Unitary Designs and their Applications in Quamtum Information Theory

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Unitary Designs and their Applications in

Quantum Information Theory

Slawi Dimitrov

June 27, 2020

Bachelor thesis Mathematics and Physics & Astronomy

Supervisors: dr. Jonas Helsen, prof. dr. Eric Opdam, dr. Michael Walter

Institute of Physics

Korteweg-de Vries Institute for Mathematics Faculty of Sciences

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Abstract

Recent years have seen the development of small scale quantum computers, but large scale implementation of quantum computing is limited by noise affecting quantum gates. In this thesis we characterize this noise by introducing the average fidelity, and we de-scribe the randomised benchmarking protocol to measure it. Uniformly random selection of unitary gates in a key element in the definition of the protocol. Second moments over the Haar measure on the unitary group are required in the protocol, and unitary designs are introduced as a solution, as they are finite sets that can reproduce the lower mo-ments. We characterize some basic mathematical properties of designs as well as prove a representation-theoretic condition for a group G ⊂ U (d) to be a design.

Title: Unitary Designs and their Applications in Quantum Information Theory Authors: Slawi Dimitrov, slawi.dimitrov@gmail.com, 11757361

Supervisors: dr. Jonas Helsen, prof. dr. Eric Opdam, dr. Michael Walter Second graders: dr. Bas Kleijn, prof. dr. C.J.M Schoutens

End date: June 27, 2020

Cover image: A graphical representation of a quantum channel. Design by A. Carignan-Dugas.

Institute of Physics University of Amsterdam

Science Park 904, 1098 XH Amsterdam http://www.iop.uva.nl

Korteweg-de Vries Institute for Mathematics University of Amsterdam

Science Park 904, 1098 XH Amsterdam http://www.kdvi.uva.nl

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1 Introduction

As classical computing has reached technological maturity and the end of Moore’s law approaches, quantum computing promises to transcend some of the limits of classical computing. However, at the moment large scale quantum computers still exist mostly in theory; current quantum computers are limited and small. The development of a real quantum computer is limited by the fragility of real quantum systems and their extreme susceptibility to outside disturbances. Any quantum system that is to be the basis of a quantum computer would be subject to this “noise”, therefore characterising this noise is essential to the construction of a physical quantum computer.

One of the ways to characterize noise in quantum computers is by measuring the “average fidelity” parameter, which can be done by implementing a protocol known as “randomised benchmarking”. The construction of this protocol requires some assump-tions however, and it has a lot of mathematical machinery behind it.

One element of the protocol is the uniformly random selection of unitary operators. Implementing this in the lab is not practical, since quantum computers have some finite set of unitary operators available for use in algorithms (the gateset). There are however specific finite sets of unitary operators that are “sufficiently random”. These sets are referred to as unitary designs, and they reproduce low moments of the unitary group. The reproduction of second moments is required for randomised benchmarking, and thus a 2-design is necessary for its implementation.

In this thesis we will prove some basic properties of designs, and prove a sufficient condition for a subgroup of the unitary group to be a design. We will then go through the assumptions behind randomised benchmarking, as well as prove any mathematics required for its validity before finally stating the protocol itself.

In chapter 2 we begin by going over the fundamental postulates of quantum mechanics. We discuss the state vector formalism and the more general density matrix formalism. We also introduce quantum channels, a general way of representing the evolution of an open quantum system. In chapter 3 we review some basic group and representation theory, including Maschke’s theorem and Schur’s lemma. In chapter 4 we introduce unitary designs, the mathematical objects on which the randomised benchmarking pro-tocol is based. We prove a representation-theoretic criterion for finding designs that are also groups. We also prove the single qubit Clifford group is such a design using said criterion.

In chapter 5 we introduce the average fidelity, the parameter measured by the ran-domised benchmarking protocol. We also prove every ”twirled” channel is a “depolarising channel”, which is required for some steps in the protocol to be justified. Then finally in chapter 6 we conclude the main text by stating the randomised benchmarking protocol and justifying it with the lemmas from chapter 5.

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We conclude in chapter 7 with an overview and an outlook on future research into unitary designs and randomised benchmarking. In this thesis we will assume an under-standing of basic linear algebra and group theory, as well as a passing familiarity with measure theory.

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2 Introductory quantum mechanics

In this chapter we will go over the fundamental postulates of quantum mechanics in order to establish the mathematical context of a quantum state and it’s evolution and measurement. We will first discuss this for the usual state vector formalisation, and then we will extend the state vector formalism into the more general density matrix formalism.

Since the foundations of quantum mechanics are well established, their introduction for the purposes of this thesis will be heavily inspired by the excellent book ”Quantum computing and Quantum information” by Nielsen and Chuang [13].

2.1 Postulates of the state vector formalism

Quantum mechanics is fundamentally a mathematical description of physical systems, and like any mathematical structure it has axioms, usually denoted as the fundamental postulates of quantum mechanics. Before we can make any meaningful statement about our physical system, be it a quantum computer or some other system, it will be necessary to specify these postulates. In this section we will discuss the fundamental postulates for the state vector formalism. We start with the first postulate:

Postulate 2.1. Any physical system can be described by a unit vector in a complex Hilbert space, the state vector of the system.

This provides us all of the machinery of linear algebra to use in our quantum descrip-tion of a physical system. Note that since our state vector is in a Hilbert space, we have an inner product and we can choose a set of orthonormal basis vectors with respect to that inner product. Thus, if we denote our state vector by |ψi and our Hilbert space by H, |ψi can be written as:

|ψi =

dim(H)

X

n=1

cn|φni, (2.1)

where {|φni : n ≤ dim(H)} is our set of orthonormal basis vectors, and cn are the

complex coefficients. Note also that the dimension of the Hilbert space need not be finite, however from now on we will assume it is and denote dim(H) = d. Often used terminology is that the state vector |ψi is a superposition of the basis vectors |φni. In

the Hilbert space we also have an inner product between two vectors |ψi, |ϕi, which we denote as hφ|ψi. Also note that our state vectors are unit vectors, and so:

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1 = hψ|ψi (2.2) = dim(H) X n=1 cnhφn| dim(H) X m=1 cn|φmi (2.3) = d X n,m=1 |cn|2hφn|φmi (2.4) 1 = d X n=1 |cn|2. (2.5)

This identity is referred to as the normalisation condition.

Now that we have a description of a state we would like to see a description of it’s evolution.

Postulate 2.2. The evolution of a closed system is described by a unitary transforma-tion.

More specifically, if at time t1 our system is in state |ψiand t2 in state |ψ0i, we

know that |ψ0i = U |ψi for some unitary operator U . This is formally a very important postulate, since it allows us to use the properties of a unitary operator when considering changing quantum systems.

Thus far however, we have only treated closed physical systems. When we consider a measurement on a quantum system, it is an interaction with another physical system (the measurement apparatus). Thus we are dealing with a regime of non-unitary evolution. For this we require another postulate:

Postulate 2.3. A quantum measurement on a physical system |ψi is given by a set of measurement operators {Mm: m ∈ I} indexed by some index set I of possible outcomes

of the measurement. For a measurement on the state |ψi, the probability of outcome m is given by:

p(m) = hψ|M_m†Mm|ψi. (2.6)

The state after the measurement is:

Mm|ψi

q

hψ|Mm†Mm|ψi

, (2.7)

and the measurement operator set satisfies:

X

m∈I

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Sometimes we define the operator Em = Mm†Mm. From postulate 2.3 and some basic

linear algebra we have that each Em is a positive operator, and that they sum to the

identity operator. We call the set {Em : m ∈ I} a Positive Operator Valued Measure,

or POVM for short.

While all of this may seem contrived, measurements in practice are usually projective, that is, their measurement operators are projection operators onto some basis. Let our basis be {|φni : n ≤ d}, then any physical state looks like:

|ψi =

d

X

n=1

cn|φni. (2.9)

The projective measurement operators for our chosen basis look like {|φnihφn|}. This set

of operators trivially satisfies the completeness relation, and the probability to measure any basis state |φni is:

p(n) = hψ|φnihφn|φnihφn|ψi = |hψ|φni|2= |cn|2. (2.10)

We now have almost all of our machinery to describe a quantum physical system. The final postulate considers two isolated physical system and their description as one big physical system, known as the composite physical system.

Postulate 2.4. The state space of a composite system is the tensor product of the state spaces of its component systems. A composite physical system |ψi with component systems {|ψii, 1 ≤ n} is described by the product state

|ψi = |ψ₁i ⊗ ... ⊗ |ψ_ni. (2.11) This is the usual formulation of quantum mechanics, but in this thesis we will use a more general formulation, where the fundamental object is the density operator, also known as the density matrix.

