digital quantum simulations of spin models in superconducting circuits · 2018-09-19 · master in...

Master in Quantum Science and Technology

Digital Quantum Simulations

of Spin Models

in Superconducting Circuits

Urtzi Las Heras

Advisor:

Prof. Enrique Solano

Department of Physical ChemistryFaculty of Science and Technology

University of the Basque Country UPV/EHU

Leioa, September 2013

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Acknowledgments

I would like to thank

Prof. Enrique Solano for giving me the opportunity to work with theQUTIS group, where I have been educated but also filled with passion forphysics.

Dr. Lucas Lamata for o↵ering me his time, his patience, and what ismore important, his great help as he did already the previous year during myBachelor Thesis.

M. Sc. Antonio Mezzacapo for wonderful discussions that had given methe way to solve all my questions.

Dr. Guillermo Romero for his big e↵ort sharing with me his huge knowl-edge.

Dr. Mikel Sanz for his amazing generosity helping me with the best smileeverytime I needed.

and the rest of QUTIS members for their support and for making this afantastic environment.

I would also like to thank my family and all my friends, who have alwaysbeen backing me with their encouragement and who make me smile at the endof the day.

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http://sites.google.com/site/enriquesolanogroup/

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Abstract

We propose a digital quantum simulation of spin chains with state-of-the-art superconducting circuits. Making use of a suitable gate decomposition, weshow that it is feasible to implement relevant spin chain models with currenttechnology. We discuss the gate fidelities needed in an experimental set up toachieve this goal. The present work paves the way for the implementation ofdigital quantum simulation methods in circuit quantum electrodynamics.

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Contents

Abstract iii

Contents v

1 Introduction 1

2 Spin systems 5

2.1 Multipole expansion . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Spin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Spin models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3.1 Ising model . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3.2 Heisenberg model . . . . . . . . . . . . . . . . . . . . . . 13

3 Circuit Quantum Electrodynamics 15

3.1 Josephson e↵ect . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2 Superconducting qubits . . . . . . . . . . . . . . . . . . . . . . . 17

3.2.1 Cooper pair box . . . . . . . . . . . . . . . . . . . . . . . 18

3.3 Transmission line resonator . . . . . . . . . . . . . . . . . . . . 21

3.4 DiVincenzo criteria . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.5 Exchange gate . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4 Quantum simulators 29

4.1 Analog and digital methods . . . . . . . . . . . . . . . . . . . . 30

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5 Simulating spin models in superconducting circuits 35

5.1 XYZ Hamiltonian (2-qubit case) . . . . . . . . . . . . . . . . . . 36

5.2 XYZ Hamiltonian (3-qubit case) . . . . . . . . . . . . . . . . . . 39

5.3 XX Hamiltonian with frustration (3-qubit example) . . . . . . . 42

5.4 Error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

6 Conclusions 51

Bibliography 53

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Chapter 1

Introduction

On the late 1940’s, the first computers were developed, among other places, inthe basements of universities. These machines have been improved every year,making their processors extremely small and powerful. However, even today,making use of the fastest classical computers, there exist numerical problemsthat are not solvable in a finite time. One of these comes from the complexityof quantum mechanical systems. The state of a quantum system is defined by awave function that lies in a vector space whose dimension grows exponentiallywith the number of particles. In this way, only to calculate the evolution ofa quantum system composed of a few particles, implies huge computationalresources.

Quantum information [1] is a high-impact field that studies quantum pro-cessing and quantum transfer of information encoded in quantum bits. Thecontrol of atomic systems allows not only to study their properties but also touse them combined with the information theory [2]. Moreover, quantum prop-erties such as linear superposition or entanglement will make possible in thefuture to realize communication and computation protocols unfeasible froma classical point of view, for instance, quantum teleportation [3], quantumcryptography [4, 5] and quantum simulations [6, 7, 8].

Quantum simulation is currently one of the most successful disciplines in-side quantum information science. This consists in reproducing the behaviorof a quantum system in another with a higher controllability. In this way, itis possible to accomplish experiments where the behavior of a system, whichis impossible to measure in the lab, is observed making use of another sys-tem that follows the same evolution. In addition, quantum simulators are notrestricted to systems that share the same dynamics. Applying digital meth-

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ods [9, 10, 11], it is possible to make the system evolve as another one with adi↵erent dynamics. These methods are based on the application of a series oflogical gates.

The success of quantum simulations is due to the power for solving prob-lems in comparison with classical simulations of quantum models. In fact,just by using about 40 qubits, it would be possible to solve problems in timesexponentially shorter than the calculations made by the fastest classical com-puters [12]. Until today, several proposals and experiments have been made inthis field, using several quantum platforms, namely, trapped ions [13, 14, 15],cold atoms [16], nuclear spins [17], quantum dots [18], photonic systems [19]or superconducting circuits [20]. With these, it has been possible to mimic allkinds of dynamics: spin chains [21], quantum phase transitions [22], relativisticquantum mechanics [23], quantum field theories [24], or bosonic and fermionicmodels [25] among others. Consequently, quantum simulation has become oneof the highest-impact fields inside quantum information.

