1 Introduction

Adiabatic Quantum Computation

by

Friederike Anna Dziemba (2728680)

A thesis submitted in partial fulfillment
of the requirements for the degree of
Master of Science in Physics

at

The Leibniz University of Hanover
Institute of Theoretical Physics
Quantum Information Group

September 25, 2014

Advisor: Prof. Dr. Tobias J. Osborne
Co-Advisor: Prof. Dr. Reinhard F. Werner

I affirm that I have written the thesis myself and have not used any sources and aids other than those indicated.

Date / Signature:  

Abstract

The quantum adiabatic theorem ensures that a slowly changing system, initially prepared in its ground-state, will evolve to its final ground-state with arbitrary precision. This fact can be exploited for a computational model with the ground-state carrying the computation information. The necessary evolution time of the adiabatic quantum computation increases with the inverse energy gap of the Hamiltonian. Currently a construction by Kitaev is the standard Hamiltonian used for simulation of an arbitrary quantum circuit via adiabatic quantum computation. The energy gap in this construction is mainly determined by the spectral gap of an underlying path graph, with the length of the simulated circuit. In this thesis, we will broaden the concept of Kitaev to a class of “standard graph Hamiltonians” which allows us to substitute the path graph in the Kitaev Hamiltonian with different graph families possessing an improved spectral gap.

However, it turns out that restrictions on the graph families will make an improvement over the Kitaev construction difficult. On the one hand, we present some Hamiltonians based on particular graphs that show the same efficiency performance as the Kitaev Hamiltonian. On the other hand, we prove some restrictions on graphs in order to use them for an efficient adiabatic quantum computation by standard graph Hamiltonians. We will show that graphs with spectral gap , , cannot be used for an efficient adiabatic quantum computation at all and that graphs with constant degree ratio and spectral gap , need, as a minimal requirement, that their vertex set grows faster than polynomial in the circuit length.

Moreover we will prove in this thesis a new quantum adiabatic theorem for projection operators that expands the statement of the original adiabatic theorem to Hamiltonians with a degenerate ground-state.

\setheadsepline

0.4pt \automark[chapter]chapter

Chapter 1 Introduction

This thesis presents a Hamiltonian model for adiabatic quantum computation with particular regard to its efficiency. Adiabatic quantum computation is based on the quantum adiabatic theorem that ensures that a slowly changing system initially prepared in its ground-state will evolve to its final ground-state with arbitrary precision. This fact motivates a computational model that encodes the computation input into the initial ground-state and ensures that the final ground-state encodes the desired computation output. The actual computation will then be achieved by time evolution. The neccessary evolution time depends, in addition to the error treshhold, on the energy gap of the Hamiltonian and the norm of its time derivatives. Since we will be interested in a particular efficient computation we will motivate a list of efficiency requirements for Hamiltonians in which the energy gap and the derivatives are the central optimization quantities.

The concept of quantum circuits will serve as a formalization of a computational task. A Hamiltonian that is capable of simulating quantum circuits via an efficient adiabatic quantum computation was introduced by Kitaev [2]. Indeed the so-called Kitaev Hamiltonian is based on the normalized Laplacian of a graph whose spectral gap mainly domiates the actual energy gap of the Hamiltonian. We take advantage of the widely explored field of spectral graph theory to extend the idea of the Kitaev Hamiltonian to derive the more general concept of a “standard graph Hamiltonian”. This Hamiltonian admits different underlying graph families and therefore the optimization of the energy gap turns into the task of finding an appropriate graph family.

However the optimization is more difficult than just choosing a graph family with a large gap, since the construction of a standard graph Hamiltonian and the efficiency requirements restrict the set of possible graphs we can use. In particular we will see that the need for a neccessary minimum diameter combined with the hope for a large spectral gap implies a problematic vertex expansion of the graph. We will actually not succeed in finding a construction that outperforms the Kitaev Hamiltonian, but we will present on the one hand some new constructions with the same efficiency and on the other hand prove some negative results, for example that standard graph Hamiltonians based on expander graphs will never be capable of an efficient adiabatic quantum computation.

Chapters 2–4 will each give an introduction to an important field: to spectral graph theory, quantum circuits and adiabatic quantum computation. Chapter 4 will also comprise a list of requirements for an efficient adiabatic quantum computation. In Chapter 5 we apply the combined knowledge of the basic chapters to define what we consider a “standard graph Hamiltonian” and specify the efficiency requirements for this particular kind of Hamiltonian. Section 5.3 will show that the important time derivatives of a standard graph Hamiltonian are always constant while Section 5.4 will contain some important restriction results. In the last three chapters we present, along with the Kitaev construction, some examples of standard graph Hamiltonians that allow an adiabatic quantum computation of comparable efficiency.