2.2 The density matrix formalism

In this section we define the density matrix formalism of quantum mechanics. Instead of a state vector |ψi in some Hilbert space H describing our physical system, our fundamental object is an operator ρ that exists in L(H), the space of linear maps on H.

We also demand that the operator ρ is positive and has the property trρ = 1 if we consider the operator in matrix form. We call this operator the density matrix.

This density matrix ρ will be the fundamental object describing our physical system. If ρ is of the form ρ = |ψihψ| for some |ψi, we say that our system is in a pure state. Otherwise our system is in a mixed state.

Note that since trρ = 1 and ρ is positive, by the spectral theorem we know that there is always some basis {|φni : n ≤ d} such that we can write ρ as:

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ρ =

d

X

n=1

pn|φnihφn|, (2.12)

where pn are the eigenvalues of ρ andP_npn= 1.

One example of a density matrix is what is referred to as the maximally mixed state. We choose a basis {|φnihφn| : 1 ≤ n ≤ d} for our Hilbert space H, the maximally

mixed state is defined as:

ρ = d X i=1 1 d|φiihφi| = I d. (2.13) We can easily see that trρ = d

d = 1, and that ρ has only positive eigenvalues. Hence

it satisfies all of the requirement of a density matrix. We now move on to the evolution of these density matrices. The unitary evolution of density matrices is easy to describe, since |ψi 7→ U |ψi, we have:

ρ =X i pi|ψiihψi| 7→ X i piU |ψiihψi|U†= U ρU†. (2.14)

The measurement postulate can also easily be adapted for the density operators. Take a POVM {Em}. We use the notation p(m|i) to mean the probability of outcome m given

initial state i. Then we have:

If the initial state of our system is |ψii, and we denote the state of the system if we

measure m by |ψ_imi, we can use postulate 2.3 to write this in terms of the initial state and the measurement operators. We use this for the case of the density operator, where we call the final density matrix after measurement (by Em) ρm:

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The only thing that remains now is the description of composite systems in terms of density matrices.

We can use postulate 2.4 and the multi-linearity of the tensor product to see that the density matrix of a compound system is the tensor product of the density matrices of its component systems. If we have a system composed of two closed subsystems; A and B:

ρAB = ρA⊗ ρB. (2.24)

Note however that not every possible state of the compound system AB is described by a density matrix of the form ρA⊗ ρB, and this only holds if A and B are not interacting.

We can however recover the density matrix ρA describing the state of A from the

density matrix of any compound system AB, where B can be any system, by taking a partial trace over the state space of B:

ρA= trB[ρAB]. (2.25)

We are now fully equipped to apply this language of quantum mechanics.

2.3 Quantum channels

A quantum channel is, in essence, a way to characterise the evolution of an open quantum system. We consider our principal system ρ and our “environment” ρenv. Here the

environment represents anything that might interact with the principal system so that the total principal and environment system is closed.

We consider these systems as isolated at first, but we know that the composite system will evolve through some unitary operator that acts on the compound system. Thus the principle system’s density matrix changes according to some map from the space of density matrices to itself, and the evolution is not necessarily unitary. We now define such a map as follows [13]:

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Definition 2.1. Λ is a Quantum Channel if and only if:

Λ : L(H) → L(H), (2.26) and for any positive ρ ∈ L(H) with trρ = 1, Λ(ρ) is positive and has trΛ(ρ) = 1.

While definition 1 is an elegant axiomatic definition, there is a different way to describe a quantum channel called the operator-sum representation that is sometimes easier to work with.

As described in section 8.2 of ”Quantum computing and Quantum information” by Nielsen and Chuang [13], we can represent any quantum channel Λ by a set of operators {E_k} such that:

Λ(ρ) =X

k

EkρE_k†, (2.27)

where the operators Ek act on the Hilbert space of the quantum system. Note also

that, since Λ(ρ) is also a density matrix:

1 = trΛ(ρ) (2.28) = trX k EkρE_k† (2.29) =X k trEkρE_k† (2.30) =X k trEkE_k†ρ. (2.31)

Since this has to be true for all ρ, we derive the completeness relation:

X

k

EkE_k† = I. (2.32)

Note that these operators are not unique.

One example of a quantum channel is the unitary channel. For any unitary operator U ∈ U (d)), we can define a map U ∈ L(L(H) such that:

U (ρ) = U ρU†. (2.33) Since U ρU† is clearly a density matrix, and thus positive and with trU(ρ) = 1, our map U is a quantum channel. One other very important example of a quantum channel is a ”depolarising channel”. It is defined like so [13]:

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Definition 2.2. A depolarising channel is a quantum channel of the form:

P(ρ) := pI

d+ (1 − p)ρ, (2.34) where d is the dimension of the state space associated with ρ

In essence, this quantum channel “scrambles” initial quantum state with probability p, that is, there is a probability p for any initial state to change into the maximally mixed state. Note that the above definition is not in the operator sum representation.

Now that we have established the mathematical context of quantum mechanics, we can explore the mathematics involved in it. To do this however a quick overview of some group and representation theory is required, and this is what we will do in the next chapter.

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3 Introductory Representation theory

As seen in the previous chapter, unitary operators are very important in quantum me-chanics. Fortunately the set of all unitary operators on a Hilbert space form a group! and whenever a unitary operator is applied onto some state vector or density matrix, the operator is actually a representation of an element in the unitary group. In this chapter we will lay out the necessary foundations in representation theory in order to properly apply it to quantum mechanics. This introductory section will be based on the book “representation theory of finite groups” by Steinberg [14].

3.1 Groups

In this section and the next we will briefly review some definitions and theorems from group theory and representation theory without necessarily proving them, the first one being the definition of a group.

Definition 3.1. A group is a pair (G, ·) where G is a set and · is an operation G × G → G that abides by the following properties:

(G1) The operation is associative, that is for all g1, g2, g3 ∈ G

g1· (g2· g3) = (g1· g2) · g3. (3.1)

(G2) There exists an identity element, that is for some e ∈ G, e · g = g · e = g for all g in G.

(G3) every element has an inverse, that is for all g ∈ G, there exists some g−1∈ G such that

g · g−1 = g−1· g = e. (3.2)

We can also find groups within groups, and we call these groups subgroups: Definition 3.2. A subgroup H ⊂ G is a subset of a group G such that:

• H contains the identity element, e ∈ H.

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Note that for example all invertible linear transformations on a Hilbert space H with dimension d form a group with the composition operation, and the unitary maps form a subgroup of that group. We denote the group of linear maps on our Hilbert space H as GL(H) and the subgroup of unitary maps as U (d).

We can also define maps between groups that preserve group structure, we call these maps homomorphisms:

Definition 3.3. For groups G, H, a homomorphism from G to H is a map f : G → H such that:

• f (e) = e,

• for all a, b ∈ G, f (ab) = f (a)f (b).

If the homomorphism is bijective we call it an isomorphism. We can now define a representation of a group, which is essentially a group homomorphism between a group and linear maps on a Hilbert space. We shall restrict ourselves to complex, finite dimensional Hilbert spaces.

3.2 Representations

The formal definition of a representation is:

Definition 3.4. A representation of a group G is a group homomorphism ϕ : G → GL(V ) for some vector space V . The dimension of V is called the degree of ϕ.

We can also define a morphism between two representations:

Definition 3.5. For two representations ϕ : G → GL(V ) and ψ : G → GL(W ), a G-morphism is a map T : V → W such that ψ(g)T = T ϕ(g). We denote the space that T belongs to as HomG(ϕ, ψ).

Since we did not specify the vector space V in our definition of a representation, we can have a huge number of representations that are very similar. We can define an equivalence relation on the set of all representations of a group:

Definition 3.6. Two representations ϕ : G → GL(V ) and ψ : G → GL(W ) are called equivalent if there exists a vector space isomorphism T : V → W such that: ψ(g)T = T ϕ(g) for all g ∈ G. We then write ϕ ∼ ψ.

Another important notion is the notion of a G-invariant subspace, we will need it later for Schur’s lemma.

Definition 3.7. For a representation ϕ : G → GL(V ), a G-invariant subspace is a linear subspace W ⊂ V such that ϕ(g)(W ) ⊂ W for all g ∈ G

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And since we can take direct sums of linear operators, we can take direct sums of representations as well. We define: ϕ ⊕ ψ : G → V ⊕ W such that, for v ∈ V ⊕ W :

(ϕ ⊕ ψ)(g)(v) := (ϕ(g) ⊕ ψ(g))(v), (3.3) where the second direct sum is the standard direct sum on the space of linear operators. There are a few more important representation theoretic definitions, most importantly irreducibility and decomposeability.