Superconducting circuits [26, 27] are one of the most studied quantumtechnologies that have been employed in the field of quantum information un-til today. Actually, they receive the name of artificial atoms because eventhough they have a length of micrometers, they are able to reproduce behav-iors proper of microscopic particles. Furthermore, this quantum technologyhas been dramatically improved during the last years [28], until having beenconsidered to be the best candidate for building a universal quantum com-puter [29]. Basic quantum algorithms [30] and fundamental tests of quantummechanics [31] have been already realized. Single and two-qubit gates [32],preparation of complex entangled states [33], and basic protocols for quantumerror correction [34] are among the quantum information tasks that can beperformed nowadays with good fidelities. As a result, circuit Quantum Elec-trodynamics (cQED) is one of the best options to make near future proposalsin quantum simulations.

In this Thesis, we first introduce in the Chapter 2 a short review of spinsystems [38, 40], going from its discovery in the experiment of O. Stern andW. Gerlach [41] until the models studied nowadays in quantum information.We focus in the Ising and Heisenberg spin models, which are two basic, andvery useful at the same time, spin models.

Circuit Quantum Electrodynamics is reviewed in Chapter 3. In this sec-tion, we explain the Josephson e↵ect [35], which is necessary in order to builta qubit from a superconducting circuit. We also show the properties of thisquantum system and we analyze which are the requirements needed for quan-

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tum computing. Finally, basic quantum gates that will be used in our proposalare studied.

In Chapter 4, we explain the basics of quantum simulators. Here, we showwhich the objective of quantum simulations is and we collect the advantagesof using quantum methods in front of classical simulations. The conditionsthat should be satisfied a by quantum system in order to realize a quantumsimulation are reviewed [65]. Finally, the principles of digital methods appliedto quantum simulation are explained.

In Chapter 5, by using the information reviewed in the previous chapters,we explain the main result of this Thesis: our proposal for the quantum simu-lation of spin chains with digital methods in superconducting circuits. Makinguse of a suitable gate decomposition studied in the third chapter, we analyzethe simulation of the Heisenberg model for two and three qubits and the frus-trated transverse field Ising model for three qubits. In other words, we makethe superconducting qubits behave like particles with spin 1/2. Given thatthe dynamics of quantum circuits sand spins is not the same, we apply digi-tal methods for changing quantum circuit natural dynamics. We discuss thefidelity needed in an experimental setup in order to achieve the experiment.Furthermore, we show how to introduce manually errors due to the imperfectimplementation of gates in a real lab.

Finally, we present our conclusions in Chapter 6.

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Chapter 2

Spin systems

In this chapter the main features of spin systems are reviewed. We startby explaining their classical origin as a magnetic dipole-dipole interaction,by making a multipolar analysis of the electromagnetic field generated by alocalized system of charges and currents. We review the discovery of the spinin the Stern-Gerlach experiment, as well as the impossibility of considering itan internal rotation of the particle, and we collect the basic properties shownby these systems. Finally, we introduce the best-known spin models: the Isingand the Heisenberg Hamiltonians.

2.1 Multipole expansion

In this section, we discuss the radiation of localized oscillating sources [37].Maxwell’s equations in free space are given by:

r · E =⇢

"0,

r · B = 0,

r ⇥ E = �@B

@t,

r ⇥ B = µoJ+ µ0"0@E

@t. (2.1)

where ⇢ and J are the charge and current distributions. Let us consider alocalized system of charges and currents varying in time. We can assumew.l.o.g. a sinusoidal behavior in time, since we can always perform a Fourier

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decomposition and handle separately each component. Therefore,

⇢(xxx, t) = ⇢(xxx)e�i!t,

J(xxx, t) = J(xxx)e�i!t. (2.2)

For the sake of simplicity, we make use of the vector potential A(xxx, t) whichdescribes both the electric and magnetic fields. This is defined by:

A(xxx, t) =µ0

4⇡

Zd3x0

Zdt0

J(xxx0, t0)

|xxx � xxx0| �✓t0 +

|xxx � xxx0|c

� t

◆. (2.3)

By replacing Eq. (2.2) in Eq. (2.3), A becomes

A(xxx, t) =µ0

4⇡

ZJ(xxx0)

eik|xxx�xxx0|

|xxx � xxx0|d3x0 (2.4)

where k = !/c is the wave number. The propagator can be expanded as

eik|xxx�xxx0|

4⇡|xxx � xxx0| = ik1X

l=0

jl(kr<) h(1)l (kr>)

lX

m=�l

Y ?lm(✓

0,�0)Ylm(✓,�), (2.5)

where Ylm(✓,�) are the spherical harmonic functions, eigenfunctions of theangular momentum, j and h are the spherical Bessel and Hankel functions,and r< (r>) are the points inside (outside) the spherical distribution of chargesand currents.

If only the first term, i.e. l = 0, is considered, we obtain in the inductionzone the electric dipole field. Thus, the vector potential is

A(xxx, t) = � iµ0!

4⇡pppeikr

r(2.6)

where ppp =Rxxx0⇢(xxx0)d3x0 is the electric dipole moment.

The next term in the expansion leads to the vector potential

A(xxx) =µ0

4⇡

eikr

r

✓1

r� ik

◆ZJ(xxx0)(n · xxx0)d3x0, (2.7)

where n = xxx/|xxx|. This vector potential can be split into a part symmetricin j and xxx0 which leads to a transverse magnetic field and an antisymmet-ric part which leads to transverse magnetic induction. Considering only themagnetization term we have the vector,

A(xxx) =ikµ0

4⇡(n ⇥ m)

eikr

r

✓1 � 1

ikr

◆, (2.8)

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where m = 1/2R(xxx ⇥ J)d3x is the magnetic dipole moment. Finally, the

symmetric part corresponding to an electric quadrupole source leads to thevector potential

A(xxx) = �µ0ck2

8⇡

eikr

r

✓1 � 1

ikr

◆Zxxx0(n · xxx0)⇢(xxx0)d3x0. (2.9)

Therefore, if the charge distribution in Eq. (2.2) is zero everywhere, everyelectric moment is automatically zero and the dominant term is the magneticdipole moment.

