). The
dimension of 𝑉 is called the degree of 𝜌, denoted by 𝑑𝜌 . (Homomorphism: The map 𝜌 is
a homomorphism if 𝜌(𝑥)𝜌(𝑦) = 𝜌(𝑥𝑦), ∀𝑥, 𝑦 ∈ 𝐺.) Note that this implies that 𝜌(𝑥 −1 ) =
𝜌(𝑥)−1 , and 𝜌(1𝐺 ) = 𝐼 where 1𝐺 is the identify element of the group 𝐺.
Example 1. Trivial representation: 𝜌(𝑥) = 1, ∀𝑥 ∈ 𝐺. Here the degree 𝑑𝜌 = 1.
Example 2. Regular representation: 𝑉 = 𝑠𝑝𝑎𝑛{|𝑥〉: 𝑥 ∈ 𝐺}. The left regular representation 𝐿
is defined by 𝐿(𝑥)|𝑦〉 = |𝑥𝑦〉, and the right regular representation 𝑅 is defined by
𝑅(𝑥)|𝑦〉 = |𝑦𝑥 −1 〉. Here the degree 𝑑𝐿 = 𝑑𝑅 = |𝐺|.
Two representations 𝜌1 and 𝜌2 are isomorphic if there is an invertible linear operator
𝑀: 𝑉 → 𝑉 s.t. 𝜌1 (𝑥)𝑀 = 𝑀𝜌2 (𝑥), ∀𝑥 ∈ 𝐺. Intuitively, it means that the two representations
are the same up to a change of basis.
For a representation 𝜌: 𝐺 → 𝐺𝐿(𝑉), a subspace 𝑊 of 𝑉 is called 𝐺-invariant if
𝜌(𝑥)𝑊 ⊆ 𝑊, ∀𝑥 ∈ 𝐺.
Claim. If 𝑊 is 𝐺-invariant, then there is another subspace 𝑊′ s.t. 𝑉 = 𝑊 ⊕ 𝑊′, and 𝑊′
is also 𝐺-invariant.
See [Ser77] for two proofs. By this result, we know that if there is a 𝐺-invariant subspace 𝑊,
then 𝑉 = 𝑊 ⊕ 𝑊′ for a subspace 𝑊′, and the representation can also be decomposed into
the direct sum of two subrepresentations, namely 𝜌 = 𝜌𝑊 ⊕ 𝜌𝑊 ′ . In matrix form, 𝜌 is a
𝜌
block diagonal matrix ( 𝑊
𝜌𝑊 ′
).
If 𝜌 doesn’t have any nontrivial 𝐺-invariant subspace, we call 𝜌 irreducible. We use 𝐺̂ to
denote the set of irreducible representations of 𝐺. We sometimes use “irrep” as a shorthand
for “irreducible representation”.
Fact. Every representation is isomorphic to a direct sum of irreducible representations.
An important result about irreducible representations is the following Schur’s lemma.
Theorem (Schur’s lemma). For two irreducible representations 𝜌1 and 𝜌2 with degree 𝑛1
and 𝑛2 , respectively, assume that there is a matrix 𝑀 ∈ ℂ𝑛1 ×𝑛2 s.t. 𝜌1 (𝑥)𝑀 =
𝑀𝜌2 (𝑥), ∀𝑥 ∈ 𝐺. If 𝜌1 is not isomorphic to 𝜌2 , then 𝑀 = 0. If 𝜌1 = 𝜌2 and 𝑛1 = 𝑛2 ,
then 𝑀 = 𝜆𝐼 for some real number 𝜆.
Proof. See [Ser77].
This lemma can be used to prove the following fundamental orthogonality theorem.
Theorem. For any two different irreducible representations 𝜌1 and 𝜌2 in matrix form, and
any 𝑖, 𝑗 ∈ [𝑛1 ], 𝑖′, 𝑗′ ∈ [𝑛2 ], it holds that
∑ 𝜌1 (𝑥 −1 )𝑖𝑗 𝜌2 (𝑥)𝑖 ′ 𝑗′ = 0.
𝑥∈𝐺
When 𝜌1 and 𝜌2 take unitary matrices, then the above equality changes to
∑ 𝜌1 (𝑥)∗𝑖𝑗 𝜌2 (𝑥)𝑖 ′ 𝑗′ = 0.
𝑥∈𝐺
Proof. See [Ser77]
2. Character theory
For a representation 𝜌, its character is simply a function 𝜆: 𝐺 → ℂ defined by 𝜆(𝑥) =
𝑡𝑟(𝜌(𝑥)), where 𝑡𝑟 is the trace---sum of eigenvalues. Note that 𝜆(1) = 𝑑𝜌 . A character is
irreducible if it is the character of an irreducible representation.
The following theorem is a simple corollary of the orthogonality of irreducible
representations.
Theorem. Different irreducible characters 𝜆 and 𝜆′ are orthogonal: ∑𝑥∈𝐺 𝜆(𝑥)∗ 𝜆′(𝑥) = 0.
Two elements 𝑥 and 𝑦 are called conjugate if ∃𝑔 ∈ 𝐺 s.t. 𝑔𝑥𝑔−1 = 𝑦. It’s an equivalent
relation and thus partitions 𝐺 into conjugacy classes. Class functions are constant for members
of the same conjugacy class. It is not hard to see that characters are class functions.