Definition 3.8. A representation ϕ : G → GL(V ) is irreducible if the only G-invariant subspaces of V are V and {0}.

Definition 3.9. A representation ϕ : G → GL(V ) is completely reducible if ϕ ∼ ϕ(1)_{⊕ ... ⊕ ϕ}(s)_{, where every ϕ}(i) _{is irreducible.}

With all of these definitions, one of the most important theorems in representation theory is Maschke’s theorem:

Theorem 3.1 (Maschke). Every representation of a compact group is completely re-ducible.

This is not the most general version of the theorem, however is it is general enough for the purposes of this thesis. We will extensively use this theorem in later sections. Note that finite groups are all compact, and the unitary group is compact. The formal definition of a compact group is not relevant for this thesis.

A corollary of Maschke’s theorem is the statement that every representation is equiva-lent to a direct sum of irreducible representations. We will not prove this, or the theorem itself. This means that for any representation ϕ : G → V of a compact group G, there exists an isomorphism of vector spaces T such that T ◦ ϕ(g) = Ls

i=1[ϕ(i)(g)]

⊕mi _{◦ T ,}

where every ϕ(i) is irreducible. The mi denotes how many times every ϕ(i) appears

in the direct sum (the multiplicity of ϕ(i)). However since vector spaces are unique up to dimension and vector space isomorphisms, we can let V = Cd, and we can let Ls

i=1[ϕ(i)]

⊕mi _{act on C}d _{as well without any loss of generality. Then the vector space}

isomrphism T is just a basis transformation. Thus if we do not specify a basis we can safely say that ϕ =Ls

i=1[ϕ(i)]⊕mi. This fact is used extensively in chapter 4, along with

Schur’s lemma.

Perhaps the most important representation theoretical result that we will use is Schur’s lemma. It allow us to greatly restrict the amount of G-morphisms between representa-tions.

Theorem 3.2 (Schur’s lemma). For two irreducible representations ϕ : G → GL(V ) and ψ : G → GL(W ),

(a) if ψ ϕ, then HomG(ϕ, ψ) = 0,

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We shall now move on to the tensor product of representations. The multi-linearity of the tensor product is a very useful property, and we will need it later to study the so called t-designs. Let ϕ : G → GL(V ), ψ : G → GL(W ) be representations and let g ∈ G. Then we define ϕ ⊗ ψ : G →GL(V ⊗ W ) such that for all u ∈ V ⊗ W :

(ϕ ⊗ ψ)(g)(u) := ϕ(g) ⊗ ψ(g)(u). (3.4) We can check that this is a representation by checking it is a group homomorphism:

• The identity is mapped to the identity:

(ϕ ⊗ ψ)(e)u = ϕ(e) ⊗ ψ(e)_{u = I ⊗ Iu = Iu = u.} (3.5)

• The group multiplication is preserved:

(ϕ ⊗ ψ)(gh)u = ϕ(gh) ⊗ ψ(gh)u (3.6) = ϕ(g)ϕ(h) ⊗ ψ(g)ψ(h)u (3.7) = (ϕ ⊗ ψ)(g)(ϕ ⊗ ψ)(h)u. (3.8)

It is very important to note that if ϕ and ψ are irreducible, their tensor product is in general not irreducible. We now have enough group and representation theoretic tools to proceed to study the unitary t-designs of the unitary group, and these tools will allow us to find a criterion to find designs to be used later in the randomised benchmarking protocol.

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4 Designs

While the previous two chapters were mostly background information, in this chapter we introduce one of our main objects of study: the unitary design. We will introduce the definition of a design, which will be an extremely powerful in defining the RB protocol later. Informally, a unitary design is a finite set that allows for the easy calculation of higher moments on the unitary group. We will introduce the mathematical definition of a design as first introduced in [2][3], and we will prove a representation theoretic criterion for establishing whether a group is a design as first done in [5]. We will do this following the treatment in [7]. We will use this criterion to prove the single qubit Clifford group is a 2 design as in [5], and so can be used for the RB protocol.

4.1 Haar measure and unitary designs

The definition of a design involves averaging a polynomial over the whole unitary group. This requires specifying the measure over which we integrate, and in our case this is the Haar probability measure, which we define for the unitary group first before proceeding to designs. We define the Haar probability measure for the unitary group U (d) as follows: Definition 4.1. The Haar probability measure on the unitary group U (d) is the unique probability measure µ such that, for all continuous functions f : U (d) → C and V, W ∈ U (d): Z f (U )dµ(U ) = Z f (V U )dµ(U ) = Z f (U W )dµ(U ). (4.1)

The proof of the existence and uniqueness of this measure on the unitary group is outside the scope of this thesis. From now on we denote every integral over the Haar measure by dU .

Note that the defining left- or right-invariance of the Haar measure mimics an index substitution in a finite sum, namely:

X g∈G f (g) =X g∈G f (hg) =X g∈G f (gh), (4.2)

where we just change the index for some fixed h ∈ G. This similarity in the algebraic properties of the Haar measure and the finite sum will be very important later in this chapter. We can now define a unitary t-design. The formal definition of a unitary t-design is as follows:

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Definition 4.2. A set D = {Uk: 1 ≤ k ≤ K} of unitary matrices on the Hilbert space

H = Cd _{is a unitary t-design if and only if:}

1 K X Uk∈D p(Uk, Uk) = Z U (d) p(U, U )dU (4.3)

for all polynomials p homogeneous of degree t in both the entries of U and U , where p(U, U ) denotes p evaluated in the entries of U .

The intuition behind designs is that the “average” outcome of the polynomial can be found by simply taking a finite sum over some finite set of unitary matrices instead of evaluating the integral. We denote the specific polynomials in use for the definition of the unitary t-design as (t,t)-homogeneous polynomials.

An alternative definition of designs that is sometimes used allows for polynomials that are not necessarily homogeneous, but only that every term has as many entries of U as of U . For this definition to be equivalent to ours would require every t-design to be an s-design for all s ≤ t. We prove this in a small lemma.

Lemma 4.1. Every unitary t-design is a unitary s-design for all s ≤ t. Proof. Note that 1 = 1_dtrU U†_{for any U ∈ U (d). Now let D = {U}

k: 1 ≤ k ≤ K} be a

unitary t-design, and let s ≤ t. If we take any (s,s)-degree homogeneous polynomial p, then: Z U (d) p(U, U )dU = Z U (d) p(U, U ) trU U † d t−s dU (4.4) = X Uk∈D p(Uk, Uk) trUkU_k† d t−s (4.5) = X Uk∈D p(Uk, Uk). (4.6)

Thus D is also a unitary s-design.

While these definitions are very useful when using designs in other mathematics, they are not very useful for determining if given set is a design. In the rest of this chapter we will prove an alternative criterion for determining if a set is a design.

4.2 The 1-design criterion

In this section we will characterize the 1-designs by proving a sufficient representation theoretic criterion for a set to be a design. The rest of this chapter will be concerned with extending this criterion to the general t-design case.

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The defining property of a 1-design D = {Uk}, where Uk ∈ Cd×d, is that for every

(1,1) degree homogeneous polynomial the fundamental property holds:

1 K X Uk∈D p(Uk, Uk) = Z U (d) p(U, U )dU. (4.7)

There is however a better way to represent this property. We know that every (1,1) degree homogeneous polynomial p is a linear combination of the entries of U and U together, that is:

p(U, U ) ∈ span{UijUkl : i, j, k, l ≤ d}. (4.8)

We can show that every such polynomial can be written as an entry of the matrix U XU†for some matrix X ∈ Cd×d, and that in fact all entries of U XU†are homogeneous (1,1)-degree polynomials.

If we take the matrix W = Ejl, the matrix with a 1 on row j and column l , and a 0

everywhere else. Then:

U W U†_ik=X

j0_,l0

Uij0W_j0_l0(U†)_l0_k= U_ij(U†)_lk= U_ijU_kl. (4.9)

And then we can see that by the linearity of matrix multiplication that we can easily construct a matrix X such that out polynomial p is the ik entry of X. In fact, linear-ity ensures for arbitrary X ∈ Cd×d that every component of U XU† is a (1,1)-degree homogeneous polynomial.

