Hidden Subgroup Problem
Representations of groups
Quantum poker
You cannot win in a quantum casino
joint work with Cristopher Moore and Alexander Russell
Piotr ‘niady
Polish Academy of Sciences and University of Wrocªaw
Bibliography
Hidden Subgroup Problem
Representations of groups
Quantum poker
Bibliography
Hidden Subgroup Problem
G is a known group (usually nite);
H ⊂ G is a hidden subgroup;
f : G → S is the oracle,
we are promised that f (a) = f (b) ⇐⇒ Ha = Hb ⇐⇒ ab
Problem: H =?
Examples
G = Z,
Simon's algorithm: G = (Z )n ,
Kuperberg's algorithm: G = Dn the dihedral group,
Graph Isomorphism Problem: G = (Sn × Sn ) o Z ,
Schor's algorithm:
2
2
−1
∈
H;
Hidden Subgroup Problem
Representations of groups
Quantum poker
Bibliography
Hidden Subgroup Problem, simplied
assume
N ⊂ G is a subgroup, [G : N ] = 2,
we are promised that either
H = {e } is trivial, or
H = {e , s } with s ∈/ N
Examples
Kuperberg's algorithm:
is the dihedral group,
G
Graph Isomorphism Problem:
is the automorphism group
of two copies of the complete
graph
G
Hidden Subgroup Problem
Representations of groups
Quantum poker
Representations
Bibliography
V is a (nite dimensional) vector space,
φ : G → End V is a group homomorphism:
φ(ab) = φ(a)φ(b)
for any a, b ∈ G ;
we say that V is a representation of a group G
representation V is irreducible if it cannot be written as a sum of
smaller representations: V 6= V ⊕ V .
1
2
Example 1: discrete Fourier transform
G
V
n
= Zn = {0, 1, . . . , − 1},
irreducible representations are indexed by
k = C is one-dimensional,
l e
φk (a + b) = φ(a)φ(b)
φk ( ) =
k ∈ Zn,
2π ikl
n
ab
for any , ∈ Zn .
Hidden Subgroup Problem
Representations of groups
Representations
Quantum poker
Bibliography
V is a (nite dimensional) vector space,
φ : G → End V is a group homomorphism:
φ(ab) = φ(a)φ(b)
for any a, b ∈ G ;
we say that V is a representation of a group G
representation V is irreducible if it cannot be written as a sum of
smaller representations: V 6= V ⊕ V .
1
Example 2
irreducible representation of the
dihedral group as symmetries of a
polygon on the plane
2
Hidden Subgroup Problem
Representations of groups
Quantum poker
Representations
Bibliography
V is a (nite dimensional) vector space,
φ : G → End V is a group homomorphism:
φ(ab) = φ(a)φ(b)
for any a, b ∈ G ;
we say that V is a representation of a group G
representation V is irreducible if it cannot be written as a sum of
smaller representations: V 6= V ⊕ V .
1
2
Example 3: the left-regular representation
G can be represented on the vector space
V = H = ` (G )
orthonormal basis: |aia G
φ(g )|ai = |gai
the group
2
∈
Hidden Subgroup Problem
Representations of groups
Quantum poker
Bibliography
Characters
if
V is an irreducible representation of G , we consider its character
χV (g ) = Tr φ(g )
N G
G N
if ⊂ is a xed subgroup and [ : ] = 2,
we say that an irreducible representation of
g
χV ( ) = 0
V G is unlucky if
for all g ∈
/N
Hidden Subgroup Problem
Representations of groups
Quantum poker
Notations
V is a representation of G
and H ⊂ G is a subgroup
if
we denote by
V H = {v ∈ V : φ(h)v = v for all h ∈ H }
the set of
if
H -invariant vectors
W ⊆ H is a vector space, we denote by
W (projection on W )
the uniform mixed state on W
ρW =
1
dim
Bibliography
Hidden Subgroup Problem
Representations of groups
Quantum poker
Coset state
Notations
G
a a G
a
we will use Hilbert space H = `2 ( ) with basis | P
i, ∈ ;
for ⊂ we denote the pure state | i = √1
a ∈X | i
|X |
X G
1
2
X
G
a
P
start with pure state | i = √1
a ∈ G | i,
|G |
compute the oracle function : →
f G S
1 X
p
|a, f (a)i ∈ ` (G ) ⊗ ` (S )
|G |
2
2
a∈G
3
measure the second coordinate, forget the output
HaihHa| = ρ
1 X
ρH \G =
|
| |
a ∈G
G
the coset state
`2 (G )H
Bibliography
Hidden Subgroup Problem
Representations of groups
Quantum poker
Burnside decomposition 1
G
M
`2 ( ) =
V
- irreducible
representation of
V ⊗V
G
∗
=
M
V
V ⊕ ··· ⊕ V
this decomposition can be used for a quantum measurement;
M
ρG \H = ρ`2 (G )H =
ρV H ⊗ ρV ∗
V
the measurement gives us:
the name of the irreducible representation
let us hope it is lucky!