Chapter 2 Graphs and Parallel Transport Networks

2.1 Basic definitions of graph theory

In this first chapter we will lay out some basics of spectral graph theory. The following definitions and lemmata can be found in most standard books on spectral graph theory. As a reference see e.g. [6, chapter 1].

Definition 1.

A graph consists of a finite, nonempty vertex set and a set of edges such that .

Definition 2.

A weighted graph is a graph with a weight function with and for all .

Actually the edge set could be omitted in the definition of a weighted graph, because it is fully implied by the weight function. But for convenience it is advantageous to have a definition of an edge set.

Graphs are visualized by drawing for each edge a line between two nodes representing the vertices and labelled respectively. For weighted graphs we add the edge weights as labels to the lines.

Example 3.

The weighted graph with

and all not defined function values of equaling , is graphically represented as

Unless otherwise stated, throughout the thesis the term “graph” refers to that of Definition 1, in some context we may say “unweighted graph” to point out that we do not talk about weighted graphs.

Most books first derive a lot of useful properties for unweighted graphs and then later generalize the proofs for weighted graphs. To avoid redundancy we will give definitions and results directly for weighted graphs in this section. Whatever property applies to weighted graphs applies of course also for an unweighted graph as the latter one can be regarded as a special case of a weighted graph with weight function

Therefore keep in mind that the fundamental graph vocabulary given by the next definition also applies for unweighted graphs accordingly:

Definition 4.

Let be a weighted graph.

  • Two vertices are called adjacent or neighbored iff .

  • A path is a sequence of two or more vertices such that . The vertices and are called connected via the path . The length of the path is defined as .

  • A connected component of a graph is a subgraph such that there is a path between any two distinct vertices and for all vertices and all vertices it holds . A graph is called connected iff it consists of just one connected component.

  • The distance of two vertices in a connected graph is defined as if the vertices are the same or otherwise as the minimum length of all paths connecting these two vertices.

  • The distance between vertex sets is defined as .

  • For a vertex set we define .

  • The diameter of a connected graph , denoted , is the largest distance between any two vertices of the graph.

  • The degree of a vertex is defined as . Moreover let and .

  • The degree matrix is the diagonal matrix with the -th entry having value . Its “inverse” is the diagonal matrix with the -th entry having value if and otherwise.

  • A graph is called -regular, iff all vertices have the same degree .

  • The volume of a vertex set is defined as .

  • is called the edge boundary of a vertex set .

  • is called the vertex boundary of a vertex set .

Note that, with the exception of the definition of “degree”, the above definitions do not even make use of the weight function. The degree of a vertex in the case of an unweighted graph reduces to the number of adjacent vertices. Similarily, in the next definition the adjacency matrix elements of an unweighted graph turn out to be simply or depending on whether the respective vertices are connected via an edge or not:

Definition 5.

The adjacency matrix of a weighted graph is a linear operator on a -dimensional vector space, called graph space, and defined as

with an orthonormal basis of the space, called the vertex basis.

We will use the conventional short-hand

for arbitrary operators and arbitrary vectors for all .

Definition 6.

The standard Laplacian of a weighted graph is defined as degree matrix minus adjacency matrix: . The normalized Laplcian is defined as .

The elements of the normalized Laplacian can be calculated as follows

Since and are symmetric, their eigevalues are real. is called normalized Laplacian, because its eigenvalues are bounded by a constant as proven by the next theorem. But first we give a helpful expression for the two Laplacians in terms of the so-called Rayleigh quotient:

Definition 7.

The Rayleigh quotient of a matrix regarding a vector , , is the expectation value of this matrix according to this vector and is given as follows:

Lemma 8.

Let be the standard Laplacian of a weighted graph and an arbitrary vector. Then it holds:

where we sum over all unordered pairs of adjacent vertices.

Proof.
Corollary 9.

The Rayleigh quotient with being the normalized Laplacian of a connected weighted graph with more than one vertex fulfills the identity

with .

Proof.

As is a connected weighted graph with more than one vertex, for all and the degree matrix is invertible.

Theorem 10.
    [label=()]
  1. The eigenvalues of the standard Laplacian of a connected weighted graph fulfill the relation and the nondegenerate null-space is spanned by the equal distribution vector .

  2. The eigenvalues of the normalized Laplacian of a connected weighted graph fulfill the relation and the nondegenerate null-space is spanned by .

  3. The eigenvectors and eigenvalues of a graph are given by the eigenvectors and -values of its connected components.

Proof.
    [label=()]
  1. If is a graph with just one vertex, the standard Laplacian is the zero matrix and the statement obviously true. So let us assume from now on that has more than one vertex.

    From Lemma 8 we know

    from which we can conclude that all eigenvalues of are non-negative. An eigenvector with the eigenvalue requires each summand on the right side to be zero. As is connected, this is only fulfilled for a multiple of the equal distribution vector . Since for all

    is indeed the nondegenerated eigenvector corresponding to the eigenvalue of .