Finally, let us remark that the spherical harmonic functions in Eq. (2.5)correspond to integer numbers of the angular momentum, l.

2.2 Spin

The spin [42, 43] is an intrinsic characteristic of not only elementary particlesbut also of their own bound states. This property is purely quantum and thereis no classical analogue. In this section, we review the first experiment in whichthe spin was observed, as well as some features, such as the quantization of itsvalue, the relationship with statistical properties, etc.

In 1922 O. Stern and W. Gerlach realized the experiment in which a prop-erty afterwards called spin, was revealed. They prepared a beam of silverneutral atoms which were deposited on a glass layer after passing through anon-uniform magnetic field. As the magnetic moment is proportional to an-gular momentum, and this is quantized, as previously was shown by N. Bohr,a discrete number of beams should be observed. Concretely, as the angularmomentum along the z direction can take the values 2l+1, where l = 0, 1, 2, ...;they expected an odd number of beams. Nonetheless, for l = 0, the beam wasnot focused on one single region, but on two. This was a clear evidence ofan additional contribution to the dipolar magnetic moment. Furthermore, theseparation between the two beams was so large that it can not be attributedto the magnetic moment of the nucleus, since it depends on the inverse ofthe mass and this should be 1000 times smaller than the observed one. Theproblem was solved by attributing an intrinsic magnetic moment to the elec-tron, which was called spin. This is a purely quantum feature. In fact, in1925, some proposals to understand the spin as a quantization of a classicalfeature, such as an internal rotation, were experimentally discarded. Consid-ering the electron as a rotating charged sphere, the radius needed to produce

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the observed splitting is aproximally R ⇠ 10�10 m, whereas electron’s radiusis re = 2.8 ⇥ 10�15 m, which should be rotating with relativistic velocity inorder to achieve the observed magnetic moment.

Figure 2.1: Graphic scheme of the Stern-Gerlach experiment. Taken from [44].

Let us now review the characteristics of this quantum magnitude:

• The spin, in the same way as the angular momentum, is quantized. Thequantum number s shows values s = 0, 1/2, 1, 3/2, ... and the componentof the spin in one direction is also quantized with eigenvalues Sn = ms ·~,where ms = �s,�s+ 1, ..., s.

• By measuring the spin in a given direction, one can only observe eigen-values of the operator corresponding to the spin along that direction. Forinstance, let us consider a measurement of the z-compontent of a spins = 1 particle. Then, the possible outcomes are Sz = �~, 0,+~.

• The angular momentum changes giving or taking energy from the en-vironment. On the contrary, the spin is an intrinsic property of everyelementary particle.

• The statistics followed by a large ensemble of particles is determinedby the spin. For an integer s, the Bose-Einstein statistics describes thecollective behavior of the ensemble, whereas for a semi-integer s, it followsthe Fermi-Dirac statistics. This is known as spin-statistics theorem [43].

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A particle of spin s = 1/2 can be considered the simplest model from thepoint of view of the quantum interaction theory. The quantum state lays in atwo-dimensional Hilbert space, and it can be decomposed in the eigenstates ofthe Pauli operator �z. Following the nomenclature where |0i corresponds tothe ground state and |1i to the excited state, the wave function of a particlewith spin 1/2 can be written as follows:

| i = ↵|0i + �|1i, (2.10)

where ↵ and � are complex and satisfy the condition of normalization. Giventhat these wave functions are characterized only by two states, they can beemployed in the field of quantum information. A quantum bit, also calledqubit, is a unit of quantum information, the quantum analogue of the classicalbit. Whereas in classical systems only the values 0 or 1 are allowed, in quantumsystems a linear superposition of them is permitted, which corresponds to thestate expressed above. For this reason, the spin is perfect to be used in thefield of quantum information.

Figure 2.2: Bloch sphere of a quantum bit. Taken from [39].

2.3 Spin models

Spin models are one of the most studied topics in many-body systems [40].The microscopic description of magnetism is probably one of the most relevant

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questions that one can study in condensed matter physics. The Ising model, amathematical model of ferromagnetism in statistical mechanics, was proposedin 1920 by W. Lenz and solved in 1-D by his student E. Ising in 1925. However,applications beyond ferromagnetism emerged and the model revealed certainrelevance in lattice gases, quantum field theory, neuroscience, etc. The classicalHeisenberg model was proposed in order to describe the magnetic dipole-dipoleinteraction on a lattice with classical interacting vectors, and contains the Isingmodel as a special case. A quantum version in which the vectors are replaced bythe Pauli matrices is currently the paradigmatic model of spin-spin interaction.On the following these two well-known spin models are reviewed.

2.3.1 Ising model

In 1925, E. Ising studied the problem of interacting nearest-neighbour dipoles,proposed by his supervisor W. Lenz, represented by discrete variables. In thismodel, the spins have one single direction, so their magnetic moment can onlytake two discrete values, �i = {+1,�1}. This is the same behavior as a spin1/2 when we restrict to the component in one direction. This is mathematicallydescribed by:

H = ~X

i

Ji�i�i+1, (2.11)

where Ji is the coupling between the spins i and i+1 that only interact alonga certain direction.