We can therefore reformulate the 1-design requirement as follows; A set of unitary matrices D = {Uk}, where Uk∈ Cd×d is a 1-design if and only if, for all X ∈ Cd×d:

1 |D| X Uk∈D UkXU † k = Z U (d) U XU†dU, (4.10)

where on the right hand side the integral refers to the integral of every component of U XU†. We proceed to the 1-design criterion.

We are going to show that if G is a finite subgroup of the unitary group U (d) such that for the defining representation of the unitary group ϕ : U (d) → Cd×d: U 7→ U , ϕ|G

is irreducible, then G is a 1-design.

Let G be a subgroup of the unitary group U (d) where ϕ : U (d) → GL(Cd), U 7→ U the defining representation for U (d) and G, and where ϕ|Gis irreducible. Note that since ϕ

is the defining representation of the unitary group, ϕ(g) is unitary for all g ∈ G. Thus ϕ(g−1) = ϕ(g)†. Let X ∈ Cd×d. If we define: X#:= 1 |G| X g∈G ϕ(g)Xϕ(g)†, (4.11)

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we can prove that X# ∈ Hom_G(ϕ|G, ϕ|G) by proving it commutes with the group action of G: X#ϕ(h) = 1 |G| X g∈G ϕ(g)Xϕ(g)†ϕ(h) (4.12) = 1 |G| X g∈G ϕ(g)Xϕ(g−1h) (4.13) = 1 |G| X p∈G ϕ(hp)Xϕ(p−1) (4.14) = ϕ(h) 1 |G| X p∈G ϕ(p)Xϕ(p)† (4.15) = ϕ(h)X#. (4.16) Since X#commutes with the group action, by definition X#∈ Hom_G(ϕ|G, ϕ|G), and

now by Schur’s lemma we know that X#_{= λI. Furthermore:}

λ = trX # tr I (4.17) = 1 dtr 1 |G| X g∈G ϕ(g)Xϕ(g−1) (4.18) = 1 d 1 |G| X g∈G trϕ(g)Xϕ(g−1₎ _(4.19) = 1 d 1 |G| X g∈G trϕ(g−1_)ϕ(g)X (4.20) = 1 d 1 |G| X g∈G trX (4.21) = trX d , (4.22) and thus X# = tr X

d . The beauty of these arguments is that due to the linearity

of the integral, and the transformation properties of the Haar measure as described in definition 4.1, they all apply to the integral over the whole unitary group as well, and we can equate the two averages. We define:

X∗:= Z

U (d)

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Then analogously we have invariance under the group action, and so we have that X∗ ∈ HomG(ϕ|G, ϕ|G), and so X∗ = µI. Similarly, we have that µ =

trX d , since

R

U (d)dU = 1. Then since X

#_{= λI = µI = X}∗ _{we have:}

1 |D| X Uk∈D UkXU_k†= Z U (d) U XU†dU (4.24)

for all X. Hence any finite subgroup G of the unitary group U (d) is a 1-design if ϕ|G

is irreducible.

While this is a very useful result, it turns out that to study the Haar average of a quantum channel, higher order designs are required. We will extend the result above to a theorem for general t-designs in the next section.

4.3 A representation theoretic characterization of t-designs

In this section we are going to be proving the generalized version of the result in the previous section, we are going to prove that for a subgroup G of the unitary group U (d), and the defining representation ϕ of U (d), if ϕ⊗t|G has the same decomposition into

irreducible representations as ϕ⊗t, then G is a t-design [7].

We are going to prove this result in an analogous way to the 1-design case, by first showing that all homogeneous (t,t)-degree polynomials can be written as components of some object, and then that a finite sum and a Haar integral of that object are equal. We will do this by proving that the finite sum and the integral commute with the group action under ϕ⊗t, and therefore by applying Schur’s lemma proving that they are identical. It will follow that the equality of these two objects implies the defining property of a t-design.

We first prove a polynomial lemma analogous to the 1-design case:

Lemma 4.2. Every homogeneous (t,t)-degree polynomial p(U, U ) can be written as a component of the matrix:

U⊗tX(U†)⊗t (4.25) for some X ∈ (Cd×d)⊗t, and in fact every component of U⊗tY (U†)⊗t for arbitrary Y ∈ (Cd×d)⊗t is such a polynomial.

Proof. For every (t,t) degree homogeneous monomial, we want to find a matrix of which it is a component, and we want for every such monomial to have a matrix. Then, by the multilinearity of the tensor product and the linearity of matrix multiplication this would be enough to claim the result for all homogeneous polynomials.

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We now take the matrix W =Nt n=1Ejnln. Then: U⊗tW (U†)⊗t = U⊗t t O n=1 Ejnln(U † )⊗t (4.26) = t O n=1 U EjnlnU †_. _(4.27)

If we want to specify a component of this object, we need to pick a component from each of the separate matrices U EjnlnU

† _{of the tensor product. Hence we need to take}

the ((i1, k1), ...., (it, kt)) component. We will label this as the IK component:

(U⊗tW (U†)⊗t)IK= ( t O n=1 U EjnlnU †₎ (i1k1,....,itkt) (4.28) = t Y n=1 (U EjnlnU †₎ inkn (4.29) = t Y n=1 UinjnU † lnkn (4.30) = t Y n=1 UinjnUknln, (4.31)

and thus we have our matrix-monomial pair as desired.

We can now , just like in the 1-design case, reformulate the t-design requirement with a more workable criterion. A set of unitary matrices D = {Uk : 1 ≤ k ≤ n}, where

Uk∈ U (d) for all k, is a t-design if and only if:

1 |D| X Uk∈D U_k⊗tX(U_k†)⊗t = Z U (d) U⊗tX(U†)⊗tdU. (4.32) for all X ∈ (Cd×d)⊗t.

Now we are going to extend the result from section 4.2 to general t-designs, but first we introduce some notation. From now on we denote the t-fold tensor product ϕ⊗t of the defining representation of the unitary group ϕ : U (d) → L((C)d) with the shorthand notation ϕt. We re-iterate that ϕt is in general not irreducible. We will use

this representation to establish a sufficient condition for subgroups of the unitary group to be designs. We now proceed onto the first lemma needed in proving this:

Lemma 4.3. let G be a finite subgroup of the unitary group U (d). Then if ϕt and ϕt|G

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1 |G| X g∈G ϕt(g)Bϕt(g−1) = Z dU ϕt(U )Bϕt(U )†. (4.33)

Proof. Just like in the 1-design case, we are first going to prove that both the integral and the finite sum commute with the group action of G under the representation ϕt,

and then we are going to prove that they are equal using Schur’s lemma. Let B# :=

1 |G| P g∈Gϕt(g)Bϕt(g −1_{). Then, for h ∈ G:} B#ϕt(h) = 1 |G| X g∈G ϕt(g)Bϕt(g−1)ϕt(h) (4.34) = 1 |G| X g∈G ϕt(g)Bϕt(g−1h) (4.35) = 1 |G| X y∈G ϕt(hy)Bϕt(y−1) (4.36) = ϕt(h)B#. (4.37)

Similarly, if we define B∗:=R dU ϕt(U )Bϕt(U†), and analogously to B# it also

com-mutes with the group action. Now we are going to use Maschke’s theorem. Since U (d) is a compact group, and G is a finite group, we have that ϕt=Lsi=1[ϕ(i)]⊕mi, were ϕ(i)are all

inequivalent, irreducible representations. Recall that we assumed that ϕt|G decomposes

into the same irreducible constituents, and so we have that ϕt|G =

Ls

i=1[ϕ(i)]

⊕mi_.We

can then write for all g ∈ G:

B# s M i=1 [ϕ(i)(g)]⊕mi = s M i=1 [ϕ(i)(g)]⊕miB#_. _(4.38)

We now want to look at what each term in the direct sum Ls

i=1[ϕ(i)(g)]

⊕mi _{does to}

different parts of B#, and therefore we split B# into blocks, where the ij-th block has deg(ϕ(i)) rows and deg(ϕ(j)) columns, where ϕ(i), ϕ(j) ∈ {ϕ(k) _{: 1 ≤ k ≤ s}. We write}

B_ij# for this block. Then, for all g ∈ G:

B_ij#ϕ(j)(g) = ϕ(i)(g)B_ij# (4.39) for some irreducible constituent representations ϕ(i)_{, ϕ}(j)_{. But then by Schur’s lemma}

we know that if ϕ(i) = ϕ(j), then B_ij# = λiI, and if ϕ(i) ϕ(j), we know that Bij# = O.

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λi = trB# ii deg(ϕ(i) (4.40) = 1 deg(ϕ(i)₎tr 1 |G| X g∈G ϕt(g)Bϕt(g−1) ii (4.41) = 1 deg(ϕ(i)₎ 1 |G| X g∈G tr s M p=1 [ϕ(p)(g)]⊕mp_B s M p=1 [ϕ(p)(g−1)]⊕mp ii (4.42) = 1 deg(ϕ(i)₎ 1 |G| X g∈G trϕ(i)_(g)B iiϕ(i)(g−1) (4.43) = 1 deg(ϕ(i)₎ 1 |G| X g∈G trϕ(i)_(g)ϕ(i)_(g−1_)B ii (4.44) = trBii deg(ϕ(i)₎, (4.45)

and thus we know that:

B#_ij =    trBii degϕ(i) if ϕ (i)_{= ϕ}(j) O if ϕ(i) ϕ(j). (4.46)

Using the exact same arguments, due to the linearity of the integral, the trace, the transformation property of the Haar measure and since we assumed that ϕt|G and ϕt

decompose into the same representations, we can follow the above reasoning for B∗ exactly. We now have that

B_ij∗ = B_ij#=    trBii

degϕ(i) if ϕ(i) = ϕ(j)

O if ϕ(i) ϕ(j)

(4.47)

for every ij block, and therefore B∗ = B#, which completes the proof. And now we can easily prove the criterion for t-designs that are also groups:

Theorem 4.1. Let G be a finite subgroup of the unitary group U (d), and let ϕ : U (d) → Cd×d be the canonical representation of the unitary group. If ϕt and ϕt|G decompose

into the same irreducible subrepresentations, then G is a t-design. Proof. By lemma 4.3 we have that for all B ∈ (Cd×d)⊗t:

1 |G| X g∈G ϕt(g)Bϕt(g−1) = Z ϕt(U )Bϕt(U )†dU, (4.48)

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and we know that every homogeneous (t,t)-degree polynomial p(U, U ) can be written as a component of the matrix:

U⊗tX(U†)⊗t (4.49) for some X ∈ (Cd×d)⊗t. And so we finally have:

1 |G| X g∈G p(ϕ(g), ϕ(g)) = Z U (d) p(U, U )dU (4.50)

for every (t,t)-degree homogeneous polynomial p(U, U ), and so G is a unitary t-design.

In the next section we will apply this criterion to prove the Clifford group is a 2-design. This will be important when justifying some steps when we define the RB protocol later.

4.4 The Clifford group

In this section we will define the family of Clifford groups Cq ⊂ U (2q), and we will use

the sufficient condition in theorem 4.1 to prove that the Clifford group for a single qubit system (where H = C2) is a 2-design [5].

We begin with the definition of the Clifford groups:

Definition 4.3. Let B = {|0i, |1i} be a basis of C2. The Clifford group is the group Cq such that Cq= hSi, Hi, Cij : i, j ≤ qi ⊂ U (2q), where

S|ki = ik|ki (4.51) H|ki = √1

2(|0i + (−1)

k_|1i) _(4.52)

C|ki ⊗ |li = |ki ⊗ |l + k mod 2i (4.53) and where Si = I⊗i−12 ⊗ S ⊗ I

⊗q−i

2 , Hi= I⊗i−12 ⊗ H ⊗ I ⊗q−i 2 .

Intuitively, Si is the S operation on the i-th copy of C2, and the same applies for

Hi. Cij applies C with the condition on |ki in the i-th copy of C2 and the flip on |li

in the j-th copy. Note that the S, H and C operators are common unitary operations used in various quantum algorithms [13], hence the potential usefulness of the Clifford group as a 2-design. It turns out that Cq is a 2-design for every q, but the proof of this

lies outside the scope of this thesis. We will however prove that C1 is a 2-design using

criterion 4.1. In the rest of this subsection we will use the notations |0i ⊗ |1i, |0i|1i and |01i interchangeably.

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Proof. We will examine the representation ϕ⊗2 : U (2) → L[(C2)⊗2], where ϕ is the defining representation of U (2). Let {|0i, |1i} be a basis for C2. Now consider the permutation operator θ : (C2)⊗2 → (C2₎⊗2_{: |vi ⊗ |wi 7→ |wi ⊗ |vi. This operator has 2}

eigenspaces, namely:

E1 = span{|00i + |11i, |00i − |11i, |01i + |10i} =: span{φ+, φ−, ψ+} (4.54)

E−1 = span{|01i − |10i} =: span{ψ−}. (4.55)

Note that these are the symmetric and antisymmetric tensors respectively. Now note that the operator θ commutes with the operator U ⊗ U for all U ∈ U (2), and thus we know that the spaces E1, E−1 are U (2) invariant subspaces under ϕ⊗2. We immedeatly

know that E−1is irreducible for the whole U (2) since it is one dimensional. We will now

prove that E1 is irreducible under the action of ϕ⊗2|C1.

First we examine a particular subgroup of the Clifford group: P1 = hX, Zi :=

hHS2_{H, S}2_{i. In matrix form the generators of P} 1 are: X =0 1 1 0 , Z =1 0 0 −1 . (4.56)

Now note that for, the action of this subgroup under ϕ⊗2, it holds that ϕ⊗2(P1)φ+ =

φ+, ϕ⊗2(P1)φ− = {±φ−} and ϕ⊗2(P1)ψ+= {±ψ+}. This means that each of the vectors

φ+, φ−, ψ+ spans an irreducible, mutually inequivalent subspace of E1 for the subgroup

P1.

Now suppose E1 is not C1-irreducible. Then, since it has 3 dimensions, it means that

it either splits into a 2 dimensional subrepresentation and a 1 dimensional subrepresen-tation, or it splits into three 1 dimensional subrepresentations. In either case there must be a one dimensional subrepresentation within E1. If there is, it must be spanned by

either φ+, φ− or ψ+, since those are the irreducible representations of P1 ⊂ C1.

Now note that S⊗2φ+= φ−, and that H⊗2ψ+= φ+. Thus span{φ+}, span{φ−}, span{ψ+}

are not invariant and thus not irreducible. Thus E1 is irreducible for the action of C1,

and therefore it is also irreducible for the action of U (2). Thus we have

ϕ⊗2 = E1⊕ E−1 = ϕ⊗2|C1 (4.57)

and thus by theorem 4.1 C1 is a 2-design.

A proof for all Clifford groups Cq can be found in [5]. 2-designs specifically will

be very important in the next two chapters, since a so called “twirled channel”, an important type of quantum channel, is in essence just a Haar average over a (2,2) degree homogeneous polynomial. This will be the focus of the next chapter.

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5 Fidelity and depolarisation

In this chapter we will discuss noise in quantum computers and how it can be expressed using the formalisms of chapter 2. We will define the fidelity and both motivate and justify its definition as a measure of the level of noise of a gate in a quantum computer [7]. This will be followed by some small theorems proving some properties of noise channels and average fidelity, which we will use in the next chapter when defining the randomised benchmarking protocol, which is a way to measure the fidelity as introduced in [12].

5.1 Noise in quantum computers

Quantum computer have the ability to prepare an initial quantum state ρ0 ∈ L(H) and

the ability to apply unitary operators Gi from some set G (the gateset) to this state,

and measure the final state.

In practice however, applying the operators in the gateset G almost never behave like the perfect unitary case, where G : ρ 7→ GρG†. In practice applying the gate would be applying a non-unitary quantum channel ˜G. We would like to know how well ˜G approximates G, and there is a measure for this called the fidelity.

5.2 The fidelity

We define the fidelity as follows:

Definition 5.1. Given the noisy gate operator ˜G = Λ ◦ G, the fidelity F is defined as F : L(L(H)) × L(L(H)) → R:

F ( ˜G, G) = Z

|||ψi||=1

trG(|ψihψ|) ˜G(|ψihψ|)d|ψi, (5.1)

where the integral is taken uniformly over the unit ball in H.

Intuitively, the fidelity can be seen as the trace of the average projection of the noisy state ˜G(|ψihψ|) onto the ideal state G(|ψihψ|). This intuition is quite helpful, since if the ideal case and the actual case are identical, our ideal state G(|ψihψ|) is simply projected onto itself and the fidelity would evaluate to 1. When noise is present, we will see the projection is smaller than 1. In that case, if we represent the noisy gate as a quantum channel ˜G, we can write:

˜

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where we define the noise channel ΛG by absorbing the second G, G† into it. We now

examine some properties of the fidelity map. Firstly, note that:

It would be nice to know what happens in less than ideal cases, like for example that it always holds that F (Λ, I) ∈ [0, 1]. This indeed the case, since:

F (Λ, I) = Z tr|ψihψ||ψiΛ(|ψihψ|)d|ψi (5.11) = Z hψ|Λ(|ψihψ|)|ψid|ψi (5.12) ≤ Z max ρ≥0,trρ=1 hψ|ρ|ψid|ψi (5.13) = Z d|ψi (5.14) = 1. (5.15)

Here ρ = Λ(|ψihψ|), but it is only important that it is some density operator. Since the density operator ρ is a positive operator, we have that hψ|ρ|ψi ≥ 0, and so F (Λ, I) ∈ [0, 1]. It is helpful to see what happens in another scenario as well. If the noise channel is maximally noisy, and it removes any information of the original gate, then we can write Λ(ρ) = I

d for all states ρ. This is the depolarising channel from chapter 2 with p = 1.