    Now we derive the upper bound on the eigenvalues. Assume that is a normalized eigenvector of corresponding to the eigenvalue . It holds that

    Since is normalized it follows directly that .

  2. If is a graph with just one vertex, the normalized Laplacian is the zero matrix and the statement obviously true. So let us assume that has more than one vertex. As is connected, for every vertex and the degree matrix is invertible.

    Since the null-space of is spanned by according to (i), the null-space of is spanned by the .

    For the upper bound let be an eigenvector of with the eigenvalue . By analogy to the calculation in (i) we can show:

  3. The statement follows directly from the fact that (or or ) is blockdiagonal with one block for each connected component.∎

Corollary 11.

The second smallest eigenvalue of the normalized Laplacian of a connected weighted graph is its lowest nonzero eigenvalue and is given by

In this thesis we are mostly interested in the eigenvalues and eigenvectors of the normalized Laplacian of a (weighted) graph. That’s why we also call them just the eigenvalues and eigenvectors of the graph. But as it is sometimes easier to derive expressions for the spectra of the standard Laplacian or the adjacency matrix of a graph, it is worthwhile to know the relationships between the different spectra:

Definition 12.

For a weighted graph we denote the increasing eigenvalues of the normalized Laplacian by , the increasing eigenvalues of the standard Laplacian by and the decreasing eigenvalues of the adjacency matrix by .

Proposition 13.

For a -regular weighted graph it holds:

Proof.

Since every vertex has degree the degree matrix is simply and according to the definitions of the different matrices

If a graph is not regular, the relationship between the spectra of normalized Laplacian, standard Laplacian and adjacency matrix are non-trivial. But one can derive the following theorem similar to [15] based on the Courant-Fischer theorem:

Theorem 14.

For a connected weighted graph with more than one vertex the following holds:

    [label=()]
  1. .

Proof.
    [label=()]
  1. As is connected and has more than one vertex, for all and is invertible. It holds that

    for some with .

    We now use the characterization of the -th eigenvalue via the Courant-Fischer theorem (min-max theorem):

    Define and for . Then it holds

    and

    since is invertible. Thus we can rewrite

    Therefore

  2. The inequality for the eigenvalues of the adjacency matrix follows analogously:

    for some with .

    and hence

2.2 The spectral gap and expander graphs

In this thesis the seond lowest eigenvalue of the normalized Laplacian of a (weighted) graph is of particular interest, justifying its own name:

Definition 15.

Let be the normalized Laplacian of a (weighted) graph . Its second smallest eigenvalue is called the spectral gap of the graph .

From now on we drop the notation and simply write or for the spectral gap of a graph . Notice that in literature sometimes the spectral gap is defined by , which equals the second largest absolute eigenvalue of the “normalized adjacency matrix” .

We are particulary interested in the shrinking behaviour of the spectral gap while looking at a whole set or family of (weighted) graphs with strictly increasing vertex sets. If the spectral gap of an unweighted graph family is lower bounded by a constant, we have a special name for this family:

Definition 16.

An -expander is an element of a infinite graph family with for all graphs .

The name “expander” graph comes from the fact that a spectral gap lower-bounded by a constant is equivalent to a constantly lower bounded edge expansion factor and vertex expansion factor , which are defined in the following way:

Definition 17.

For a nonempty set of vertices of a connected graph with more than one vertex define

The edge expansion factor or Cheeger constant of the graph is defined as .

Definition 18.

For a nonempty set of vertices of a connected graph with more than one vertex define

The vertex expansion factor of the graph is defined as .

The proof of the next theorem is based on [6, chapter 2].

Theorem 19.

For any connected graph with more than one vertex the following holds:

Proof.

The first inequality sign is easy to verify since every nonempty subset fulfills

and hence .

For the second inequality sign let’s take a look at the vector with

with the vertex set that achieves the Cheeger constant: .

is orthogonal to , thus according to Lemma 11

We have shown that eigenvalue expansion () implies edge and vertex expansion (). As mentioned above the opposite also holds, but since for our purposes only the first direction is interesting, we omit to proof the other direction and refer the interested reader to [6, chapter 2].

The next technical lemma is a consequence of vertex expansion and will help us later to derive some important restrictive results for the graph families in our constructions:

Lemma 20.

Let be a graph family depending on the parameter . Let be a positive constant and such that . Then it holds

Proof.

From Theorem 19 about vertex expansion we know that for any , the following inequality holds:

and hence

With this equality we can derive successively

We can conclude from the above lemma that the diameter of an expander graph is always upper bounded by a logarithmic function in the size of the vertex set (consider in the above lemma as diameter). In addition to this result we will also use later the following diameter bound by Alon and Milman [3] since it offers an even stricter bound for graphs with constant degree ratio:

Theorem 21.