The Ising model shows a very di↵erent behavior depending on the sign ofthe coupling Ji:

• Ji < 0: The interaction is called “ferromagnetic”, and the spins tend toalign in the same direction.

• Ji > 0: The interaction is called “antiferromagnetic”, and the spins tendto align in opposite direction.

• Ji = 0: The spins are noninteracting. Notice that the coupling is zero incase the corresponding sites are not nearest neighbors.

By considering identical positive coupling terms, Ji = J > 0, 8i, a very de-generate ground state emerges. For instance, assuming a spin chain composedof three spins with periodic boundary conditions, the ground state is six-fold

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degenerate and the state is two-fold degenerate. This provides the possibilityto achieve highly entangled ground states.

Furthermore, in case we take the coupling constants equal and positive,Ji = J > 0, a Hamiltonian with frustration may be obtained depending on thestructure of the lattice. For example, let us consider a 2-D triangular lattice.As the model is antiferromagnetic, one can choose two nearest-neighboringspins with opposite directions to minimize the energy. However, for the thirdspin, which closes the triangle, it is impossible to locally minimize the en-ergy with the other two at the same time, and this phenomenon is namedfrustration [45, 46].

Figure 2.3: Frustration in a triangular lattice with antiferromagnetic coupling.Taken from [45].

Ising studied the problem by adding an external magnetic field with thesame coupling constant for all sites:

H = ~JX

i

�i�i+1 +BX

i

�i, (2.12)

where B is the magnitude of the magnetic field. With this additional term,the degeneracy is broken and the frustration removed.

The free energy of a one dimensional Ising model on a chain with N siteswith free boundary conditions has the analytical solution:

f(�, B) = � limN!1

1

�Nln(Z(�))

= � 1

�

⇣e�J cosh(�B) +

pe2�J(sinh �B)2 + e�2�J

⌘, (2.13)

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being Z(�) the partition function, � = 1/kBT , kB the Boltzmann constantand T the temperature.

In 1944, L. Onsager studied the problem but on a two dimensional squarelattice [47]. Given that the partition function of the Ising model in the absenceof external magnetic field in two dimensions on this lattice can be mapped toa two-dimensional free fermion model, the specific heat can be analyticallycomputed. Onsager obtained the analytical solution for the free energy andalso showed that there was a phase transition in the magnetization, somethingwhich does not occur in the 1-D case. The expression that characterizes themagnetization as a function of the temperature is the following:

M =

1 � sinh

log(1 +

p2)Tc

T

��4!1/8

, (2.14)

where the critical temperature is given by

Tc =2J

kB log(1 +p2). (2.15)

The Ising model has a quantum analogue known as quantum Ising model.In this model the spins ”up” or ”down” must be replaced by linear superposi-tion of both states. In this way, the Hamiltonian is defined as follows:

H = ~X

i

Ji�xi �

xi+1 +B

X

i

�yi . (2.16)

where �xi is the Pauli operators for the site i. Spins interact with a coupling

constant Ji only in the x-direction, and they are also a↵ected by a transversemagnetic field along y-direction. Notice that the notation used for the tensorproduct is defined as

�xi �

xj = 1 ⌦ 2 ⌦ ... ⌦ �x

i ⌦ �xj ⌦ ... ⌦ N . (2.17)

Even though the Ising model seems to be very simple and rough, it hasmany applications nowadays, namely, in the field of neuroscience. The neuronsin the brain can be statistically modeled. Given that each neuron can be eitheractive or inactive, they behave like spins in excited or ground states [48].Moreover, as the neural activity is modeled by independent bits, the Isingmodel provides a good first approximation to the neural network.

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2.3.2 Heisenberg model

The Heisenberg model can be understood as a generalization of the Ising inter-action. This model describes the interaction between magnetic dipoles. Theclassical model can be represented as follows:

H = ~X

i

Jisssi · sssi+1, (2.18)

being sssi the normalized vector that characterizes the direction of a magneticdipole in the i-site, Ji the coupling constant between the dipoles in sites iand i+ 1, and · the inner product. Notice that the inner product ensures therotational symmetry of the Hamiltonian.

In order to solve the one-dimensional antiferromagnetic Heisenberg Hamil-tonian, H. Bethe conceived a method for finding the exact solutions of cer-tain one-dimensional quantum many-body models. This was named the BetheAnsatz [50]. Making use of this method, other models, namely, Bose gas andHubbard model among others, were solved. The Heisenberg Hamiltonian couldbe solved by numerical methods, but using the Bethe’s method, the Hamilto-nian could be diagonalized making it possible to find the eigenvectors of thesystem.

Figure 2.4: Heisenberg Hamiltonian’s ground states: ferromagnetic model (up)and antiferromagnetic model (down). Taken from [49].

In the quantum version of this model, the spin vectors are again replaced bythe Pauli matrices. Following the same notation used above, the Hamiltoniancan be written as follows:

H = ~X

i

(Jxi �

xi �

xi+1 + Jy

i �yi �

yi+1 + Jz

i �zi �

zi+1). (2.19)

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where Jai are the coupling constants for qubits i and i+1 along the a-direction.

It is essential to mention that for every two qubits, there are three couplingconstants, one for each direction. By tuning the coe�cients, di↵erent physicalproperties may be observed. The most studied models are the XYZ, wherethe coupling constants are equal, Jx = Jy = Jz and therefore the rotationalsymmetry SU2 is preserved, and the XXZ model, where one direction has adi↵erent coupling constant, Jx = Jy 6= Jz.