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F (Λ, I) = Z tr|ψihψ| · I dd|ψi (5.16) = 1 d Z d|ψi (5.17) = 1 d. (5.18)

Thus the fidelity indeed behaves like a measure of the distortion of the ideal operator G, where a fidelity closer to 1 indicates a less distorted operation. In the next section we will examine two very important cases of possible noise channels: the twirled channel and the depolarising channel.

5.3 Channel twirling and depolarising channels

In this section we will define the twirled channel, and prove that it is always a depolarising channel [12], along with some small lemmas. We will do this using Schur’s lemma. While a lot of this section may seem arbitrary at first, the results will be extremely important when discussing randomised benchmarking later. We start by defining a twirled channel: Definition 5.2. For any channel Λ we define the twirl of Λ as:

T (Λ)(ρ) := Z

U (d)

U†Λ(U ρU†)U dU. (5.19)

First we prove a small lemma:

Lemma 5.1. Twirling a quantum channel Λ does not change its fidelity: F (T (Λ), I) = F (Λ, I).

Proof. Let Λ be any quantum channel and let ρ ∈ Cd×da density operator. Note that for any unitary U ∈ U (d), if we use the operator sum representation for Λ: U†Λ(U ρU†)U = P

kU †_E

kU ρU†E_k†U , by lemma 4.2 we know every entry of U†EkU is a homogeneous (1,1)

polynomial with entries of U , and thus every entry of U†Λ(U ρU†)U is a homogeneous (2,2) polynomial with entries from U . Thus is we take the Clifford group Cq(or any other

2 design) we know that T (Λ)(ρ) = R_{U (d)}U†Λ(U ρU†)U dU = _|C1

q|

P

G∈CqG

†_Λ(GρG†_)G.

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F (T (Λ), I) = Z tr|ψihψ|T (Λ)(|ψihψ|)d|ψi (5.20) = Z tr|ψihψ| 1 |Cq| X G∈Cq G†Λ(G|ψihψ|G†)Gd|ψi (5.21) = 1 |Cq| X G∈Cq Z

trG|ψihψ|G†_{Λ(G|ψihψ|G}†_)d|ψi _(5.22)

= 1 |Cq| X G∈Cq Z tr|ψihψ|Λ(|ψihψ|)d|ψi (5.23) = 1 |Cq| X G∈Cq F (Λ, I) = F (Λ, I). (5.24)

Now we move on to a larger and more important result; proving every twirled channel is a depolarising channel. Before we can do this however, we need to examine a particular representation of the unitary group:

We define Φ : U (d) → L(Cd×d_):

Φ(U )(ρ) := U (ρ) := U ρU†. (5.25) Φ is linear, we prove that it is a representation as well, let U, W ∈ U (d):

Φ(U W )(ρ) = U W ρ(U W )†= U W ρW†U†= Φ(U )(Φ(W )(ρ)), (5.26) and thus it is a representation. It is however not irreducible. In patricular, it has two orthogonal U (d)-invariant subspaces (We will later see orthogonal with respect to which inner product). First take V = {λI : λ ∈ C}. This is clearly a vector space. Then, for any U ∈ U (d):

Φ(U )(λI) = U (λI)U†= λU U†= λI, (5.27) and thus we have V as one U (d)-invariant subspace. Now take W = {M ∈ Cd×d : trM = 0}. Due to the linearity of the trace this is also a vector space. Then for U ∈ U (d):

trΦ(U )(M ) = trU M U†_{= trU}†_{U M}_{= trM = 0,} _(5.28)

and thus we have that W is also a G-invariant subspace. Note that V ∩ W = {0} We also have that V ∪W = Cd×d_{, since if M ∈ C}d×d_{, then M = M −trM}I

d+trM dI,

and M − trM I

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We can introduce a natural inner product on the space Cd×d, which we define as follows. For A, B ∈ Cd×d:

hA, Bi := trA†_B.

(5.29) Lemma 5.2. h·, ·i : Cd×d× Cd×d _{→ C is an inner product.}

This is proven in [9]. Let A = λI ∈ V and let B ∈ W . Then:

hA, Bi = trA†_B_{= λtrIB = λtrB = 0} _(5.30)

And thus we have that L(Cd) = Cd×d = V ⊕ W , and since these subspaces are G-invariant, we have that Φ decomposes into two constituents ϕV : U (d) → L(V ) and

ϕW : U (d) → L(W ), where Φ = ϕV ⊕ ϕW. Since V is one dimensional we already know

that ϕV is irreducible. ϕW Is irreducible as well (we prove this in lemma 5.3), and thus

we can use Schur’s lemma.

If we take some arbitrary quantum channel Λ. Then for any unitary X ∈ Cd×d:

T (Λ) ◦ Φ(X)(ρ) = Z U†Λ(U XρX†U†)U dU (5.31) = Z XU0†Λ(U0ρU0†)U0X†du0 (5.32) = XT (Λ)(ρ)X† (5.33) = Φ(X) ◦ T (Λ)(ρ). (5.34) Thus T (Λ) ∈ Hom(Φ, Φ), and by Schur’s lemma we conclude that T (∆) = λPV+µPW.

Note that if we plug in the maximally mixed state, T (∆)(I

d) = λPV(dI)+µPW(dI) = λId.

Thus, since we want T (Λ)( to be a quantum channel, T (∆)(I

d) must be a density matrix

and therefore have trace 1, and so we know that λ = 1. Thus, since ρ = ρ +tr[ρ]_d −tr[ρ]_d , we have: T (Λ)(ρ) = PV(ρ) + µPW(ρ) = trρI d + µ ρ −trρI d = µρ + (1 − µ)I d, (5.35) and thus T (Λ)(ρ) = P(ρ) ; the twirl of an arbitrary quantum channel is always a depolarising channel. We now prove that ϕW is irreducible to complete this argument.

Lemma 5.3. The representation ϕW : U (d) → W is irreducible.

Proof. Note that B = B1∪ B2 = {Eij : i, j ≤ d, i 6= j} ∪ {Eii− E(i+1)(i+1) : i ≤ d − 1} is a

basis for W . Let W0 be a nontrivial G-invariant substance of W . We are going to prove that PW0, the projection onto W0 must be the identity operator on W , thus proving ϕ_W

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First we prove that PW0 is diagonal in the basis B. We know that by assumption of the

G-invariance of W0 that [PW0, Λ_U] = 0 for all U ∈ U (d). Now let V be a nondegenerate,

unitary, diagonal matrix. Then we have that, for i 6= j, Φ(V )(Eij) = V EijV† = λijEij

for some nonzero λij. Since V is nondegenerate all these λij will also be different for

every pair of (i, j), i 6= j. And since [PW0, Φ(V )] = 0 for the nondegenerate diagonal

superoperator Φ(V ), and due to their commutativity they must share an eigenbasis, we can conclude that PW0 is diagonal on B₁ = {E_ij : i 6= j}. When considering P_W0 as a

matrix acting on W we will refer to PW0|B₁ as the first block.

Similarly, we define the matrices Ti+, the unique matrices such that Ti+ei = ei+1, Ti+ej =

ej for i 6= j 6= i + 1. There are exactly d − 1 of them. We then have that Φ(Ti+)(Ejj−

E(j+1)(j+1)) = (−1)δij(Ejj− E(j+1)(j+1)). Thus the eigenvalues of the superoperator Ti+

include both -1 and 1. If we expand [PW0, Φ(T_i+)] = 0, we can see that as a matrix, for

PW0 it holds that: (−1)δ1i_P 1, ... , (−1)δ(d−1)iPd =   (−1)δ1i_P1 ... (−1)δ(d−1)i_Pd  , (5.36)

where Pk denotes the k-th column of PW0 and Pk the k-th row. Since this must hold

for all d − 1 different choices for i, we can conclude that PW0 is diagonal on B₂ as well.