For two complex-valued functions 𝜙 and 𝜓 on 𝐺, define two inner products
(𝜙, 𝜓) =
1
∑
𝜙(𝑥)𝜓(𝑥 −1 )
|𝐺| 𝑥∈𝐺
1
and 〈𝜙, 𝜓〉 = |𝐺| ∑𝑥∈𝐺 𝜙(𝑥)𝜓(𝑥)∗
where the superscript * is the complex conjugate. These two inner products are the same if
one of the functions is a character, due to the following fact.
Fact. For a character 𝜒 of 𝐺, 𝜒(𝑥 −1 ) = 𝜒(𝑥)∗.
∗
Proof. 𝜒(𝑥 −1 ) = tr(𝜌(𝑥 −1 )) = ∑𝑖 𝜆𝑖 (𝜌(𝑥 −1 )) = ∑𝑖 𝜆𝑖 (𝜌(𝑥)−1 ) = ∑𝑖 𝜆𝑖 (𝜌(𝑥)) = 𝜒(𝑥)∗ ,
where 𝜆𝑖 (𝑀)’s are the eigenvalues of 𝑀.
3. Regular representation and Fourier transform
Recall the left regular representation 𝐿(𝑥)|𝑦〉 = |𝑥𝑦〉 and the right regular representation
𝑅(𝑥)|𝑦〉 = |𝑦𝑥 −1 〉. The following equalities hold.
𝐿 ∼ ⨁𝜌∈𝐺̂(𝜌 ⊗ 1𝑑𝜌 ) and 𝑅 ∼ ⨁𝜌∈𝐺̂(1𝑑𝜌 ⊗ 𝜌∗ ).
(1)
In fact, these hold for the same isomorphism, which is the Fourier transform over 𝐺.
Formally, define the Fourier transform by the following.
|𝑥〉 ↦ |𝑥̂〉 ≝
1
√|𝐺|
∑𝜌∈𝐺̂ 𝑑𝜌 |𝜌, 𝜌(𝑥)〉, where |𝜌(𝑥)〉 =
1
√𝑑𝜌
𝑑
𝜌
∑𝑗,𝑘=1
𝜌(𝑥)𝑗,𝑘 |𝑗, 𝑘〉.
In other words, the Fourier transform is
𝐹𝐺 = ∑|𝑥̂〉〈𝑥| .
𝑥∈𝐺
Note that 𝐹𝐺 depends on the choice of basis for each irrep of dimension greater than 1.
Theorem. The Fourier transform 𝐹𝐺 simultaneously decomposes the left and right regular
representations 𝐿 and 𝑅 into their irreducible components.
Proof. We verify for left regular representation, and the case for 𝑅 is similar.
The identity Eq.(1) also implies the following basic fact.
Fact. |𝐺| = ∑𝜌∈𝐺̂ 𝑑𝜌2 .
4. Abelian groups
𝑥𝑦
For a cyclic group ℤ𝑛 , the irreps are 𝜌𝑦 : |𝑥〉 ↦ 𝜔𝑛
𝜌𝑘 : |𝑥〉 ↦
1
∑ 𝜔 𝑥𝑦 |𝑦〉.
√𝑛 𝑦 𝑛
for 𝑦 ∈ [𝑛]. The Fourier transform is
For a finite Abelian group, suppose it is isomorphic to ℤ𝑛1 × … × ℤ𝑛𝑡 , then
𝑥 𝑦1
the irreps are 𝜌𝑦1 …𝑦𝑡 : |𝑥1 … 𝑥𝑡 〉 ↦ 𝜔𝑛11
𝑥 𝑦𝑡
… 𝜔𝑛𝑡𝑡
𝜌𝑦1 …𝑦𝑡 : |𝑥1 … 𝑥𝑡 〉 ↦
1
where 𝑥𝑖 , 𝑦𝑖 ∈ [𝑛𝑖 ], and the Fourier transform is
𝑥 𝑦1
∑ 𝜔𝑛11
√𝑛 𝑦
𝑥 𝑦
… 𝜔𝑛𝑡𝑡 𝑡 |𝑦1 … 𝑦𝑡 〉.
1 …𝑦𝑡
For Abelian groups, all irreps are one-dimensional. The converse is also true: Any non-Abelian group
has an irrep with degree strictly larger than 1.
One can also note that Abelian groups 𝐺 have the property that 𝐺 ≅ 𝐺̂ , which non-Abelian groups
do not enjoy.
Note
A good (and concise) reference for group representation is [Ser77]. [CvD10] also has an
appendix for some basic knowledge about group representation, though it takes a matrix
(instead of operator) view. For general introduction of algebra, see [Art10]. For more
extensive introduction of abstract algebra, a standard textbook is [DF03].
Reference
[Art10] Michael Artin, Algebra, 2nd edition, Pearson, 2010.
[CvD10] Andrew M. Childs and Wim van Dam, Quantum algorithms for algebraic problems, Reviews
of Modern Physics, Volume 82, January–March 2010.
[DF03] David S. Dummit and Richard M. Foote, Abstract Algebra, Wiley, 2003.
[Ser77] Jean-Pierre Serre, Linear representations of finite groups, Springer-Verlag, 1977.
Exercise
1. Prove that characters are class functions, namely 𝜒(𝑥𝑦𝑥 −1 ) = 𝜒(𝑦). Also prove another property:
𝜒(𝑥 −1 ) = 𝜒(𝑥)∗ .
2. Check that the Fourier transform defined is unitary.