The Heisenberg model shows the dynamics of magnetic dipoles in threedimensions but its physical relevance increased when L. F. Mattheiss [51] andD. I. Paul [52] showed that both the ground states and the elementary excitedstates of every chain of molecules, where the valence electrons occupy nonde-generate s-orbitals, are well described by an e↵ective Heisenberg Hamiltonian.This contains antiferromagnetic couplings among nearest-neighbour spins andneglects all the motional degrees of freedom. On the other hand, if two ormore valence states are allowed to conduct electrons on each atom, one mightexpect that an e↵ective Heisenberg Hamiltonian with ferromagnetic nearest-neighbour interactions describes the magnetic degrees of freedom of the chain(but not the electronic ones).

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Chapter 3

Circuit QuantumElectrodynamics

Circuit Quantum Electrodynamics [26, 27, 28] is one of the quantum tech-nologies that has advanced more during the past decade. It describes theinteraction between superconducting qubits and superconducting resonators.Other quantum technologies such as trapped ions or photonics systems havea microscopic scale, so circuit Quantum Electrodynamics is one of the firsttechnologies showing quantum properties at a macroscopic size thanks to su-perconductivity, which allows one to observe macroscopic quantum e↵ects.

Superconducting (SC) quantum circuits are often called artificial atoms. Infact, the physics of an atom within a cavity, that is, cavity Quantum Electro-dynamics, is exactly the same as the cQED one. The superconducting circuitplays the role of the atom with non-linear energy levels while the cavity isrepresented by a transmission line resonator that acts as a quantum harmonicoscillator. Given that the same physics is described by two di↵erent devices, itis important to check other features. For instance, given that cQED elementshave a macroscopic size, it is easier to couple several elements. However, thisis not always an advantage. As a consequence of the high coupling achievedbetween the di↵erent elements of the setup, it is a hard task to isolate them. Inany case, this technology has really improved achieving a high controllabilityand long coherence times and it is expected to be one of the most promisingplatforms for the universal quantum computer. Furthermore, cQED may o↵era good platform to implement interesting proposals in the field of quantumsimulators.

Below we present a brief review of cQED where we show the basics of

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superconducting circuits and how they can be used for our proposals.

3.1 Josephson e↵ect

The basic element needed to construct superconducting quantum circuits usedfor the field of quantum information is the Josephson junction (JJ) [35]. An LCcircuit acts as a harmonic oscillator because all the energy levels are equidis-tant. In order to employ SC circuits, a non-linear component, namely the JJ,is needed for modifying the di↵erence between the energy levels. A JJ is com-posed of two superconductors separated by a thin insulating layer. B. Joseph-son made a complete analysis of this configuration in 1962 and he was awardedthe 1973 Nobel Prize in Physics for this work. Josephson demonstrated thatCooper pairs, which are two electrons coupled due to superconductivity con-ditions, could tunnel the insulating layer without dissipation.

The wave function of a superconductor is given by the number of chargecarriers, ns, and the phase, ', in each position; that is:

s =pns e

i'. (3.1)

Josephson showed that a JJ stores energy that depends on the phase di↵erencebetween the wave functions of the two superconductors '1 � '2:

U = �EJ cos', (3.2)

where ' = '1 � '2 and

EJ = �0IC/2⇡, (3.3)

where we have that constants �0 = h/2e are the flux quantum with h thePlanck constant, e the charge of the electron, and IC the critical current, whichis the maximum current that can flow through the JJ. The latter depends onthe resistance of the junction, Rn, and on the electric potential gap betweenboth superconductors, Vg. Moreover, the current which flows through the JJdepends on the di↵erence between the phases of the wave functions of thetwo superconductors. The following equation is known as the first Josephsonequation:

Is = IC sin'. (3.4)

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The second Josephson equation gives the relation between the voltage acrossthe junction and the phase di↵erence:

V =�0

2⇡

d'

dt. (3.5)

Combining Eq.(3.4) and Eq.(3.5) it is possible to obtain the Josephson induc-tance:

LJ =�0

2⇡IC cos'. (3.6)

One can describe a JJ by using the resistively and capacitively shuntedjunction (RCSJ) model. It considers the parallel combination of a non-linearinductor LJ , a capacitor given by the parallel-plate geometry of the junctionCJ , and a resistor that models the voltage losses, R(V ). The JJ is modeledas a resonator whose frequency is !p = 1/

pLJCJ and the quality factor is

Q = !pRCJ .

V

I

R(V )LJ CJ

Figure 3.1: RCSJ model of a Josephson junction.

3.2 Superconducting qubits

Making use of di↵erent geometrical configurations of Josephson junctions, sev-eral SC qubits can be created. Originally, three basic types of SC qubits weremade: charge [53], flux [54] and phase [55] qubits. Each one has advantagesand disadvantages, but they all have evolved during the last years, according

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to improvements in fabrication, measurement fidelities and coherence times.Moreover, there has been a variety of other designs, such as quantronium [56],transmon [57], fluxonium [58] and hybrid qubits [59], which are composed ofthe same elements but the new designs allow to reduce their sensitivity todecoherence mechanisms. Below we analyze the first SC qubit: the chargequbit.

3.2.1 Cooper pair box

The charge qubit, also known as Cooper pair box, is made of a tiny islandcoupled to a reservoir. This is achieved unifying a Josephson junction, a ca-pacitor and a voltage source. At temperatures of 20 mK, the system becomessuperconductor and the electrons couple to each other creating Cooper pairs.In addition, it is possible to set the di↵erence between the number of Cooperpairs staying in the island and the ones in the rest of the superconductor.