Now note that since we proved that PW0 is both diagonal in B₁ and B₂, we know that

P (B1) ⊂ B1 and P (B2) ⊂ B2. Thus the off-diagonal blocks of PW0 must be 0. This

means that PW0 is a diagonal superoperator. Since it is also a projector, we know that

all the eigenvalues on the diagonal must be either 1 of 0.

Now suppose that PW0(E_ij) = 0 for some E_ij ∈ B. We define T_ik ∈ Cd×dto be

the permutation matrix that exchanges the basis vectors ei and ek. It then holds that

Φ(TjlTik)(Eij) = TjlTikEijTikTjl = TjlEkjTjl = Ekl, and therefore we can say that

PW0(E_kl) = P_W0(Φ(T_jlT_ik)(E_ij)) = Φ(T_jlT_ik)(P_W0(E_ij)) = Φ(T_jlT_ik)(0) = 0. And thus

we have that PW0(E_kl) = 0 for all k 6= l. And thus P_W0 is the zero operator on B₁.

We define the matrix H = I+E12+E(21−2E22. This is effectively a Hadamard operator

on the top right of the identity matrix. Note that Φ(H)(E12+ E21) = 2(E11− E22).

Thus, if PW0(E_ij) = 0, then P_W0(E₁₂+ E₂₁) = 0. Then since P_W0 and Φ(H) commute

we also have PW0(Φ(H)(E₁₂+ E₂₁)) = 0. And thus P_W0(E₁₁− E₂₂) = 0.

Then if we take the matrix X such that Xei = ei+1 mod d for all i ≤ d, we have

that Φ(X)(E11− E22) = E22− E33 etc. Thus by repeatedly applying X we can make

all Eii− E(i+1)(i+1) ∈ B2. Then again as above we can conclude that PW0 = 0 on B₂,

and thus PW0 is the zero superoperator. This is a contradiction, since our G-invariant

subpace W0 must be nonzero.

If PW0(E_ii− E_(i+1)(i+1)) = 0 for some i, then by considering Φ(X) as above we can

conclude that PW0 = 0 on B₂. But then by considering Φ(H) and Φ(T_ij) as above, we

can analogously conclude that then PW0 = 0 the zero superoperator. This is the same

contradiction. Thus we must conclude that all the eigenvalues of PW0 must be 1, thus

PW0 is the identity, and therefor W0 = W . This is again a contradiction, and so we have

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Now we have all of the physical and mathematical tools to define and justify the randomised benchmarking protocol, which we will do in the next chapter.

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6 Randomised benchmarking

In this chapter we will combine the results from all the other chapters and define the randomised benchmarking protocol as done in [10] and [11]. This protocol allows for the measurement of the average fidelity of any given gateset. Conventionally, a quantum computer can use/manipulate of three objects: an initial state ρ0 ∈ L(H), a set of

possible unitary operators (gateset ) G = {Gi}, and POVM E = {E1, ..., Em}. One very

important feature of the protocol is that the measurement of F (Λ, I) is independent of both ρ0 and E. This is in fact one of the biggest advantages of the protocol. We will

explicitly state the assumptions and requirements of the protocol as in [7], and we will prove any large steps in its derivation.

6.1 Initial configuration and assumptions

As defined in the previous chapter we have a quantum computer with initial state ρ0,

gateset G and POVM E = {E1, ..., En}. For the randomised benchmarking protocol to

work we need to implement the Clifford group Cqwith our gateset G, and we can restrict

E to E = {E1, I − E1} since we need only one measurement operator in the protocol.

Note also that the state space H has dim(H) = d = 2q.

Recall that in practice when applying unitary operator G from the gateset G to the initial state ρ0, Our final state looks like ˜G(ρ0) = ΛG(Gρ0G†), where ΛG is the noise

channel

We will now make an assumption before deriving the protocol: this noise channel will be the same for all gates in the gateset. This assumption needs justification. If we take Λ = _|G|1 P

iG˜i, then we can write:

˜

G = ΛG◦ G = Λ ◦ G + (ΛG− Λ) ◦ G, (6.1)

where if the second term is small enough (the deviation from the average), we can neglect it. We are effectively assuming that the deviation from the average noise channel is a small 1st order perturbation. We will do this for the rest of the derivation of the randomised benchmarking protocol. The goal is to use the fidelity map to learn something about the average noise channel Λ. We will measure F (Λ, I) by only applying gates from the Clifford group and by preforming a measurement on the state of the system after applying those gates.

In the RB protocol a certain variable is being measured using the operations of a quantum computer. If we (uniformly) randomly pick Gn ∈ Cq, the quantity to be

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trE1G˜RG˜m· · · ˜G1(ρ0), (6.2)

where for GR := (Gm· · · G1)† we define ˜GR as ˜GR(ρ) := Λ(GRρG †

R). Intuitively this

is a random walk of length m on the Clifford group, where with every step the noise channel is applied. If there would be no noise ˜GRG˜m· · · ˜G1(ρ0) = ρ0, and so the effect of

the noise can be isolated. We now claim that by averaging over all random walks this variable can be used to measure the fidelity. We define:

pm := 1 |Cq|m X G1,...,Gm∈Cq trE1G˜RG˜m· · · ˜G1(ρ0). (6.3)

In the rest of this chapter we will be rewriting pm until it is expressed in some values

that we can use to derive the fidelity from.

6.2 Derivation of the randomised benchmarking protocol

In this section we will derive the average fidelity from pm. We start with a lemma:

Lemma 6.1. pm is of the form:

pm= Aλm+ B, (6.4)

Where A and B are independent of m. Proof. Recall the definition of pm:

pm := 1 |Cq|m X G1,...,Gm∈Cq trE1G˜RG˜m· · · ˜G1(ρ0). (6.5)

Then if we let G†ρG =: G†, we can rewrite this as

pm = 1 |C_q|m X G1,...,Gm∈Cq trE1ΛG₁†· · · Gm†ΛGmΛGm−1· · · ΛG1(ρ0). (6.6)

Then if we let Xm = ΛGim−1· · · ΛGi1(ρ0), we can see that:

pm= 1 |Cq|m X G1,...,Gm∈Cq trE1ΛG1†· · · G † mΛGm(Xm) (6.7) = 1 |Cq|m−1 X G1,...,Gm−1∈Cq trE1ΛG1†· · · G † m−1 1 |Cq| X Gm∈Cq G†_mΛ(GmXmG†m)Gm . (6.8)

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Now recall that in lemma 5.1 we saw that the entries of U†Λ(U ρU†)U are all (2,2) degree homogeneous polynomials. Thus we can use the 2-design property of the Clifford group Cq as proven in theorem 4.2, and write pm as:

pm = 1 |C_q|m−1 X G1,...,Gm−1∈Cq trE1ΛG1†· · · G † m−1 Z U (d) U†Λ(U XmU†)U dU (6.9) = 1 |Cq|m−1 X G1,...,Gm−1∈Cq trE1ΛG1†· · · G † m−1T (Λ)(Xm). (6.10)

We know that every twirled channel is a depolarising channel, and thus we can say that T (Λ)(ρ) = pΛρ + (1 − pΛ)_dI =: P(ρ). Note that P commutes with every G, since we

have: P ◦ G(ρ) = p_ΛGρG†+ (1 − pΛ)I d (6.11) = pΛGρG†+ (1 − pΛ) GG† d (6.12) = GP(ρ)G†= G ◦ P(ρ), (6.13) and thus we can proceed to write pm as follows:

pm = 1 |C_q|m−1 X G1,...,Gm−1∈Cq trE1ΛPG₁†· · · G_m−1† (Xm). (6.14)

Now recall that Xm= ΛGim−1· · · ΛGi1(ρ0). We can define Xm−1 := ΛGim−2· · · ΛGi1(ρ0)

so that Xm = ΛGm−1(Xm−1) and thus:

pm = 1 |Cq|m−1 X G1,...,Gm−1∈Cq trE1ΛPG1†· · · G † m−1ΛGm−1(Xm−1). (6.15)

Now we can repeat what we did in the Xm case m separate times, once for every

Xm−i, until we have:

pm = trE1ΛPm(ρ0). (6.16)

Now we proceed to expand Pm(ρ0). Note that P(_dI) = _dI. Thus we can see that

P2(ρ0) = pΛP(ρ0) + (1 − pΛ)I d (6.17) = pΛ(pΛρ0+ (1 − pΛ)I d) + (1 − pΛ) I d (6.18) = p2_Λρ0+ (1 − p2)I d. (6.19)

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From there it is quite easy to see that Pm(ρ0) = pmΛρ0+ (1 − pmΛ)dI and thus: pm= trE1Λ(pmΛρ0+ (1 − pmΛ) I d) (6.20) = pm_ΛtrE1(Λ(ρ0) − I d) + tr E1 d . (6.21) And now we have derived the sought after relation between pm and and pΛ, namely

pm = ApmΛ + B.

