The Hidden Subgroup Problem and Other Algorithms

The hidden subgroup problem and other algorithms

In the previous lecture two examples of problem that could be efficiently solved using the quantum Fourier transform were shown. Are there any more? The answer is yes and now a class of problems that can be solved efficiently using the quantum Fourier transform and phase estimation will be described. Any problem that can reformu- lated in such a way that it involves the solution of the hidden subgroup problem can be solved efficiently. In order to understand what this problem is some fundamental concepts from group theory must be introduced.

Deﬁnition A group, G, is a set on which a binary multiplication operation is deﬁned such that

• g1 g2 = g1g2 ∈ G where g1, g2 ∈ G.

• is associative ⇔ g1 (g2 g3) = (g1 g2) g3. • There exists a unique identity element e ∈ G such that for all g ∈ G, g e = e g = g and g−1 ∈ G, g−1 g = g g−1 = e.

Some further deﬁnitions for groups:

• If the -operation also is commutative G is called an Abelian group. • The cardinality of this group, |G|, is the number of elements in G (for finite groups). • A subgroup of G is a subset, H, which in itself is a group, this usually denoted H ≤ H. • If any element in G can be written as a product between elements in a specific subset of G called H then the elements of H are called the generators of G. Some examples of groups: • Setting G = {0, 1} and defining a b = a + b mod 2 gives a group. The group is Abelian and often is replaced with ⊕. The group is finite and |G| = 2.

• Setting G = Z (set of integers) and to regular addition gives a group. The group is infinite but is finitely generated by {−1, 1} for example. Examples of subsets include the sets of all even integers (with zero) but not the set of all odd integers (with zero). The set of all odd integers (with zero) is a set of generators since adding two odd numbers gives an even number. • The set of all quantum gates, G, form a group when corre- sponds to serial application of gates. This is an infinite group that is finitely generated. Any set of generators is called a universal set of gates. The Pauli gates {I,X,Y,Z} is a subgroup of G. This group is non-Abelian.

2 The final concept of group theory needed is that of the coset. Definition The left- and right cosets of H ≤ G determined by g ∈ G is the set {gh|h ∈ H} and {hg|h ∈ H} respectively. It can be shown that cosets can define an equivalence relation, and thus a partition, on G. This means that if you take two cosets, g1H and g2H, they will either have all elements in common or no elements in common. With this new knowledge of the notation and language of groups the hidden subgroup problem can be formulated in the following manner: Definition of the hidden subgroup problem Let f : G 7→ X where G is a finitely generated group and X is a finite set such that f is constant for cosets of K ≤ G and distinct on each coset. Find a generating set of K using the unitary operation U|gi|hi = |gi|h ⊕ f(g)i where g ∈ G, h ∈ H and ⊕ an appropriate direct sum on X. Using quantum computation this is a problem that can be solved in a very similar fashion to phase estimation. The Fourier transform can be generalized to groups. To describe how G G consider a function ρ : G 7→ Mn where Mn is a set of matrices of dimension n × n that behaves like a group if denotes matrix multiplication. Such a set is usually called a matrix group. If ρ also has the property that

g1g2 = g3 ⇒ ρ(g1)ρ(g2) = ρ(g3) it is called a representation of G. Let G be an Abelian finite group of order |G| = N. Represent the element with g ∈ [0,N − 1] and represent with addition. Then −2πigh a representation ρ(g) = e N of G can be defined. This allows the 1 Fourier transform of a function f : G 7→ C, fˆ, to be defined as 1A reader who is already acquainted with group theory knows there are more general definitions of both representations and the Fourier transform, as these would raise the complexity of the discussion without clarifying it the less general version were chosen.

3 |G|−1 1 X 2πilg ˆ − |G| |f(g)i = p e |f(g)i (1) |G| l=0 and the inverse Fourier transform as

|G|−1 1 X 2πilg |G| ˆ |f(g)i = p e |f(g)i (2) |G| l=0 Here |gi represent vectors in a Hilbert space with as many basis states as there are elements in G. The operator U can be used to create the state |ψi = √1 P |gi|f(g)i |G| g∈G in the same way as it was used for phase estimation. From (2) it is clearly possible to express |f(g)i using |fˆ(g)i a basis. The function f was constant and distinct on each coset of K ≤ G meaning that

g = hk, h ∈ G, k ∈ K ⇒ |f(g)i = |chi This means that (1) can be written as:

|G|−1 |K|−1 1 X 2πilg 1 X X 2πilk ˆ − |G| − |G| |f(g)i = p e |f(g)i = p e |cki |G| l=0 |G| H∈gK k=0

|K|−1 2πilk P − |G| Note that if k = m|G|, m ∈ Z then k=0 e = |K| and else |K|−1 − 2πilk P |G| ˜ k=0 e = 0. This means that for K = {k = m|G|; k ∈ K} (2) can be written:

|G|−1 |G|−1 1 2πilg 2πilh |K˜ ||K| 2πilg X |G| X − |G| X |G| |f(g)i = e e |chi = e |chi p|G| p|G| l=0 h∈K˜ l=0

l where |G| can be found using phase estimation in the way described for Shor’s algorithm.

4 A large variety of problems has the hidden subgroup problem as a part of them, notable examples are ﬁnding the unknown period of a periodic function (the period-ﬁnding problem) or given a and b = as determining s (the discrete logarithm problem), and for many of them the quantum algorithm used to solve it is exponentially faster than the classical counterpart.

Some examples of hidden subgroup problems: • Period-finding Suppose f(x) is a periodic function such that f(x) = f(x + r) for some unknown 0 ≤ r ≤ 2L. G = Z with product +. X is any finite set. K = {0, r, 2r, . . .}, r ∈ G. f(x) = f(x + r). • Order-finding Find the smallest integer r such that xr = 1 mod N where N is an integer and 1 ≤ x ≤ N − 1. G = Z with product +. j r X = {a }, with j ∈ Zr and a = 1. K = {0, r, 2r, . . .}, r ∈ G. f(x) = ax, f(x) = f(x + r). • Discrete logarithm Given integers a and b, find s such that s a = b if it exists. G = Zr × Zr (pairs if integers smaller than or equal to r) with product + mod r. j r X = {a }, with j ∈ Zr and a = 1. K = {(l, −ls)}, l, ls ∈ Zr. sx1+x2 f(x1, x2) = a , f(x1 + l, x2 − ls) = f(x1, x2).

The hidden subgroup problem is not the only problem that quantum computation algorithms can solve faster than their classical counter- parts. One example is Grover’s search algorithm √which ﬁnds the shortest route through every node in a graph in O( N) operations where N is the amount of possible routes while the fastest classical algorithms is O(N). Another algorithm can decrease the complexity

5 of counting the number of solutions to a search problem in the same way.