V,
state ρV H (useful, but encrypted quantum information),
state ρV ∗ (useless!)
Bibliography
Hidden Subgroup Problem
Representations of groups
Quantum poker
Burnside decomposition 2
ρG \H = ρ`2 (G )H =
M
V
ρV H ⊗ ρV ∗
Probability distribution:
H = {e }
P (V ) = (dim|GV| ) , Plancherel measure
if H = {e , s }
(dim V ) dim V + χV (s )
P (V ) =
,
|G |
if
2
V
if is unlucky, there is no dierence
if we are unlucky we should do something more. . .
Bibliography
Hidden Subgroup Problem
Representations of groups
Quantum poker
Bibliography
Quantum poker
V , ρV H ), (V , ρV H ),. . .
unlucky? we can iterate this procedure: (
tensor product of representations:
M
1 ⊗ 2 =
V
V V
1
1
2
V ⊕ ··· ⊕ V
state:
ρV H ⊗ ρV H = ρV H ⊗V H =
1
2
1
2
M
V
ρ(some subspace of V H )
measurement gives us:
the name of the irreducible representation
let us hope it is lucky!
state ρ(some subspace of V H ) ,
and if we are unlucky. . .
V,
2
Hidden Subgroup Problem
Representations of groups
Quantum poker
Bibliography
Quantum poker, simplied
whenever you like, you can ask the dealer for a random
irreducible representation of ,
whenever you like, you can also. . .
V G
1
take two irreducible representations
V1 , V2 from your deck of
cards, give them to the dealer,
2
the dealer gives you back the random irreducible representation
from the tensor product
V1 ⊗ V2 ,
if you have a lucky irreducible representation, you WON!;
otherwise you have to keep playing
Hidden Subgroup Problem
Representations of groups
Quantum poker
Bibliography
Quantum poker
Theorem (Kuperberg)
if G is the dihedral group, there is an explicit strategy to win in a
quantum poker quickly with high probability
Theorem (Moore, Russell, ‘niady)
if G is the automorphism group of two copies of the complete
graph, there is no strategy to win in a quantum poker quickly with
high probability
proof: very strong estimates on the characters of the symmetric
groups
Hidden Subgroup Problem
Representations of groups
Quantum poker
Bibliography
Bibliography
Cristopher Moore, Alexander Russell, and Piotr ‘niady.
On the impossibility of a quantum sieve algorithm for graph
isomorphism.
In
, pages 536545. ACM,
New York, 2007.
STOC'07Proceedings of the 39th Annual ACM
Symposium on Theory of Computing
Cristopher Moore, Alexander Russell, and Piotr ‘niady.
On the impossibility of a quantum sieve algorithm for graph
isomorphism.
, 39(6):23772396, 2010.
SIAM J. Comput.
Valentin Féray and Piotr ‘niady.
Asymptotics of characters of symmetric groups related to
Stanley character formula.
, 173(2):887906, 2011.
Ann. of Math. (2)