Figure 3.2: Graphic scheme of the Cooper pair box. Taken from [60].

In order to quantize the degrees of freedom of quantum circuits, we needquantum mechanical operators that obey commutation relations of the positionand momentum. This behavior is found in the charge (number) operator andflux (phase) operator. In fact, the number operator is given by:

N =~i

@

@'. (3.7)

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The classical Hamiltonian of the system is

H = 4EC(N � Ng)2 � EJ cos', (3.8)

where the charging energy, the gate charge and the Josephson energy are

EC =(2e)2

8C⌃, Ng =

CgVg

2e, EJ =

IC�0

2⇡, (3.9)

with C⌃ = Cg + CJ , and N 2 Z . Let us now put it in terms of the numberstates:

H =X

N

4Ec(N � Ng)

2|NihN | � EJ

2(|NihN + 1| + |N + 1ihN |)

�. (3.10)

We consider the energy scales of EC and EJ . For the charge qubit we takeEJ ⌧ EC .

ECEJ

Ng0 1

E N=

0

N=

1

Figure 3.3: Energy diagram of the Cooper pair box at E ⇠ EC . Takenfrom [61].

As can be seen in Fig. 3.3, the two lowest energy levels exhibit an an-ticrossing, and the energy di↵erence between them is the Josephson energy.

19

The anticrossing points are at Ng = n + 1/2 with n 2 Z . So at these points,the states |ni and |n+ 1i are coupled. Solving the Hamiltonian, one can findthe energies and corresponding eigenstates of each lowest energy anticrossingpoint:

" = EC ± EJ/2,

| i =1p2(|ni ± |n+ 1i). (3.11)

Fixing the number of Cooper pairs it is possible to restrict the system toa unique anticrossing point, for instance n = 0, in which the Hilbert spacehas dimension 2. The eigenstates of the Hamiltonian can be coupled with afrequency ! = EJ/~ in order to isolate them from other energy levels. Thequantum bit at the point n = 0 is composed of the states:

|0i =1p2(|0i + |1i),

|1i =1p2(|0i � |1i). (3.12)

By this method, we have shown that it is possible to create a qubit, whichcan be used in the field of quantum information, from a system that initiallyhad several energy levels. The remaining Hamiltonian after restricting to thistwo level system is:

H =EJ

2�x. (3.13)

This analysis can also be realized for other type of superconducting qubitssuch as the flux, phase or transmon among others. Nevertheless, all of themhave common properties. For example, the energy di↵erence between the low-est levels is about 10 GHz, and all of them have a similar size, approximatelyfrom 1 to 100 µm2. On the other hand, the flux, phase and transmon qubitsneed a Josephson energy much higher than the charging energy, EJ � EC .In any case, the coherence time makes the di↵erence between the best qubits.Fig. 3.5 shows the improvements in qubit lifetimes since 1998 until 2013. Ascan be seen, there is a clear improvement during the last decade. Moreover,not only the coherence times have been improved, the fidelity of quantum gateshas also been increased, and the time required to apply them has decreased.

20

As a result, the newest SC qubits, whose coherence times are about 100 µs,allow to apply of the order of 1000 quantum gates before decoherence a↵ectsthem [62].

routing microwave photons with transmissionlines [now known as circuit QED (12, 44, 60)]might make on-chip versions of these schemeswith superconducting circuits an attractive alter-native. Although this strategy can be viewed asless direct and requires a variety of differing parts,its advantage is that stringent quality tests are easierto perform at the level of each module, and hid-den design flaws might be recognized at earlierstages. Finally, once modules with sufficient per-formance are in hand, they can then be programmedto realize any of the other schemes in an addi-tional “software layer” of error correction.

Finally, the best strategy might include ideasthat are radically different from those consideredstandard fare in quantum information science.Much may be gained by looking for shortcutsthat are hardware-specific and optimized for theparticular strengths and weaknesses of a partic-ular technology. For instance, all of the schemesdescribed above are based on a “qubit registermodel,”where one builds the larger Hilbert spaceand the required redundancy from a collection ofmany individual two-level systems. But for su-perconducting circuits, the “natural units” are os-cillators with varying degrees of nonlinearity,rather than true two-level systems. The use of

noncomputational states beyond the first two lev-els is of course known in atomic physics, and hasalready been used as a shortcut to two- and three-qubit gates in superconducting circuits (23, 61).Under the right conditions, the use of nonlinearoscillators with many accessible energy levelscould replace the function of several qubitswithout introducing new error mechanisms. Asa concrete example of the power of this approach,a recent proposal (62) for using a cavity as aprotected memory requires only one ancillaand one readout channel—a real decrease incomplexity.

How architectural choices like these affectour ability to perform error-corrected informationprocessing will be a key scientific question occu-pying this field in the near future, and will prob-ably take several years to resolve. The knowledgegarnered in this process has the potential to sub-stantially change the resources required for build-ing quantum computers, quantum simulators, orquantum communication systems that are actual-ly useful.