Let be a connected graph with and be the second smallest eigenvalue of the standard Laplacian. Then

Knowing the relationship between the spectra of standard and normalized Laplacian from Theorem 14 we can conclude directly

Corollary 22.

Let be a connected graph with and its spectral gap. Then

2.3 Cayley graphs

In this section we introduce Cayley graphs as a special kind of unweighted graphs whose eigenvectors and eigenvalues can be calculated with some background knowledge about group theory. In order to do so we first need to recall some properties about characters of a group:

Definition 23.

A character of a finite abelian group is a group homomorphism .

As every element of a finite group has finite order according to Langrange’s theorem, the image of a character are roots of unity and thus actually even comprised in .

Lemma 24.

Distinct characters of a finite abelian group are linearly independent.

Proof.

We follow the induction proof of [7]: Let be distinct characters . For the character(s) is trivially linear independant. So let and suppose are linearly independent characters. Consider the linear independence relation

(2.1)

As this equation should hold for any group element, we can equivalently write

We subtract the equation

to get the condition

As are linear independent according to the induction assumption, has to vanish for all and all . But as for every and are distinct characters, there exists an element , such that . Consequently for all and according to (2.1) also . This means that are linearly independent. ∎

Lemma 25.

A finite abelian group has exactly distinct characters.

Proof.

For now let be cyclic, a generator of the group and a character. Because of , the image of the generator has to be a -root of unity, of which there are exactly . As each possible image of the generator fully defines a distinct character, the group has exactly distinct characters.

Now assume that is an arbitrary finite abelian group. According to the structure theorem of finite abelian groups (see as reference theorem 14.2 in [11]), is isomorphic to a direct product of cyclic groups with :

Let be the characters of . It can easily be checked that with are the characters of . Thus has distinct characters. ∎

Now we can define Cayley graphs and use the above results to derive a useful theorem about their eigenvectors and -values:

Definition 26.

For a finite abelian group and a symmetric subset (i.e. ) the Cayley graph is definded as the graph with vertex set and edge set .

Notice that a Cayley graph is indeed a correctly defined undirected graph and that it is -regular.

The following theorem adapted from [12] gives the promised result about the eigenvectors and -values of the normalized Laplacian of a Cayley graph:

Theorem 27.

Let be a character of the finite abelian group . Then is an eigenvector of the normalized Laplacian of the Cayley graph with eigenvalue

Proof.

Let A be the adjacency matrix of the Cayley graph and a character of the group . As we understand as both, as function and vector, we identify with .

Thus is an eigenvector of the adjacency matrix corresponding to the eigenvalue . As is -regular, is also an eigenvector of the normalized Laplacian to the eigenvalue . ∎

With the knowledge from Lemmata 24 and 25 that the group has linear independent characters, we can conclude that they form an eigenbasis of the normalized Laplacian.

Interestingly the eigenvectors of a Cayley graph are fully defined by the group , whereas the specific choice of a subset only affects the eigenvalues.

We will finish this section by giving a formula for the eigenvectors and -values of a certain family of Cayley graphs, namely those with the group . We will understand the elements of the group as bit strings of length and hence the group operation, which we will denote by , is just bitwise addition or bitwise XOR.

Theorem 28.

Let be the group with bitwise addition on the set of bit strings with length and a symmetric subset. Then the eigenvectors of the normalized Laplacian of have the form

with the corresponding eigenvalues

where denotes the scalar product of two bit vectors modulo .

Proof.

For all the above defined is a character of since for all the following holds:

It is left to check that we have really defined distinct characters. Assume , hence w.l.o.g. and for some bit position . Let denote the bit string with all zeroes and and the only in position . Then

So are indeed distinct characters and hence linearly independant according to Lemma 24. With Theorem 27 it follows that is an eigenbasis of with the corresponding eigenvalues

2.4 Contractions and coverings

In this section we return to weighted graphs and introduce covering and contraction, two useful tools that allow us to transform a weighted graph into another one with a related spectrum.

Definition 29.

Let and be weighted graphs. is called a contraction of , iff there exists a surjective contraction function such that for all

If comprises more than one element, we say that the vertices of are contracted to the vertex .

Lemma 30.

Let be a contraction of via the contraction function . The degree of a vertex in be denoted by and the degree of a vertex in by . Then it holds for all :

Proof.
Theorem 31.

Let and be connected weighted graphs with more than one vertex and a contraction of . Then the spectral gaps fulfill the following inequality:

Proof.

We denote the contraction function by , the degree of a vertex in by , the degree of a vertex in by and the degree matrices by and , respectively. Let be the eigenvector of with the eigenvalue . Define by

The fact that implies directly since

Hence according to equation (11) the spectral gap of can be bounded by: