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Quantum Gate Fidelities in AtomicEnsembles

BACHELORS THESIS PHYSICS AND ASTRONOMYSIZE 12 EC, RESEARCH CONDUCTED BETWEEN

MAY 10, 2010 AND JULY 8, 2010

Author:D.R.M. Pijn

Student number:5802008

Supervisor:Dr. R.J.C. Spreeuw

Second Reviewer:Prof. Dr. H.B. van Lin-den van den Heuvell

Daily Supervisor:A. Tauschinsky

August 27, 2010

Faculty of ScienceVan der Waals-Zeeman Institute

Valckenierstraat 651018 XE AmsterdamThe Netherlands

Abstract

This thesis analyzes the fidelity of quantum gates on ensembles of neutral atoms trappedon the surface of a magnetic atom chip. Each magnetic trap forms one qubit andis loaded with an ensemble of several neutral 87Rb atoms. The Rydberg blockade isused to generate the interaction between qubits that is needed for a two-qubit CNOTgate. An uncertainty in the number of atoms in each trap causes an error factor in theapplication of quantum gates on this system. The error factor is analyzed as a functionof mean atom number for two atom number distributions; for a Poisson distribution anda distribution with Fano factor 0.6. Results indicate that the error increases with theFano factor and decreases with the mean atom number for both single qubit and CNOTgates. The typical error reduction factor for ensembles with a mean atom number 10 dueto a decrease in Fano factor from f = 1 to f = 0.6 is 0.6. The fidelities of five differentimplementations of a CNOT gate are analyzed, and it turns out that the fidelity isnegatively related to the number of |0〉 state involving operations. Finally, compositepulses are successfully used to further increase the fidelity. For ensembles with a meanatom number 20, error reduction factors as good as 0.007 are found.

iii

Contents

Abstract iii

1 Introduction 11.1 General Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Dutch summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.4 Qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.5 Quantum Gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.5.1 Single Qubit Gates . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.5.2 Two Qubit Gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.6 Fidelity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.7 Using neutral atoms and the Rydberg blockade . . . . . . . . . . . . . . . 8

1.7.1 Computational basis . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2 Fidelity of single qubit rotations 112.1 Rotation matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.2 Rotation of an undetermined number of atoms . . . . . . . . . . . . . . . 122.3 Fidelity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Fidelity of CNOT Gates 173.1 Creating CNOT gates from single qubit rotations . . . . . . . . . . . . . . 173.2 Fidelity of different implementations of a CNOT gate . . . . . . . . . . . 233.3 Comparing effects on pure states . . . . . . . . . . . . . . . . . . . . . . . 25

4 Improving the fidelity with the use of composite pulses 274.1 Composite pulses in NMR . . . . . . . . . . . . . . . . . . . . . . . . . . . 274.2 Using composite pulses in single qubit rotations . . . . . . . . . . . . . . . 29

4.2.1 Using composite pulses to improve a Ramsey interferometer . . . . 324.3 Using composite pulses in CNOT gates . . . . . . . . . . . . . . . . . . . . 34

Discussion 37

Conclusion 39

Acknowledgements 41

Bibliography 43

v

1 Introduction

The manipulation and transmission of information is today carried out by machinesthat operate according to the laws of classical mechanics. But the laws of classicalmechanics do not form the final physical theory of the world. Since the laws of classicalmechanics emerge as a special limit of quantum mechanics, it is only reasonable toassume that machines which work according to the laws of quantum physics would havefar greater computational power. This makes research into quantum information theoryand quantum mechanical systems much more interesting. The full computing power of aquantum system is not yet known, but several algorithms have been developed that canperform specific tasks much more efficient than a classical machine ever could. Two ofthe most famous algorithms are Shor’s algorithm, which speeds up the task of factoringan n-digit number exponentially, and Grover’s algorithm, which speeds up the task offinding a specific entry in a database of n entries quadratically. These tasks are justsped up, but there are even tasks possible which could never be done on a classicalmachine: the fact that the measurement of a quantum system collapses it makes a levelof secrecy in communication possible that a classical machine could never reach. All ofthis sounds very promising, but although a lot of theory has been developed in the fieldof quantum computing, actually making a working quantum computer has proven to bequite hard. It’s not easy to produce a scalable, isolated quantum system that can befully controlled and maintained for a significant time, and has the right properties to doquantum computations with. A lot of research has been done on several possible systems,but it’s not yet clear which one is the best. Some candidate systems are nuclear spin(NMR), electron spin, ion traps, neutral atom traps and quantum dots. Each system hasit’s advantages, but the basic requirements to do quantum computations, which werelisted by DiVincenzo[8], can often only partially be met.

1.1 General Outline

This thesis is the result of a two-month research project at the Van der Waals-ZeemanInstitute in Amsterdam. The main target of the project was to analyse the fidelity ofqubit operations in the particular qubit system used by the Quantum Gases & Infor-mation group at the Van der Waals-Zeeman Institute. Most of the work consisted ofcomputations in Mathematica, where earlier acquired experimental data about the dis-tribution of atoms and the form of errors was used to apply general quantum informationmathematics to the particular system that we were interested in.Chapter 1 gives an introduction to the field of quantum computing and provides thereader with the necessary background information to understand the rest of this thesis.In chapter 2, I introduce the mathematical tools to analyse the fidelity and I analyse the

1

1 Introduction

fidelity of single qubit operations. In chapter 3, single qubit operations are combinedto form quantum gates. These gates are then combined in several ways to form threedifferent versions of a CNOT gate. Finally, the fidelities of the different CNOT gatesare compared. In chapter 4, the concept of composite pulses is first introduced andthen applied to both single qubit operations and CNOT gates to improve the fidelity.The possibility to improve a Ramsey interferometer with composite pulses is also brieflystudied.

1.2 Motivation

Personal motivationDuring my bachelor in physics there was no proper introduction in quantum informa-tion. However, the subject was mentioned briefly by a few teachers that happened to beworking in that area of research. Since I first heard of it, I have always found it very in-teresting. It’s a relatively new, rapidly developing area of research where very advancedtechnologies are being used to manipulate nature at the atomic level. It’s not clear ifquantum computation has a big future, but the research offers a lot of new insights inthe world of quantum physics and atoms anyway, which I find fascinating. I chose thisproject in an attempt to get to know the field of research better and to help myself indeciding whether I want to continue my career in this direction.

Scientific motivationThe Quantum Gases & Information group at the Van der Waals-Zeeman Institute isdoing research on a system in which neutral atoms in magnetic traps are used[1]. Com-pared to systems like NMR, the research is still in its infancy, but the system shows alot of promise. When the understanding and control of this system has reached a levelwhere it becomes possible to perform quantum computation algorithms, it will be inter-esting to know what the best set of gates to use and the best way to implement themis. To shed some light on this matter, I studied the fidelity of different gates and gateimplementations. Additionally, I tried to improve the fidelities with the use of compositepulses.

2

1.3 Dutch summary

1.3 Dutch summary

Computers worden steeds maar kleiner en sneller. In het huidige tempo zal het niet lang meerduren voor de meest fundamentele componenten van computers de grootte van enkele atomenhebben. Voor die tijd zullen we echter al tegen een probleem aanlopen: de wereld van atomenwerkt op een heel andere manier dan de wereld om ons heen. Atomen gedragen zich namelijkvolgens de wetten van de quantumfysica. Deze vertellen dat er altijd een bepaalde onzekerheidverbonden is aan o.a. de positie, snelheid en energie van een deeltje. Atomen bevinden zich dusniet op een plek, maar zijn verspreid over de ruimte, zoals golven. Pas als je gaat kijken waar hetatoom is kiest het een plek. Dit gedrag van deeltjes op atomair niveau klinkt ingewikkeld, maarkan ook erg nuttig zijn. Je kunt een computer maken die volgens de wetten van de quantumfysicawerkt en daardoor veel sneller is dan een klassieke computer.De meest fundamentele informatie-eenheid van een klassieke computer is een bit, met de waarde0 of 1. In een quantumcomputer nemen qubits deze plaats in. Qubits kunnen behalve 0 of 1ook een beetje 0 en een beetje 1 tegelijk zijn. Hierdoor kan je met qubits meerdere berekeningentegelijkertijd doen. In het algemeen kan een quantumcomputer met n qubits 2n berekeningentegelijk uitvoeren. De quantumcomputer zal een zoekopdracht dus kwadratisch sneller uitvoerendan een klassieke computer.Een quantumcomputer heeft veel voordelen, maar het bouwen ervan blijkt niet zo gemakkelijk tezijn. In de praktijk zijn er verschillende systemen waar je een quantumcomputer mee kan bouwenen het is nog niet duidelijk wat het beste systeem is. Aan het Van der Waals-Zeeman instituutdoet men onderzoek naar een systeem waarin de qubits gevormd worden door groepjes neutrale87Rb atomen die gevangen zitten in het veld van een magnetische atoomchip. De berekeningenworden gedaan door de qubits te manipuleren met laserpulsen. Voor een betrouwbare bereken-ing moeten de laserpulsen wel precies afgestemd zijn op het aantal atomen waar de qubit uitbestaat. Dit aantal atomen is echter onzeker doordat er af en toe een atoom uit de magnetischevallen ontsnapt. Een onzekerheid in het aantal atomen per qubit leidt tot een onzekerheid in deberekening. In deze scriptie wordt onderzocht hoe je berekeningen met de minste onzekerheidkan uitvoeren ondanks de onzekerheid in het aantal atomen. Verder wordt er aangetoond dat erextra zekerheid kan worden verkregen door bepaalde laserpulsen te vervangen door een speciaalgekozen reeks pulsen.

Figuur 1.1: Hier zie je de structuur van de magnetische atoomchip. Op plekken waar het mag-neetveld minimaal is zitten groepjes atomen gevangen op een afstand van 10 mi-crometer boven de chip. Elk groepje atomen vormt een qubit.

3

1 Introduction

1.4 Qubits

In quantum information, the fundamental units of information are qubits. They are thequantum mechanical analogue of bits in classical computers. The difference between bitsand qubits is that, where bits can only take on the values 0 or 1, qubits can be in anysuperposition of the quantum states |0〉 and |1〉. The state of a qubit is described by

|ψ〉 = α|0〉+ β|1〉 (1.4.1)

where α and β are complex numbers. When a qubit is measured, it collapses to one ofthe basis states |0〉 or |1〉 with probability |α|2 and |β|2. Naturally, since the probabilitiesmust add up to one, the values for α and β are restricted to |α|2 + |β|2 = 1.

Apart from this restriction, the coefficients can take on any value so there are aninfinite number of possible states. But although there are an infinite number of possiblestates, you will always only measure one of the two basis states. It is therefore impossibleto determine the exact state of a qubit. But you can manipulate it without measuringit, and thereby you can still use the infinite complex vector space to do calculations.That’s why quantum computers will be able to do much more than classical machines.

The best way to visualize the state of a qubit, is by looking at the Bloch sphere. Because|α|2 + |β|2 = 1, equation 1.4.1 can be rewritten to

|ψ〉 = cosθ

2|0〉+ eiφ sin

θ

2|1〉 (1.4.2)

where θ and φ are real numbers. These numbers define a point on the unit three-dimensional sphere, as shown in figure 1.2. This sphere is called the Bloch sphere. Aqubit state is now simply a unit vector in the Bloch sphere. For a state |ψ〉 = |0〉 thevector points along the +z axis, for a state |ψ〉 = |1〉 the vector points along the −zaxis and for a superposition state |ψ〉 = 1√

2(|0〉+ |1〉) the vector points along the x axis.

Operations that change the state of a qubit can now be seen as rotations on the Blochsphere.

1.5 Quantum Gates

Quantum gates are the analog to logic gates in a classical computer. They changethe state of one or more qubits. Unlike many classical logic gates, quantum gates arereversible. The state of a qubit can be represented by a vector, where the coefficients ofthe basis states form the coefficients of the vector components. Operations that changethe state of a qubit can then be represented by unitary matrices. In this section I willdiscuss the most commonly used quantum gates.

1.5.1 Single Qubit Gates

We’ll start with the quantum NOT gate. Like the classical logic NOT gate, the quantumNOT gate is an operation that takes the state |0〉 to the state |1〉 and vice versa. But

4

1.5 Quantum Gates

x

y

z

φ

θ

1

0

ψ

Figure 1.2: A Bloch sphere. The state of a qubit |ψ〉 = cos θ2 |0〉 + eiφ sin θ2 |1〉 can be

represented by a unit vector with angles θ and φ. Single qubit operationscan be represented by rotations. Source: Wikipedia.

this does not tell us what happens to a superposition of the states |0〉 and |1〉. In fact,the quantum NOT gate just swaps the coefficients of both states. So a state α|0〉+ β|1〉is taken to a state β|0〉 + α|1〉 and vice versa. On the Bloch sphere, a quantum NOTgate would amount to the rotation of the state vector over an angle π about the x axis.The most convenient way to work with quantum gates, is by using matrices and vectors.If the qubit state is written as a vector

|ψ〉 = α|0〉+ β|1〉 =

(αβ

)(1.5.1)

then the quantum NOT gate can be written as a unitary matrix

NOT =

(0 11 0

)(1.5.2)

which correctly swaps the coefficients of the basis states:

NOT|ψ〉 =

(0 11 0

)(αβ

)=

(βα

)(1.5.3)

Some other important single qubit gates are the Z gate, the Hadamard gate and thePhase gate. The Z gate gives only the coefficient of the |1〉 state a phase of -1, theHadamard gate turns |0〉 into 1√

2(|0〉+ |1〉) and |1〉 into 1√

2(|0〉 − |1〉) and the Phase

gate just adds a phase factor to the coefficient of the |1〉 state. On the Bloch sphere, aZ gate would be a π rotation about the z axis, a Hadamard gate would be a π

2 rotationabout the x axis and a Phase gate would be an arbitrary rotation about the z axis (so aPhase gate with a phase π is the same as a Z gate). The Z-, Hadamard and Phase gate

5

1 Introduction

can also be represented by unitary matrices, which are

Z =

(1 00 −1

)(1.5.4)

H =1√2

(1 11 −1

)(1.5.5)

P(φ) =

(1 00 eiφ

)(1.5.6)

It’s also possible to define a rotation matrix for an arbitrary rotation in the Bloch sphere.The rotation matrix for an arbitrary rotation about the x axis is[6]

R(θ) =

(cos (θ/2) −i sin (θ/2)−i sin (θ/2) cos (θ/2)

)(1.5.7)

When you combine this with the Phase gate, you have an arbirary rotation about anarbitrary axis.

1.5.2 Two Qubit Gates

If qubit A is in the state |ψ〉A and qubit B is in the state |φ〉B, the state of the compositesystem is given by the tensor product

|ψ〉A ⊗ |φ〉B (1.5.8)

For the system of two qubits there are now four basis states and the state can berepresented by the vector

|ψφ〉 = α|00〉+ β|01〉+ γ|10〉+ δ|11〉 =

αβγδ

(1.5.9)

When applying single qubit gates to the individual qubits, the rotation matrix that workson the 4-dimensional vector can be acquired by taking the tensor product between the2× 2 matrices. For example, the rotation matrix for a NOT gate on the second qubit is

NOT2 = I ⊗NOT =

(1 00 1

)⊗(

0 11 0

)=

0 1 0 01 0 0 00 0 0 10 0 1 0

(1.5.10)

Quantum gates can also work on two qubits. A very important gate that works ontwo qubits is the controlled NOT or CNOT gate. This gate has two input qubits, knownas the control and the target qubit. If the control qubit is set to 0, then the gate changes

6

1.6 Fidelity

nothing. If the control qubit is set to 1, then the target bit is flipped. The rotationmatrix for the CNOT gate is

CNOT =

1 0 0 00 1 0 00 0 0 10 0 1 0

(1.5.11)

Of course, there are many more interesting quantum gates. However, in a sense theCNOT gate and single qubit gates are the prototypes for all other gates because theyform a universal set. This means that any operation can be described by a combinationof the gates that make up the universal set. For a proof, see Nielsen en Chuang[6].

1.6 Fidelity

When doing quantum computations on a physical quantum computer, there is alwaysthe problem of errors. Some errors are random, for example: qubits might get entangledwith their environment, or they might even be lost completely. And some errors aresystematic: the rotations intended by the quantum gates might have a small deviation.Fortunately, there are error correction algorithms, so for a small error rate the errors canbe corrected faster than they occur. This means that the error rate has to stay belowa certain threshold. Often there are several ways to implement a quantum algorithm,with different error rates depending on the particular physical system used. So we wouldhave to compare different combinations of gates. The gates can be modified to includeerror factors, but how do you compare two gates?

For this, the fidelity is defined. It measures the closeness of two quantum states, inthis case the final state after applying a faulty gate to the final state after applying anideal gate. The fidelity of a quantum gate X, applied to a state |ψ〉 is defined as[6]

F(Xfaulty) =∣∣〈ψ|X†ideal .Xfaulty|ψ〉

∣∣2 (1.6.1)

Since the gate matrices are unitary, the product of two identical gates would give theidentity matrix. The fidelity of an ideal gate applied to a state |ψ〉 = α|0〉+ β|1〉 is then

F(Xideal) =∣∣〈ψ|X†ideal .Xideal|ψ〉

∣∣2=∣∣〈ψ|I|ψ〉∣∣2

=∣∣〈ψ|ψ〉∣∣2

=∣∣|α|2 + |β|2

∣∣2= 1

For faulty gates the fidelity is always lower than 1 and the fidelity ranges from 0 to 1.For an ideal gate the error rate should be 0, so the error rate is defined as 1− F.

7

1 Introduction

1.7 Using neutral atoms and the Rydberg blockade

The particular physical system that is being researched at the WZI consists of neutralatoms in magnetic traps. Every physical qubit consists of a small group of ultra-cold87Rb atoms trapped in the magnetic potential of a specially designed magnetic chip.Figure 1.3 shows a picture of the magnetic lattice of such a chip. The 87Rb atoms arecooled with lasers and then loaded into the magnetic traps indicated in figure 1.3.

The advantage of using neutral atoms instead of ions is that neutral atoms are rel-atively immune to interfering electromagnetic fields and have therefore a much longercoherence time (the time a qubit can be used before interaction with it’s environmentmakes the qubit unusable). Using small ensembles of atoms as qubits instead of singleatoms also prolongs the time a qubit can be used: If an atom is lost, in the worst casethe qubit state is changed, but the qubit is not lost.

The transition from one state to another is achieved by sending out a laser pulse witha resonant frequency, for a certain amount of time. The laser pulse induces stimulatedabsorption and emission, which makes the atom oscillate between two states at the so-called Rabi frequency. For certain pulse durations, the oscillation will stop while theatom is in the target state and the transition is achieved. For each atom there is acertain probability per unit of time that the atom will interact with the laser. So toexcite one atom from an ensemble a laser pulse has to have a certain duration, but thereis a chance that another atom will also be excited, or that no atoms will be excited.This gives an uncertainty in the number of atoms excited, and thus in the effect of thegate. This is the uncertainty that we will analyse.

The two hyperfine ground states of 87Rb are used as qubit states. Atoms in the higherhyperfine ground state take much longer to decay than atoms in an excited state, sothis prolongs the coherence time even further. A third state, called a Rydberg state,is used to achieve the interaction between two qubits that we need for a CNOT gate.A Rydberg state is a state where one electron is excited to a much higher level thanthe others. The other electrons effectively shield part of the core charge to the oneelectron. The electronic structure is now similar to that of hydrogen and the atomstarts to behave hydrogen-like. Most importantly, the atom starts to have a very strongdipole moment which changes the energy levels of the surrounding atoms. Differentenergy levels have different Rabi frequencies, so the laser that was tuned to the old Rabifrequencies won’t interact with any of the atoms that surround an atom in a Rydbergstate. The surrounding atoms are effectively blocked for as long as the one atom stays ina Rydberg state. This mechanism is called the Rydberg blockade. It lets the interactionof one qubit depend on the state of another. We can use that to make a CNOT gate. [5]

1.7.1 Computational basis

Ensembles of 87Rb atoms in the two hyperfine ground states are used as qubit states.But we only need two energy levels, so we should change the state of only one atom.Consequently, we will choose the following states as the basis states of our qubit: Forthe qubit state |0〉, all the atoms of the ensemble will be in the lower hyperfine ground

8

1.7 Using neutral atoms and the Rydberg blockade

state. For the qubit state |1〉, only one atom will be in the higher hyperfine ground stateand the rest will be in the lower hyperfine ground state. A qubit in the Rydberg statewill also only have one atom in a Rydberg state. Figure 1.4 shows the distribution ofatoms over the atomic states for each qubit state. [7]

Figure 1.3: An electron microscope photograph of the atom chip that produces the mag-netic traps. Several trapping sites are indicated by dashed ellipses. Theensemble of atoms trapped on a site forms one qubit. Source: Website WZIinstitute.

Figure 1.4: The distribution of atoms over the atomic states for each qubit state for aqubit consisting of four atoms. g0, g1 and r refer to the lower and higherhyperfine ground states and the Rydberg state.

9

2 Fidelity of single qubit rotations

The atoms that make up a qubit are during the quantum computations in a superpositionof the ground state |0〉, the excited state |1〉 and the Rydberg state |r〉.

|ψ〉 = α|0〉+ β|1〉+ γ|r〉 (2.0.1)

If the atom state is represented by a vector, the operations can be represented byrotation matrices that work on the atom state vector. Depending on the rotation angleθ, the rotation matrix will put the atom in a different pure state or leave it between twostates, which means in a superposition of these two states. (θ = π

2 puts the atom ina superpostion between the initial and target state, θ = π puts the atom in the targetstate and θ = 2π gives a rotation back to the initial state, with only a phase differenceof -1). The pulse length determines the rotation angle, so θ depends on t.

Figure 2.1: A state scheme for the ground state |0〉, the Rydberg state |r〉, the excitedground state |1〉 and the rotations between those states. Rotations R0r andR1r are achieved by sending out a light pulse with respective Rabi frequenciesΩ0r and Ω1r.

2.1 Rotation matrices

To study the fidelity of single qubit operations, the state of the atoms can be representedby a vector

|ψ〉 = α|0〉+ β|1〉+ γ|r〉 =

αβγ

(2.1.1)

11


The rotation matrices for the two operations shown in figure 2.1 can then be repre-sented by

R0r(θ0) =

cos (θ0(t)/2) 0 −i sin (θ0(t)/2)0 1 0

−i sin (θ0(t)/2) 0 cos (θ0(t)/2)

(2.1.2)

R1r(θ1) =

1 0 00 cos (θ1(t)/2) −i sin (θ1(t)/2)0 −i sin (θ1(t)/2) cos (θ1(t)/2)

(2.1.3)

2.2 Rotation of an undetermined number of atoms

If Ω0↔r and Ω1↔r are the Rabi frequencies needed for the rotations R0 and R1, thenthe rotation angles θ0 and θ1 are given by

θ0(t) =

∫ t

0Ω0↔r(τ) dτ (2.2.1)

θ1(t) =

∫ t

0Ω1↔r(τ) dτ (2.2.2)

While rotations between the Rydberg state |r〉 and the single excited state |1〉 alwaysinvolve just one atom, the rotations which involve the lowest ground state have to dealwith an ensemble of n atoms. Thus, a modified definition of θ0 is needed.For an ensemble of n atoms with collective Rabi frequency

√nΩ0↔r the rotation angle

is given by

θn0 (t) =√n

∫ t

0Ω0↔r(τ) dτ (2.2.3)

A full rotation between the |0〉 state and the Rydberg state can be achieved by a π-pulse,for which θn0 = π. This gives the relation∫ tnπ

0Ω0↔r(τ) dτ =

π√n

(2.2.4)

For an undetermined number of atoms the rotation angle needs another modification.Equation 2.2.3 shows that when dealing with an ensemble of n atoms, the rotation angleis multiplied by a factor

√n. According to equation 2.2.4, for the full rotation of one

atom, a pulse with frequency Ω0↔r and duration t1π would be needed:∫ t1π

0Ω0↔r(τ) dτ =

π√1

= π (2.2.5)

If this pulse would actually hit an ensemble of m atoms instead of one atom, the rotationangle would be multiplied by a factor

√m

θm0 (t1π) =√m

∫ t1π

0Ω0↔r(τ) dτ = π

√m (2.2.6)

12

2.3 Fidelity

Now, if the pulse was originally intended for an ensemble of n atoms instead of one atom,the duration t1π is changed to tnπ. So for a pulse tuned to give an ensemble of n atoms aπ rotation, which hits an ensemble of m atoms, the effective rotation angle is

θm0 (tnπ) =√m

∫ tnπ

0Ω0↔r(τ) dτ = π

√m

n(2.2.7)

For a an arbitrary rotation angle θ this becomes

θm0 (tnθ ) = θ

√m

n(2.2.8)

For a rotation without uncertainty in the number of atoms, m equals n, and the anglereduces again to

θm0 (tmθ ) = θ

√m

m= θ (2.2.9)

2.3 Fidelity

A rotation between a |0〉 state to a |1〉 state consists of a R0r rotation and a R1r rotation.Since there is no uncertainty in the number of atoms for a R1r rotation the pulse forthat rotation can be exactly defined and the fidelity will be one. For this reason, whencalculating the fidelity, a rotation between |0〉 and |1〉 states can be treated as a singlepulse and the state |r〉 can be left out of the basis states. This gives a rotation matrix

R =

(cos (θm0 (tnθ )/2) −i sin (θm0 (tnθ )/2)−i sin (θm0 (tnθ )/2) cos (θm0 (tnθ )/2)

)(2.3.1)

working on a qubit state

|ψ〉 = α|0〉+ β|1〉 =

(αβ

)(2.3.2)

The fidelity of a rotation R over an angle θ is then given by

F(θ) =∣∣〈ψ|R†ideal .Runcertain|ψ〉

∣∣2=∣∣〈ψ|R (θm0 (tmθ ))† .R (θm0 (tnθ )) |ψ〉

∣∣2=∣∣〈ψ|R(θ)†.R

(θ

√m

n

)|ψ〉∣∣2

=

∣∣∣∣(|α|2 + |β|2)

cos

[θ

2

(√m

n− 1

)]− iαβ∗ sin

[θ

2

(√m

n− 1

)]− iα∗β sin

[θ

2

(√m

n− 1

)]∣∣∣∣2 (2.3.3)

If the qubit starts out in a pure state, for which one of the coefficients α and β is 0and the other one is 1, the last two terms reduce to zero. This gives an expression forthe smallest possible value of the fidelity, and thus the highest possible error.

F(θ) ≥ cos2[θ

2

(√m

n− 1

)](2.3.4)

13


While the actual number of atoms in each qubit is undetermined, S. Whitlock et alshowed[1] that in our system, for a mean atom number between 50 and 300, a sub-Poissonian distribution with Fano factor1 f = 0.6 may be assumed for the actual atomnumber m. Therefore, all of the mean fidelity calculations in this thesis assume either aPoissonian or a sub-Poissonian distribution of actual atom number m around the meanatom number n.

The expected error for the rotation of an ensemble of atoms with a certain distributionin actual atom number is calculated by

1− 〈F〉 = 1−∑m

PD(n,m) · F(n,m) (2.3.5)

where PD(n,m) is the (sub-)Poisson distribution with mean atom number n and actualatom number m, and F(n,m) is the fidelity as defined in equation 2.3.3.To analyse the fidelity of single qubit operations, the maximum expected error wascomputed with the function FindMaximum in Mathematica. Findmaximum searches fora local maximum, starting with random coefficients α, β and the constraint |α|2 + |β|2 =1. FindMaximum[1− 〈F(n)〉] was computed while varying n, to determine the maximumexpected error as a function of mean atom number.

The maximum expected error as a function of mean atom number n is shown in figure2.2 for a Poisson distribution and a sub-Poissonian distribution with Fano factor 0.6.The ratio of the two is shown in figure 2.3. Finally, the expected error as a functionof the Fano factor for ensembles with mean atom number 10 is shown in figure 2.4.The figures show that the error increases with the Fano factor and decreases with atomnumber. For n = 10, the error of a π

2 pulse is 61% lower for sub-Poissonian distributedensembles.

1The Fano factor is a measure of the dispersion of a probability distribution and is defined as f = σ2

µ.

A Poisson distribution is a normal distribution with Fano factor 1. Distributions with a Fano factorbetween zero and one are sub-Poissonian.

14

2.3 Fidelity

1.0 10.05.02.0 3.01.5 15.07.0

0.01

0.02

0.05

0.10

0.20

n

1-<

F>

f = 0.6

Poisson

1.0 10.05.02.0 3.01.5 15.07.0

0.05

0.10

0.20

0.50

n

1-<

F>

f = 0.6

Poisson

Figure 2.2: The expected error of the rotation of a Poisson distribution(blue) comparedto a sub-Poissonian distribution with Fano factor 0.6(red). To the left for aπ2 -rotation and to the right for a π-rotation.

1.0 10.05.02.0 3.01.5 15.07.0

0.61

0.62

0.63

0.64

0.65

n

1-

<F

0.6

>

1-

<F

Pois

son

>

1.0 10.05.02.0 3.01.5 15.07.0

0.62

0.63

0.64

0.65

0.66

0.67

n

1-

<F

0.6

>

1-

<F

Pois

son

>

Figure 2.3: The ratio of the expected error of single rotations on a Poissonian distributedensemble and a sub-Poissonian distribution with Fano factor 0.6 (The ratioof the lines in figure 2.2). To the left for a π

2 -rotation and to the right for aπ-rotation.

Figure 2.4: The expected error of the rotation of a sub-Poissonian distribution with meanatom number 2 as a function of the Fano factor. To the left for a π

2 -rotationand to the right for a π-rotation.

15

3 Fidelity of CNOT Gates

In the previous chapter we analysed the fidelity of single qubit gates. Since a set ofuniversal gates can consist of a CNOT gate and some single qubit gates, it would beinteresting to discover the most error-free method of creating a CNOT gate. The truthtable for a CNOT gate is

CNOTINPUT OUTPUT|00〉 |00〉|01〉 |01〉|10〉 |11〉|11〉 |10〉

where the notation “|ψφ〉” refers to the control qubit A and the target qubit B being ina product state |ψ〉A ⊗ |φ〉B.

To compute the fidelity of a CNOT gate in our system, we would have to know how toincorporate the error factor

√mn in the rotation matrix of a CNOT gate. Since we know

how to incorporate the error factor in the rotation matrix of single qubit operations, wehave to find out how to build a CNOT gate starting from single qubit operations.

3.1 Creating CNOT gates from single qubit rotations

The Rydberg blockade mechanism blocks operations on qubits that are close to a qubitin the Rydberg state. One qubit in a Rydberg state, which blocks the application of aNOT gate on a neighboring qubit, can thus be used as the control qubit of a CNOT gate.A CNOT gate blocks the application of a NOT gate on the target qubit if the controlqubit is in the state |0〉, but in our system a qubit in the state |0〉 doesn’t block anything.So to make a control qubit we first need an operation that rotates qubits in the |0〉 stateto the |r〉 state and doesn’t affect qubits in the |1〉 state. Unfortunately, the rotationR0r(π) wouldn’t work here because of our defintion of the state |1〉: a qubit in the state|1〉 has one atom in the hyperfine excited ground state and the rest of it’s atoms in thelowest hyperfine ground state. R0r(π) would leave the one atom in the excited groundstate unaffected, but would still excite another atom from the lowest ground state tothe Rydberg state. So R0r(π) would always activate the Rydberg blockade, regardlessof the qubit state.

Without a fitting rotation it seems impossible to directly make a CNOT gate. Butan inverse CNOT gate, which only blocks the application of a NOT gate on the targetqubit if the control qubit is in the state |1〉, can be made: because the excited ground

17


state is only occupied for qubits in a state |1〉, a R1r(π) rotation will only activate theRydberg blockade for qubits in a state |1〉. The truth table for an inverse CNOT gate is

CNOT*

INPUT OUTPUT|00〉 |01〉|01〉 |00〉|10〉 |10〉|11〉 |11〉

This inverse CNOT gate can be turned into a standard CNOT gate by applying a NOTgate on the control qubit before and after it. Another way to make a CNOT gate is witha sequence of two Hadamard gates on the target qubit and in between a controlled-Zgate on both qubits. While the same problem arises for the controlled Z-gate, an inversecontrolled-Z gate can be made and can be turned into a standard controlled-Z gate byadding some extra single qubit gates.

Rotation schemes which show how all the needed gates are constructed from singlequbit rotations are shown below. The rotations are numbered and shown from top tobottom in the order in which they are applied. The resulting truth tables are shownnext to each rotation scheme.Note: The Hadamard gate H∗2 maps pure states to superpositions of states, which aredenoted by |0+〉 = 1√

2(|00〉+ |01〉) and |0−〉 = 1√

2(|00〉 − |01〉).

CNOT*

INPUT OUTPUT|00〉 |01〉|01〉 |00〉|10〉 |10〉|11〉 |11〉

H2*

INPUT OUTPUT|00〉 |0+〉|01〉 |0−〉|10〉 |1+〉|11〉 |1−〉

CZ*

INPUT OUTPUT|00〉 |00〉|01〉 −|01〉|10〉 −|10〉|11〉 −|11〉

18


NOT1

INPUT OUTPUT|00〉 |10〉|01〉 |11〉|10〉 |00〉|11〉 |01〉

NOT2

INPUT OUTPUT|00〉 |01〉|01〉 |00〉|10〉 |11〉|11〉 |10〉

NOT1,2

INPUT OUTPUT|00〉 |11〉|01〉 |10〉|10〉 |01〉|11〉 |00〉

Z1

INPUT OUTPUT|00〉 |00〉|01〉 |01〉|10〉 −|10〉|11〉 −|11〉

Z2

INPUT OUTPUT|00〉 |00〉|01〉 −|01〉|10〉 |10〉|11〉 −|11〉

19


Z1,2

INPUT OUTPUT|00〉 |00〉|01〉 −|01〉|10〉 −|10〉|11〉 |11〉

The rotation matrices for the gates are obtained by multiplying the rotation matricesfor the pulses that make up the gates in the right order and taking the tensor productbetween rotations on the first and second qubit. Starting again with the matrices forsingle qubit rotations between the lowest ground state or the excited ground state andthe Rydberg state

R0r(θ0) =

cos (θ0/2) 0 −i sin (θ0/2)0 1 0

−i sin (θ0/2) 0 cos (θ0/2)

(3.1.1)

R1r(θ1) =

1 0 00 cos (θ1/2) −i sin (θ1/2)0 −i sin (θ1/2) cos (θ1/2)

(3.1.2)

working again on a state

|ψ〉 = α|0〉+ β|1〉+ γ|r〉 =

αβγ

(3.1.3)

The formulas which define the gates in terms of these rotations are

CNOT∗ = RA1r(π) RB

1r(π) RB0r(π) RB

1r(π) RA1r(π) (3.1.4)

H∗2 = RB1r(π) RB

0r(π

2) RB

1r(π) (3.1.5)

CZ∗ = RA1r(π) RB

1r(2π) RA1r(π) (3.1.6)

NOT1 = RA1r(π) RA

0r(π) RA1r(π) (3.1.7)

NOT2 = RB1r(π) RB

0r(π) RB1r(π) (3.1.8)

NOT1,2 = RB1r(π) RB

0r(π) RB1r(π) RA

1r(π) RA0r(π) RA

1r(π) (3.1.9)

Z1 = RA1r(2π) (3.1.10)

Z2 = RB1r(2π) (3.1.11)

Z1,2 = RB1r(2π) RA

1r(2π) (3.1.12)

where RAx (θ) represents a rotation of qubit A with rotation matrix RA

x (θ) = Rx(θ)⊗ Iand RB

x (θ) represents a rotation of qubit B with rotation matrix RBx (θ) = I ⊗Rx(θ).

20


Since this is matrix multiplication, the rotations are applied from right to left.The resulting matrices have as a 9-dimensional basis the states

|ψ〉 = α00|00〉+α01|01〉+α0r|0r〉+α10|10〉+α11|11〉+α1r|1r〉+αr0|r0〉+αr1|r1〉+αrr|rr〉(3.1.13)

However, when taking a closer look at the rotation matrices it can be seen that initialstates which don’t involve a Rydberg state never map to a Rydberg state and vice versa.See for example the H∗2 matrix:

H∗2 =

1√2− 1√

20 0 0 0 0 0 0

− 1√2− 1√

20 0 0 0 0 0 0

0 0 −1 0 0 0 0 0 00 0 0 1√

2− 1√

20 0 0 0

0 0 0 − 1√2− 1√

20 0 0 0

0 0 0 0 0 −1 0 0 00 0 0 0 0 0 1√

2− 1√

20

0 0 0 0 0 0 − 1√2− 1√

20

0 0 0 0 0 0 0 0 −1

(3.1.14)

Initial states |00〉, |01〉, |10〉 and |11〉 are mapped to a superposition of states that don’tinvolve the Rydberg state. For this reason, it’s possible to leave the Rydberg states outof the basis and construct 4× 4 matrices working on states

|ψ〉 = α|00〉+ β|01〉+ γ|10〉+ δ|11〉 =

αβγδ

(3.1.15)

The final matrices for the gates are then

CNOT∗ =

0 1 0 01 0 0 00 0 1 00 0 0 1

(3.1.16)

H∗2 =1√2

1 −1 0 0−1 −1 0 00 0 1 −10 0 −1 −1

(3.1.17)

CZ∗ =

1 0 0 00 −1 0 00 0 −1 00 0 0 −1

(3.1.18)

Z1 =

1 0 0 00 1 0 00 0 −1 00 0 0 −1

(3.1.19)

Z2 =

1 0 0 00 −1 0 00 0 1 00 0 0 −1

(3.1.20)

Z1,2 =

1 0 0 00 −1 0 00 0 −1 00 0 0 1

(3.1.21)

21


NOT1 =

0 0 1 00 0 0 11 0 0 00 1 0 0

(3.1.22)

NOT2 =

0 1 0 01 0 0 00 0 0 10 0 1 0

(3.1.23)

NOT1,2 =

0 0 0 10 0 1 00 1 0 01 0 0 0

(3.1.24)

The Hadamard*, the CNOT* and the CZ*-gate differ from the standard definition bya few minus signs, but this can be corrected by adding a few single qubit gates in thefollowing way

CZ = −NOT1,2 CZ∗NOT1,2 = CZ∗ Z1,2 (3.1.25)

H2 = Z2 H∗2 Z2 (3.1.26)

CNOT = NOT1 CNOT∗NOT1 (3.1.27)

In the end there are three ways to construct a CNOT gate. By using the inverseCNOT gate, or by using the CZ- and Hadamard gates (which can be done in two ways)

CNOTa = H2 CZ H2 = Z2 H∗2 Z2 CZ∗ Z1,2 Z2 H∗2 Z2 (3.1.28)

CNOTb = H2 CZ H2 = −Z2 H∗2 Z2 NOT1,2 CZ∗NOT1,2 Z2 H∗2 Z2 (3.1.29)

CNOTc = NOT1 CNOT∗NOT1 (3.1.30)

By combining some of the gates this can be simplified to

CNOTa = H∗2 CZ∗ Z2 H∗2 (3.1.31)

CNOTb = −H∗2 NOT1 CZ∗NOT1 H∗2 (3.1.32)

CNOTc = NOT1 CNOT∗NOT1 (3.1.33)

These CNOT gates meet the standard definition of a CNOT gate, but the extra singlequbit gates add some uncertainty for every lowest ground state pulse that they contain.Therefore, it might be better to change the algorithm of a certain quantum computationto make direct use of the slightly different CNOT gates. To investigate the effect of theextra single qubit gates, two more CNOT gates are defined by

CNOTe = H∗2 CZ∗H∗2 (3.1.34)

CNOTf = CNOT∗ (3.1.35)

For every rotation R0r(θ) that involves the lowest ground state the uncertainty in atomnumber is again introduced by substituting θ → θ

√mn . The matrices with uncertainty

22

3.2 Fidelity of different implementations of a CNOT gate

then become dependent on n and m. For example, the H∗2 matrix becomes

H∗2(n,m) =

cos[π4

√mn

]− sin

[π4

√mn

]0 0

− sin[π4

√mn

]− cos

[π4

√mn

]0 0

0 0 cos[π4

√mn

]− sin

[π4

√mn

]0 0 − sin

[π4

√mn

]− cos

[π4

√mn

] (3.1.36)

The elements of the CNOT gates are all modified in this way to make the uncertainCNOT gates.

3.2 Fidelity of different implementations of a CNOT gate

The fidelity of a certain CNOTx gate is given by

Fx =∣∣〈ψ|CNOT†x,ideal .CNOTx, uncertain|ψ〉

∣∣2=∣∣〈ψ|CNOT†x .CNOTx(n,m) |ψ〉

∣∣2 (3.2.1)

The mean or expected error is given by

1− 〈Fx〉 = 1−∑m

PD(n,m) · Fx(n,m) (3.2.2)

where PD(n,m) is again a (sub-)Poisson distribution with mean atom number n andactual atom number m. The input state with the maximum error was again automati-cally selected by using the function FindMaximum.

The maximum expected errors of the three different CNOT gates as a function of meanatom number are compared in figure 3.1 on page 24. The error is again lower for a sub-Poissonian distribution: f = 0.6 reduces the error by a factor 0.6 (for n = 10). Since theCNOTa gate shows the highest fidelity, this gate is compared to the two non-standardCNOTe and CNOTf gates in figure 3.2 on page 24. Apparently the CNOTa, CNOTe

and CNOTf gate have an equal fidelity as function of mean atom number.

23


1.0 10.05.02.0 20.03.01.5 15.07.0

0.02

0.05

0.10

0.20

0.50

n

1-<

F>

CNOTc subPoisson

CNOTb subPoisson

CNOTa subPoisson

CNOTc Poisson

CNOTb Poisson

CNOTa Poisson

Figure 3.1: The maximum expected error of the three CNOT gates applied on a Poissondistribution and a normal distribution with Fano factor 0.6, as a function ofmean atom number n.

Figure 3.2: The maximum expected error of the CNOTa, CNOTe and CNOTf gates ona Poisson distributed ensemble of qubits as a function of mean atom numbern.

24

3.3 Comparing effects on pure states

3.3 Comparing effects on pure states

To further investigate the difference between the different CNOT gates, the expectedprobability to get a certain output state with a certain input state is plotted in figure3.3 on page 26. This expected probability was calculated as

〈Px〉 =∑m

PD(n,m) · 〈ψ|CNOTx(n,m) |φ〉 (3.3.1)

for an input state |ψ〉 and an output state |φ〉. The probabilities were computed for anensemble of only two atoms because the higher error makes it easier to see the differencebetween the gates.

Again, the CNOTa gate shows the highest probabilities. But apparently there’s an-other difference between the CNOTa gate and the other CNOT gates. For the CNOTa

gate input states |00〉 and |01〉 stay totally unaffected and states |10〉 and |11〉 have acertain probability not to be affected, while both the CNOTb and the CNOTc gate havea probability to give each output state for all four input states. Since the CNOTa gate isthe only gate without single qubit NOT gates, the NOT gate might be the cause of thisbehavior. The expected probability to get certain output states after applying a NOTgate is shown in figure 3.4.

As could be expected, figure 3.4 shows that a NOT gate simply has a small probabilityto leave the qubit unaffected for all four input states. This is not too much of a surprise,but it does explain our earlier discovery in a certain way. For the CNOTb and CNOTc

gates, there is a small probability that the first NOT gate doesn’t invert the controlqubit, after which the inverted CZ∗ or CNOT∗ gate works on the target qubit while itshould have left the target qubit unaffected.

We can’t conclude much more from figures 3.3 and 3.4 because they only show whathappens to pure states. But the fact that the CNOTa gate leaves the two states |00〉and |01〉 totally unaffected gives this gate a big advantage over the other gates. TheCNOTa gate can be considered the ideal CNOT gate for the states |00〉 and |01〉.

25


È00>

È01>

È10>

È11>

InputÈ00>

È01>

È10>

È11>

Output

0.0

0.2

0.4

0.6

0.8

<P>

(a) CNOTa

È00>

È01>

È10>

È11>

InputÈ00>

È01>

È10>

È11>

Output

0.0

0.2

0.4

0.6

<P>

(b) CNOTb

È00>

È01>

È10>

È11>

InputÈ00>

È01>

È10>

È11>

Output

0.0

0.2

0.4

0.6

<P>

(c) CNOTc

Figure 3.3: The expected probabilities of the CNOT gates on a Poisson distributed en-semble of qubits with mean atom number 2. For each pure state as inputthe probability to get any pure state as output is shown.

È00>

È01>

È10>

È11>

InputÈ00>

È01>

È10>

È11>

Output

0.0

0.2

0.4

0.6

<P>

Figure 3.4: The expected probabilities of a NOT gate on a Poisson distributed ensem-ble of qubits with mean atom number 2. For each pure state as input theprobability to get any pure state as output is shown.

26

4 Improving the fidelity with the use ofcomposite pulses

In the previous chapter we have seen that H∗2CZ∗Z2H∗2 is the sequence of gates that forms

the CNOT gate with the highest fidelity. To further increase this fidelity we turn to thecomposite pulse technique used in NMR quantum computing and medicine.

4.1 Composite pulses in NMR

A composite pulse is a sequence of consecutive, phase-shifted radiofrequency pulses. Thecomposite pulse emulates the effect of a single radiofrequency pulse, but has a mechanismwhich makes it less sensitive to some sort of pulse imperfection. The concept has beenwidely used in NMR ever since it was invented in 1978 by Malcolm Levitt1.

A common source of pulse imperfections in NMR is a magnetic field strength thatis not uniform over the whole sample, which leads to pulse length errors: The pulseduration will be tuned to the nominal magnetic field strength to achieve the desiredrotation. At places where the magnetic field strength is below it’s nominal value, thepulse duration will then be too short and at places where the magnetic field strengthis higher, the pulse duration will be too long. The result is a spread in the achievedrotation angle.In our system, the source of pulse imperfections is the undetermined number of atomsin each qubit. The pulse duration will be tuned to the mean atom number and will betoo long for qubits with a higher than average atom number, and too short for qubitswith a less than average atom number. So fluctuations in the magnetic field in NMRare analog to fluctuations in qubit atom number in our system.

The individual rotations that make up a composite pulse are denoted by θφ, where θ isthe rotation about a rotation axis ex cosφ+ey sinφ. Figure 4.1 shows how the geometri-cal mechanism of a 909018009090 composite pulse counters the spread of a 180 rotation.The trajectories of different unit vectors, that correspond to different magnetic fieldstrength values, are tracked during the pulse sequence. All shown vectors start alongthe z axis and correspond to magnetic field strength values weaker than the nominalvalue to which the rotations are tuned. The mechanism that counters the rotation dif-ference in this case is hidden in the 1800 pulse: this pulse rotates the vectors with thesmallest field strength values closest to the -z axis and thereby inverses the differences(with respect to the -z axis). Together with the last pulse, this leaves all the vectorsaligned along the -z axis with only a small difference in phase.

1For an overview of composite pulses used in NMR, see M.H. Levitt, ”Composite Pulses”, Encyclopediaof Nuclear Magnetic Resonance, vol. 2, p.1396.

27

4 Improving the fidelity with the use of composite pulses

The 909018009090 pulse only works for the 180 rotation of a vector that starts alongthe z axis and it generates a small phase distortion, but there are many more types ofcomposite pulses. Since the starting states of qubits are usually not known in the middleof complex quantum computations, we are only interested in fully compensating pulsesequences, which work for any initial state.

S. Wimperis described several ”phase distortionless” pulse sequences in which the over-all rotation angle remains a variable and for which the initial state doesn’t matter[2]. Hedemonstrated three basic types of these composite pulses: the narrowband, broadbandand passband type. A simple pulse would give a desired rotation to a nominal range ofmagnetic field strength values. The narrowband type composite pulse would give thedesired rotation to a narrower range, the passband to a somewhat broader range andthe broadband to a much broader range of magnetic field strength values. The differenttypes all have their advantages, but since we would like a high fidelity for as wide arange (of the actual atom number distribution around the mean atom number of thequbits) as possible, we will be using the broadband composite pulse Wimperis referredto as BB1.

BB1(θ) = 180φ1360φ2180φ1θ0 (4.1.1)

with φ1 = cos−1(−θ/720) and φ2 = 3φ1.

Another suitable composite pulse was derived by H.K. Cummins et al[3]. He modeledsingle-qubit gates, which are simply rotations on the Bloch sphere, by quaternions2. Eachfaulty pulse would be modeled by a quaternion with an error factor incorporated intoit. By taking the product of three quaternions and comparing it to a single quaternionwithout the error factor (representing the ideal rotation) he managed to derive relationsfor the six variables (three for the rotation angles and three for the phases) which coun-tered the errors. By adapting the form of the error factor the composite pulse could betuned to counter various pulse imperfections. He did this for off-resonance errors (errorswhere the pulse would be slightly out of resonance with the relevant transition, resultingin a rotation about some tilted axis) and for pulse-length errors. For pulse-length errorshe found a sequence to which he referred to as ”a short composite rotation for undoinglength over and under shoot” or SCROFULOUS sequence.

SCR(θ) = θ1,φ1180φ2θ3,φ3 (4.1.2)

2A quaternion is a four-dimensional vector where the first coordinate is a real number and the other threecoordinates form the imaginary coordinates of a three-dimensional vector. Because the imaginaryvector part of a quaternion is a vector in R3, the geometry of R3 is reflected in the algebraic structureof quaternions. A rotation can thus be modeled by a quaternion qθφ = s,v where s = cos(θ/2)depends only on the rotation angle θ and v = sin(θ/2)a depends on both the rotation angle θ and aunit vector along the rotation axis a.

28

4.2 Using composite pulses in single qubit rotations

where

θ1 = θ3 = arcsinc

(2 cos(θ/2)

π

)φ1 = φ3 = arccos

(−π cos θ1

2θ1 sin(θ/2)

)φ2 = φ1 − arccos(−π/2θ1)

and the function sinc(x) is defined as sinc(x) = sin (πx)πx .

Figure 4.1: The geometrical mechanism of a 9090 1800 9090 composite pulse. Source: [4].


To compute the fidelity of a BB1 or SCR pulse sequence the rotation matrix (equation2.3.1) has to be adapted to include phase rotations. Physically, rotating with a certainphase φ means waiting untill the quantum system has evolved over a time t = ~

Eφ

and picked up a phase factor e−iE~ t = e−iφ. Mathematically, the rotation matrices we

used before were constructed from the rotation operator about the x axis[6]. A rotationmatrix about an arbitrary axis can be constructed by adding two matrices: a phasematrix that shifts the x axis to the desired rotation axis before the rotation and a phasematrix that shifts it back after the rotation (otherwise the next rotation will start withan offset in phase). A Phase gate matrix is given by

P(φ) =

(e−i

φ2 0

0 eiφ2

)(4.2.1)

29


Note that this defintion has an extra global phase factor e−iφ2 with respect to the defini-

tion in the introduction. This doesn’t matter though, since a global phase factor doesn’tproduce any measurable effects. For an explanation of this, see [6].The new rotation matrix then becomes

T

(θ

√m

n, φ

)= P(−φ).R

(θ

√m

n

).P(φ)

=

(eiφ2 0

0 e−iφ2

).

(cos(θ2

√mn

)−i sin

(θ2

√mn

)−i sin

(θ2

√mn

)cos(θ2

√mn

) ) .(e−iφ2 0

0 eiφ2

)

=

(cos(θ2

√mn

)−ieiφ sin

(θ2

√mn

)−ie−iφ sin

(θ2

√mn

)cos(θ2

√mn

) )(4.2.2)

Note that there is no uncertainty added to the phase angle φ. This is because all theatoms evolve in the same way. They will all pick up the same phase factor e−iφ after atime t = ~

Eφ, independent of the number of atoms in a qubit. The fidelity of a certainpulse is again computed by

F(θ, φ) =∣∣〈ψ|T (θ, φ))† .T

(θ

√m

n, φ

)|ψ〉∣∣2 (4.2.3)

In figure 4.2, the fidelity of a simple π2 pulse as a function of actual atom number m,

with mean atom number n = 20, is compared to the fidelity of a BB1 and a SCR π2

pulse. The composite pulses clearly show a high fidelity for a much broader range thana simple pulse.

The maximum expected error as a function of mean atom number n for a simple π2

and π pulse is compared to the maximum expected error of the corresponding BB1 andSCR pulses in figure 4.3. The function FindMaximum was again used to select the inputstate with the highest error.

0 20 40 60 80 1000.0

0.2

0.4

0.6

0.8

1.0

m

F

Normal distr.

SCR

BB1

simple

Figure 4.2: The fidelity of a simple, a BB1 and a SCR π2 pulse as a function of actual

atom number m, with a mean atom number n = 20. There is a Poissondistribution plotted around the mean atom number for clarity.

30


For both a π2 and a π pulse, the decrease in error as a function of mean atom number

is much faster for composite pulses. For higher atom numbers, the BB1 pulse performssignificantly better than the SCR pulse.

1.0 10.05.02.0 20.03.01.5 15.07.0

0.01

0.02

0.05

0.10

0.20

0.50

n

1-

<F> SCR

BB1

simple

(a) for a π2

pulse

1.0 10.05.02.0 20.03.01.5 15.07.0

0.01

0.02

0.05

0.10

0.20

0.50

n

1-

<F

>

SCR

BB1

simple

(b) for a π pulse

Figure 4.3: The maximum expected error as a function of mean atom number n forsimple, BB1 and SCR π

2 and π pulses.

31


4.2.1 Using composite pulses to improve a Ramsey interferometer

Another application of composite pulses that might be interesting is using it to improve aRamsey interferometer, which is used in atomic clocks. In an atomic clock, an ensembleof Cs-133 atoms in the ground state is subject to two π

2 pulses, separated by a certaintime during which the quantum state of the atoms is allowed to evolve. After the firstπ2 pulse, the atoms will start to oscillate between states with a frequency that is uniqueto the energy (and type) of the atoms. The probability of finding the atoms in a certainstate after the second π

2 pulse depends on the time between the two pulses. By varyingthe time and doing a lot of measurements, the frequency of the oscillation can be found.An atomic clock uses this frequency as a very accurate measure of time3. Because a shiftin energy level changes the oscillation frequency, we could use a Ramsey interferometerin our system to detect interference. This could be electromagnetic fields, nearby atomsin a Rydberg state, or anything else that shifts the energy levels of the atoms in a qubit.An important aspect of a Ramsey interferometer is the contrast of the oscillation. Thehigher the contrast, the easier it is to detect any change. It’s interesting to find out ifthe use of composite pulses will enhance the contrast of a Ramsey interferometer in oursystem.

However, there is one big difference between a standard Ramsey interferometer andour system. In a standard Ramsey interferometer there is no Rydberg blockade, so aπ2 pulse puts the whole ensemble in a superposition of states. As a consequence, whenall the atom states are measured, a distribution of atoms over states is found. Thisdistribution is used to tune the time between two pulses and is thus needed to find theoscillation frequency (and thereby the energy difference). In our system the Rydbergblockade blocks the excitation of more than one atom, so only one atom is put in asuperposition of states. When the state of this one atom is measured, we can only speakabout a probability and not about a distribution. This makes it much harder to use oursystem as a Ramsey interferometer. We could forget about the Rydberg state and exciteatoms directly to the |1〉 state, but then we would have to develop a new expression forthe error factor. So we will look at single-atom excitations from the |0〉 to the |1〉 state,via the Rydberg state (even though we don’t know if it’s useful). Effectively, we aredealing with π

2 rotations between the |0〉 and |1〉 state. Between the two π2 rotations,

the atoms are in a superposition of the states |0〉 and |1〉 and are allowed to evolve for atime t = ~

Eφ. Depending on this time, the second π2 rotation will either put the atoms

in the |0〉 or the |1〉 state. If the state of the atoms is denoted by a vector

|ψ〉 = α|0〉+ β|1〉 =

(αβ

)(4.2.4)

and the matrix that performs the pulses and lets the system evolve during a time t = ~Eφ

is given by

G(φ, n,m) = T

(π

2

√m

n, 0

).

(e−i

φ2 0

0 eiφ2

).T

(π

2

√m

n, 0

)(4.2.5)

3The second is in fact defined as 9192 631 770 cycles of the standard Cs-133 transition.

32


Then the probability of finding the atoms of Poisson distributed ensembles in the |0〉state after the two pulses is given by∣∣α(φ, n)

∣∣2 =∑m

PD(n,m) ·∣∣〈0|G(φ, n,m)|0〉

∣∣2 (4.2.6)

This probability is shown as a function of the phase φ in figure 4.4. The contrast of theoscillation is calculated by

contrast =max−min

max + min(4.2.7)

The contrasts of a simple pulse and a BB1 composite pulse as a function of mean atomnumber n are shown in figure 4.5. Using a BB1 pulse clearly improves the contrast,especially for lower atom numbers.

0 Π 2 Π 3 Π 4 Π 5 Π 6 Π 7 Π 8 Π 9 Π 10 Π

0

1

Φ

ÈΑ2

BB1

simple

Figure 4.4: The probability of finding an ensemble of qubits with a mean atom numbern = 20 in the ground state after applying two π

2 pulses, as a function of thephase φ, which the atoms are allowed to pick up between the pulses.

0 20 40 60 80 100

0.92

0.94

0.96

0.98

1.00

n

Con

tras

t BB1

simple

Figure 4.5: The contrasts of the functions shown in figure 4.4 as a function of mean atomnumber.

33


4.3 Using composite pulses in CNOT gates

To study the effect of composite pulses on CNOT gates, the rotation matrices need tobe redefined to include a phase.

R0r(θ0, φ0) =

cos(θ02

√mn

)0 −ieiφ0 sin

(θ02

√mn

)0 1 0

−ie−iφ0 sin(θ02

√mn

)0 cos

(θ02

√mn

) (4.3.1)

R1r(θ1, φ1) =

1 0 00 cos (θ1/2) −ieiφ1 sin (θ1/2)0 −ie−iφ1 sin (θ1/2) cos (θ1/2)

(4.3.2)

The simple CNOT gates are then made in the same manner as in the previous chapter,with phases φ = 0. The Composite CNOT gates are made by replacing every rotationthat involves the lower ground state with the rotation matrix for a composite pulse. Forexample, the sequence of rotations that defines a simple NOT1 gate is

NOT1 = [R1r(π, 0) R0r(π, 0) R1r(π, 0)]⊗ I (4.3.3)

and the sequence that defines a BB1 composite NOT1 gate is

NOTBB11 = [R1r(π, 0)BB1(π, 0) R1r(π, 0)]⊗ I

= [R1r(π) R0r(π, φ1) R0r(2π, φ2) R0r(π, φ1) R0r(π, 0) R1r(π)]⊗ I (4.3.4)

where φ1 = cos−1(−π/4π) and φ2 = 3φ1. The other BB1 and SCR gate matrices arecreated in the same way.

The fidelity as a function of actual atom number m of a simple CNOTa gate iscompared to fidelity of the corresponding composite gates in figure 4.6. The maximumexpected errors of the simple CNOT gates and the composite CNOT gates as a functionof mean atom number are compared in figure 4.7. The figure shows that for a mean atomnumber of 5, using BB1 composite pulses instead of simple pulses reduces the error by afactor 0.13. This factor decreases rapidly with atom number: further calculations showthat for a mean atom number of 20 the error is reduced by a factor 0.007. Apparently,composite pulses greatly improve the fidelity of CNOT gates.

34

4.3 Using composite pulses in CNOT gates

0 20 40 60 80 1000.0

0.2

0.4

0.6

0.8

1.0

m

F

Normal distr.

SCR

BB1

simple

Figure 4.6: The fidelity of a simple, a BB1 and a SCR CNOTa gate as a function ofactual atom number m, with a mean atom number n = 20. There is aPoisson distribution plotted around the mean atom number for clarity.

1.0 10.05.02.0 3.01.5 15.07.0

0.02

0.05

0.10

0.20

0.50

n

1-

<F> SCR

BB1

simple

Figure 4.7: The maximum expected errors of CNOTa gates using simple, BB1 and SCRpulses as a function of mean atom number.

35

Discussion

For single qubit gates the error is apparently lower for a sub-Poissonian distribution ofatom number and increases with the Fano factor (when the Factor comes closer to one,the distribution comes closer to a Poisson distribution). The error of a π

2 pulse on a sub-Poisson distributed ensemble with mean atom number 10 and Fano factor 0.6 is 61% ofthe error of a π

2 pulse on a Poisson ditributed ensemble. This was to be expected becausefor a narrower distribution, there will be more qubits with an actual atom number mclose to the mean atom number n, which makes the average shift in rotation anglesmaller (∆θ =

∣∣θ − θ√mn

∣∣). For the different CNOT gates (equations 3.1.31 through3.1.33) it’s nice to see that, as would be expected, the gate which contains the leastamount of pulses that involve the |0〉 state has the highest fidelity. This would suggesta relation between the fidelity of a CNOT gate and the number of |0〉 state involvingpulses. Surprisingly though, the two non-standard CNOT gates (equations 3.1.34 and3.1.35) showed no higher fidelity than the CNOTa gate. The difference between theCNOTa and the CNOTe gate is only a Z2 gate, which does not involve the |0〉 state.That would explain why the CNOTe gate doesn’t have a higher fidelity. However, theCNOTf gate has a lower amount of |0〉 state involving pulses and so a higher fidelitythan the related CNOTc gate, but apparently not higher than the CNOTa gate. This isa bit unexpected and would contradict the theory that there is a direct relation betweenthe fidelity and the number of |0〉 state involving pulses because the CNOTa gate hastwo |0〉 state involving pulses, whereas the CNOTf gate has only one.

The use of composite pulses to increase the fidelity works apparently quite well. For aπ2 pulse on Poisson distributed ensemble with mean atom number 20, the best compositepulse(BB1) decreases the error by a factor 0.007. Using BB1 composite pulses in theCNOTa gate on a Poisson distributed ensemble with mean atom number 20 also reducesthe error by a factor 0.007. In this thesis I only tested composite pulses that wereoriginally derived for use in NMR, but deriving a new sequence would result in a similarcomposite pulse since the mathematics of both two-state systems are much alike. Thefidelity as a function of actual atom number (figure 4.2) is clearly broadened by thecomposite pulses. For the composite CNOT gates the fidelity is also broadened, althoughless than for single qubit pulses. The BB1 composite pulse shows the highest fidelity inall cases, but the difference with a SCR composite pulse is significantly smaller for a πpulse than for a π

2 pulse or a CNOTa gate. It would be interesting to further study thereason for this difference in fidelity of a BB1 pulse and a SCR pulse. The BB1 compositepulse also improves the contrast of a Ramsey interferometer. For a Poisson distributedensemble with mean atom number 20, the contrast is improved by a factor 1.06.

The maximum expected errors in this thesis are computed using a method thatsearches for a local maximum. There might be a higher global maximum, which would

37


not be found with this method. The shown maximum expected errors might thus notbe entirely realistic, although nonrealistic maxima should lead to jumps in the graphs,and those are not seen. Nevertheless, these error values could be improved by using adifferent computation method or by solving the functions analytically.

Finally, it would be interesting to expand figure 3.3 on page 26 in some way to showthe effects of the CNOT gates on non-pure states.

38

Conclusion

For single qubit pulses, the maximum expected error decreases with the qubit mean atomnumber and increases with the Fano factor of the qubit atom number distribution. Theerror of a π

2 pulse is comparable to the error of a π pulse. The typical error reductionfactor for ensembles with a mean atom number 10 due to a decrease in Fano factor fromf = 1 to f = 0.6 is 0.61. For a π pulse this reduction factor is 0.62. For a CNOT gate,the maximum expected error appears to increase with the number of |0〉 state involvingpulses that it contains: applied to a Poisson distributed ensemble with a mean atomnumber 20, the CNOTa gate has only 23% of the eror of a CNOTb and 20% of the errorof a CNOTc gate. Again, the error decreases with atom number and increases withFano factor: for ensembles with a mean atom number 10 and a decrease of Fano factorfrom f = 1 to f = 0.6 the error reduction factor is 0.62. The CNOT gate with thelowest error is made up of the sequence of gates CNOTa = H∗2CZ∗Z2H

∗2. Two additional

non-standard CNOT gates were defined without the single qubit gates to correct for theminuses, but they do not show a lower error. An additional advantage of the CNOTa

gate and the two non-standard CNOT gates is that the CNOTa gate leaves the purestates |00〉 and |01〉 untouched and the two non-standard CNOT gates leave the states|10〉 and |11〉 untouched. This makes these three gates ideal gates for two of the fourpure states.

The BB1 and SCR composite pulses, which were originally developed for use in NMR,improve the fidelity of single pulses and CNOT gates significantly. The BB1 compositepulse shows the highest fidelity, although the difference in fidelity of a BB1 and a SCRpulse depends on the rotation angle. The difference is larger for a π

2 pulse than for a πpulse. The typical error reduction factor when replacing a simple π

2 pulse (on a Poissondistributed ensemble with mean atom number 20) with a BB1 composite pulse is 0.007.When replacing this simple pulse with a SCR composite pulse, the error reduction factoris 0.23. By replacing |0〉 state involving pulses with composite pulses in CNOT gatesthe fidelity of these gates can also be improved. The typical error reduction factor forreplacing a simple CNOTa gate (applied on a Poisson distributed ensemble with meanatom number 20) with a BB1 CNOTa gate is also 0.007. Composite pulses can also beused to improve the contrast of a Ramsey interferometer, although it is doubtful if thiskind of interferometer can be used (since only a probability can be measured instead ofa distribution).

39

Acknowledgements

I would like to thank Atreju Tauschinsky and Robert Spreeuw for their daily support,helpful papers, suggestions and feedback.

41

Bibliography

[1] S. Whitlock, C.F. Ockeloen, R.J.C. Spreeuw, “Sub-Poissonian Atom-Number Fluc-tuations by Three-Body Loss in Mesoscopic Ensembles”, Physical Review Letters,March 2010, 104,120402.

[2] S. Wimperis, “Broadband, Narrowband, and Passband Composite Pulses for Usein Advanced NMR Experiments”, Journal of Magnetic Resonance, 1994, Series A109, 221-231.

[3] H.K. Cummins, G. Llewellyn, J.A. Jones, “Tackling systematic errors in quan-tum logic gates with composite rotations”, Physical Review Letters A, April 2003,67,042308.

[4] M.H. Levitt, “Composite Pulses”, Encyclopedia of Nuclear Magnetic Resonance,vol. 2, 1996, p.1396.

[5] L.H. Pedersen, “Neutral Atom Quantum Computing with Rydberg Blockade”, July2008, PhD thesis University of Aarhus, Denmark.

[6] M.A. Nielsen, I.L. Chuang, “Quantum Computation and Quantum Information”,Cambridge University Press, 2004.

[7] B.S. Rem, “Fidelity of mesoscopic atomic ensemble qubits”, August 2009, BachelorThesis University of Amsterdam, The Netherlands.

[8] D.P. DiVincenzo, “The Physical Implementation of Quantum Computation”, Ex-perimental Proposals for Quantum Computation, February 2008.

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