The Path ForwardThe field of QIP with superconducting circuitshas made dramatic progress, and has already dem-

onstrated most of the basic functionality withreasonable (or even surprising) levels of perform-ance. Remarkably, we have not yet encounteredany fundamental physical principles that wouldprohibit the building of quite large quantum pro-cessors. The demonstrated capabilities of super-conducting circuits, as in trapped ions and coldatoms, mean that QIP is beginning what maybe one of its most interesting phases of devel-opment. Here, one enters a true terra incognitafor complex quantum systems, as QEC becomesmore than a theoretical discipline. As in the past,this era will include new scientific innovationsand basic questions to be answered. Even if thisstage is successful, there will remain many furtherstages of development and technical challengesto be mastered before useful quantum informa-tion processing could become a reality. However,we think it is unlikely to become a purely techno-logical enterprise, like sending a man to the Moon,in the foreseeable future.After all, even theMoore’slaw progression of CMOS integrated circuits overthe past four decades has not brought the end ofsuch fields as semiconductor physics or nano-science, but rather enabled, accelerated, and steeredthem in unanticipated directions. We feel that fu-ture progress in quantum computation will always

A B

1

10

100

1000

10,000

Num

ber

of b

its p

er q

ubit

lifet

ime

201420122010200820062004Year

3D-transmon+JPC+P-filter

CPB+HEMT

Transmon+JBA

100

101

102

103

104

105

106

107

Qub

it lif

etim

e (n

s)

2012200820042000Year

1

10

100

1000

10,000

100,000

Ope

ratio

ns p

er e

rror

T1

T2

Tcav

CPB

Charge echo

Quantronium

Transmon

Fluxonium

3D transmon

3D cavities

Improved3D transmon

cQED

Fig. 3. Examples of the “Moore’s law” type of exponential scaling in performanceof superconducting qubits during recent years. All types have progressed, but wefocus here only on those in the leftmost part of Fig. 2C. (A) Improvement ofcoherence times for the “typical best” results associated with the first versions ofmajor design changes. The blue, red, and green symbols refer to qubit relaxation,qubit decoherence, and cavity lifetimes, respectively. Innovations were introducedto avoid the dominant decoherence channel found in earlier generations. So faran ultimate limit on coherence seems not to have been encountered. Devicesother than those in Fig. 2C: charge echo (63), circuit QED (44), 3D transmon (43),and improved 3D transmon (64, 65). For comparison, superconducting cavitylifetimes are given for a 3D transmon and separate 3D cavities (66). Even longertimes in excess of 0.1 s have been achieved in similar 3D cavities for Rydberg atomexperiments [e.g., (67)]. (B) Evolution of superconducting qubit QND readout. Weplot versus time themain figure of merit, the number of bits that can be extracted

from the qubit during its T1 lifetime (this number combines signal-to-noise ratioand speed). This quantity can also be understood as the number of measurements,each with one bit of precision, that would be possible before an error occurs. Datapoints correspond to the following innovations in design: a Cooper-pair box readby off-resonance coupling to a cavity whose frequency is monitored by a micro-wave pulse analyzed using a semiconductor high–electron mobility transistoramplifier (CPB+HEMT) [also called dispersive circuit QED (68)], an improvedamplification chain reading a transmon using a superconductor preamplifierderived from the Josephson bifurcation amplifier (transmon+JBA) (49), and fur-ther improvement with another superconductor preamplifier derived from theJosephson parametric converter (51) combined with filter in 3D transmon cavityeliminating Purcell effect (3D-transmon+JPC+P-filter). Better amplifier efficiency,optimal signal processing, and longer qubit lifetimes are expected to maintainthe rapid upward trend.

www.sciencemag.org SCIENCE VOL 339 8 MARCH 2013 1173

SPECIALSECTION

on

Augu

st 2

8, 2

013

ww

w.s

cien

cem

ag.o

rgD

ownl

oade

d fro

m

Figure 3.4: Improvements in qubit coherence times for the ”typical best” re-sults associated with the first versions of major design changes. The blue,red and green symbols refer to qubit relaxation, qubit decoherence and cavitylifetimes, respectively. Taken from [28].

3.3 Transmission line resonator

Until now we have shown how to create a two-level artificial atom with SCcircuits, but for quantum computing we need to control several qubits bythe application of single and two-qubit gates. For this, a transmission lineresonator is one of the best approaches.

21

This device is composed of two external ground plates and a central su-perconductor, which is split on the far sides, making three di↵erent parts thatare known as input line, central conductor, and output line. The gaps cre-ated between them, the capacitance points, have a very high impedance andaccordingly, they play the role of a mirror for the voltage and current wavespropagating along the central line.

Given that there are two boundary conditions in the central part, the elec-tromagnetic waves created by the superconductor become stationary. Fur-thermore, the continuum of modes, proper of the electromagnetic field in freespace, is restricted to a discrete set of modes dependent on the length of theresonator. Taking a resonator whose central conductor has a length of 10 mm,the frequency of the electromagnetic waves within the resonator is about 10GHz, which corresponds to the microwave regime. In addition, this is thefrequency that separates the energy levels of SC qubits, thus it is perfect tocouple them to the resonator. In fact, given that the plates are connected toearth, the electric field is focused on a very small area: notice that the sepa-ration between the external and the central plates is 10 µm. This provides anintense electric field in the resonator, where the SC circuits are placed, and astrong coupling is achieved as result.

Figure 3.5: A coplanar transmission line resonator used for circuit QED. Takenfrom [63].

Selecting one of the modes, the resonator behaves as a harmonic oscillator.The energy of the cavity is given by the number of photons within it, knowingthat all of them are at the same frequency !r,

Hr = ~!r(a†a+ 1/2), (3.14)

where !r = 1/pLC. The inductance and capacitance of the transmission line

resonator can be calculated from the well known inductance and capacitanceper unit length : l = 0.1nF/m and c = 0.2µH/m. For a typical resonatorof 20mm, the total inductance and capacitance are L = 4nH and C = 2pF,giving a characteristic impedance of Z =

pL/C = 50⌦.