Note that both A and B are independent of m, and that both E1 and ρ0 are absorbed

in A and B. Thus by varying m in the lab, by fitting the exponential accordingly one can experimentally determine pΛ without concern for ρ0 or E1. This has been done in

We have found the average fidelity of our gateset G, as F (Λ, I) = (d−1)pΛ+1

d . This

completes the derivation of the randomised benchmarking protocol. We proceed to some final remarks.

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7 Conclusion and outlook

In this chapter we will summarize the main material in each chapter, and we will finish with a short outlook on future research into designs and randomised benchmarking.

We started this thesis in chapter 2 by establishing the density matrix formalism of quantum mechanics, the mathematical setting for quantum computing. We motivated and defined quantum channels, and we gave two examples, namely the unitary and depolarising channels.

Then in chapter 3 we provided the necessary background in group and representation theory, stating important theorems using Schur’s lemma and Maschke’s theorem, as well as discussing the tensor product of representations.

In chapter 4 we defined designs as a set that reproduces low moments over the Haar measure of the unitary group, and then we proved a criterion to establish when a sub-group of the unitary sub-group is a t-design: a sub-group G ⊂ U (d) is a t-design if the t-fold tensor productof the defining representation of the unitary group U (d) decomposes into the same irreducible representations as when restricted to G. We proved this using Schur’s lemma and the fundamental properties of the Haar measure. We used this criterion to prove that the Clifford group on a single qubit is a 2-design.

In chapter 5 we quantified the noise in a quantum computer with the fidelity F (Λ, I), and we proved that every Haar twirled channel is a depolarising channel, an important fact in the derivation of the randomised benchmarking protocol.

We brought everything together in chapter 6: discussing the randomised benchmarking protocol, which is an experimental protocol to measure the average fidelity F (Λ, I) of a quantum computer. This protocol requires a 2-design to be a subset of the gateset of the quantum computer in question.

The mathematics of group designs, specifically the criterion proved in chapter 4, might seem as the way to classify all designs, however there is a limit to group designs. It has been proven that there are no group t-designs for t ≥ 5 [1], and for t ≥ 2 all group designs have been classified for dimension d ≥ 2 [1]. This means that one needs to go beyond groups to find higher t-designs. For our purposes in the randomised benchmarking protocol however group designs are sufficient. The uses for higher t-designs in quantum computing and quantum information theory are a subject of active research [6].

Randomised benchmarking is a well established protocol, and it is routinely used in labs around the world [4] [8] to measure the average fidelity of the quantum computers involved. Perhaps the most important factor as to why this protocol is so widely used is the fact that, as can be seen in formula 6.20, both the measurement operator E1 and

the initial state ρ0 are absorbed in the quantities A and B, both of which do not affect

the measurement of the fidelity F (Λ, I). This robustness of the protocol, its resistance to state preparation and measurement (SPAM) errors, is why it is such a widely used

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procedure. The assumption made in this thesis that the noise channel is the same for every gate in the gateset is quite strong and is in general invalid. Under certain circumstances however it is not necessary for randomised benchmarking to work [15]. Extending the validity of randomised benchmarking by weakening the initial assumptions is currently an active area of research as well, one which will hopefully accelerate the development of large scale quantum computers.

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Bibliography

[1] E. Bannai, G. Navarro, N. Rizo, and P. H. Tiep. Unitary t-groups. J. Math. Soc. Japan, 2020. Advance publication.

[2] C. Dankert. Efficient simulation of random quantum states and operators. arXiv preprint quant-ph/0512217, 2005.

[3] C. Dankert, R. Cleve, J. Emerson, and E. Livine. Exact and approximate unitary 2-designs and their application to fidelity estimation. Physical Review A, 80(1):012304, 2009.

[4] J. P. Gaebler, T. R. Tan, Y. Lin, Y. Wan, R. Bowler, A. C. Keith, S. Glancy, K. Coakley, E. Knill, D. Leibfried, et al. High-fidelity universal gate set for be 9+ ion qubits. Physical review letters, 117(6):060505, 2016.

[5] D. Gross, K. Audenaert, and J. Eisert. Evenly distributed unitaries: On the struc-ture of unitary designs. Journal of mathematical physics, 48(5):052104, 2007. [6] D. Gross, S. Nezami, and M. Walter. Schur-weyl duality for the clifford group with

applications: Property testing, a robust hudson theorem, and de finetti representa-tions. arXiv preprint arXiv:1712.08628, 2017.

[7] J. Helsen. Quantum information in the real world: Diagnosing and correcting errors in practical quantum devices. PhD thesis, Delft University of Technology, 2019. [8] C. D. Herold, S. D. Fallek, J. Merrill, A. M. Meier, K. R. Brown, C. Volin, and

J. M. Amini. Universal control of ion qubits in a scalable microfabricated planar trap. New Journal of Physics, 18(2):023048, 2016.

[9] P. Igodt and W. Veys. Lineaire algebra. Universitaire Pers Leuven; Leuven, 2011. [10] E. Knill, D. Leibfried, R. Reichle, J. Britton, R. B. Blakestad, J. D. Jost, C. Langer,

R. Ozeri, S. Seidelin, and D. J. Wineland. Randomized benchmarking of quantum gates. Physical Review A, 77(1):012307, 2008.

[11] E. Magesan, J. M. Gambetta, and J. Emerson. Characterizing quantum gates via randomized benchmarking. Physical Review A, 85(4):042311, 2012.

[12] M. A. Nielsen. A simple formula for the average gate fidelity of a quantum dynamical operation. Physics Letters A, 303(4):249–252, 2002.

[13] M. A. Nielsen and I. Chuang. Quantum computation and quantum information. American Association of Physics Teachers, 2002.

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[14] B. Steinberg. Representation Theory of Finite Groups: An Introductory Approach. Springer-Verlag New York, 2011.

[15] J. J. Wallman. Randomized benchmarking with gate-dependent noise. Quantum, 2:47, 2018.

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Popular Summary

Quantum computers are machines that can run specific algorithms (like calculations), but work very differently from regular computers. Regular computers store and process information in bits: They manipulate a series of 0’s and 1’s to do some calculation. Quantum computers have the ability to do many of the same things, but they process and store information in qubits, a series of values that are not restricted to 0 or 1. This technology promises to revolutionise computers in the future, with algorithms for fast searches in large databases or algorithms to break common encryption codes in seconds. However as of today no robust and reliable large scale quantum computer has been built yet. This is because the quantum systems which make up the qubits are exceptionally fragile: they change with the slightest disturbance from the outside world, be it a stray background magnetic or electric field or something bumping against the machine; they are very susceptible to “noise”. This not noise in the literal sense but a way to talk about all these unwanted outside disturbances that can influence our quantum computer.

When doing things (like implementing algorithms) with our potential quantum com-puter, the answers that come out might be accurate or be complete nonsense, depending on the amount of noise affecting the quantum computer. In this thesis we study a way to assign a number to this noise, where the larger the number is the less noisy and more accurate the potential computer is. We call this “noise number” the average fidelity of our quantum computer.

There is a small problem with this average fidelity however, and that is that it is quite an abstract quantity, its not that easy to either measure or calculate. This is due to the limitations of any quantum computer. Quantum computers can do three things: prepare some initial state ρ0, which is for example a set of qubits that are all in the 0 state. A

quantum computer can also change the initial state into another state with a specific type of operation called a unitary operation, and a quantum computer can measure the final state. Every time a unitary operation is applied some noise also inevitably comes in, it is for this noise that we want to find the average fidelity.

How do we measure the average fidelity in practice? There is a protocol to measure this average fidelity called the randomised benchmarking protocol. The conceptual overview of the protocol is that you apply n random unitary operations to your initial state, with the noise, one by one, and then at the end you apply the unitary operation that is the reverse of all the operations you applied before, at once. This way you essentially only have your initial state which has only gone through the noise n + 1 times, and then you measure the outcome. By varying the number of random operations (changing n), it turns out that you can see how “far away” from the initial state the n + 1-times noisy state is, and how much noisier it gets with every new unitary operation. The average fidelity is actually hidden within this information and can be found by doing some math,

Unitary Designs and their Applications in Quamtum Information Theory