22

Input and output lines are used to introduce radio frequency pulses and dccurrents into the cavity in order to control the qubit. In fact, it is completelynecessary to have a system with high controllability if it is going to be used forquantum computing. The lines are also used for measurements of the statesof the qubits, by detecting the electric field coupled to them.

The Hamiltonian presented above seems to be simple, but we need to addthe Hamiltonian of each qubit placed within the cavity and what is more, theHamiltonian coming from interaction between the qubits and the resonator.Neglecting the constants from each expression the total Hamiltonian of j qubitsin the resonator has the form:

H = ~!ra†a+

X

j

~!aj

2�zj �

X

j

~gj(µj � cj�zj + sj�

xj )(a

† + a), (3.15)

with !aj

=q

E2Jj

+ [ECj

(1 � 2Ng,j)]2 the transition frequency of a qubit j,

gj = e(Cg,j/C⌃,j)V 0rms/~ the coupling strength of the resonator to qubit j

where C⌃ = Cg + CJ and V 0rms =

p~!r/2C, being C the capacitance of the

resonator. We define µj = 1 � 2Ng,j, cj = cos ✓j and sj = sin ✓j, where✓j = arctan[EJ

j

/ECj

(1 � 2Ng,j)] is the mixing angle.

We consider the system at its charge degeneracy point: ✓j = ⇡/2. We alsoconsider that the frequencies of the qubits are the same, !a

j

= ⌦, and thecouplings are the same too, gj = g. Then the Hamiltonian becomes:

H = ~!ra†a+

X

j

~⌦2�zj �

X

j

~g�xj (a

† + a). (3.16)

We move into an interaction picture where the operators �j and a become:

��j (t) = ��

j (0)e�i⌦t,

a(t) = a(0)e�i!r

t. (3.17)

In this way, the Hamiltonian turns into:

HI(t) = �X

j

g(a†�+j e

+i(!r

+⌦)t + a�+j e

�i(!r

�⌦)t

+a†��j e

+i(!r

�⌦)t + a��j e

�i(!r

+⌦)t). (3.18)

Now, we apply the rotating wave approximation: we neglect the fastestoscillating terms taking into account that !r +⌦ � g and |!r �⌦| ⌧ !r +⌦,

23

where the typical experimental values are g/2⇡ = 0.3GHz, !r/2⇡ = 10GHzand ⌦/2⇡ = 5GHz. The result is:

HI(t) = �X

j

g(a�+j e

�i(!r

�⌦)t + a†��j e

+i(!r

�⌦)t). (3.19)

This is the Jaynes-Cummings model, which transfers the field excitations ontothe qubits and viceversa.

3.4 DiVincenzo criteria

In order to do quantum computing in a physical system there are some require-ments that must be fulfilled. The criteria established as standard was givenby DiVincenzo at 2001 [64]. In the following we list the di↵erent conditionsthat superconducting qubits must satisfy.

(1) It must be a scalable physical system with well characterized qubits: it ispossible to excite only two energy levels of the superconducting circuit,becoming in this way what is known as superconducting qubit. Further-more, several SC qubits can be coupled making use of a transmission lineresonator.

(2) It must be possible to initialize the qubits: by cooling the system, onecan put the qubits in the ground state with zero photons in the resonator,providing a state to initialize the system with high fidelity.

(3) The qubits must have a coherence time much longer than the operationtime: the latest architectures for superconducting qubits have decoher-ence times 105 ns whereas single and two-qubit gates are near to 102 ns,so it is possible to make several logical gates without losing coherence.

(4) There has to be a universal set of quantum gates: In the most simplecase, it is possible to make single-qubit rotations and also entanglingtwo-qubit gates. This is enough for a universal set of quantum gates.

(5) A qubit-specific measurement must be attainable: there are di↵erentmeasurement techniques available in cQED setups. For example, disper-sive measurements, dual path, or superconducting tunneling in the caseof the phase qubit. In the latter, making use of a driving in the resonatorwithout changing the state of the qubit, the tunneling takes place if the

24

state is |1i whereas it does not if the state is |0i. By this method, itis possible to calculate the probabilities of the respective correspondingstate and obtain them with high accuracy.

Moreover, in order to build a quantum network, it is required:

(6) The ability to interconvert stationary and flying qubits: superconductingcircuits are coupled to resonators where photons are created. Thesephotons can be used to transfer information from one device to another.

(7) The ability to faithfully transmit flying qubits between specified loca-tions: the resonators can be coupled by means of photons. This allowsthe communication between two resonators and their corresponding cou-pled qubits.

All the conditions of the DiVincenzo criteria are fulfilled making supercon-ducting qubits an excellent quantum technology for quantum computation.

3.5 Exchange gate

In this section, we present the exchange gate which is a two-qubit gate straight-forward in cQED [26]. As it is shown below, this gate has several applications,e.g., implementing the iSwap gate. Furthermore, we will use it in Chapter 5for the proposal of quantum simulations of spin models in superconductingcircuits.

Our starting point is the Hamiltonian in the Eq. 3.16. We consider naturalunits, ~ = 1 for simplicity, and we restrict to the case of two qubits:

H = !ra†a+

X

j=1,2

⌦

2�zj �

X

j=1,2

g�xj (a

† + a). (3.20)

In order to change into the interaction picture, we decompose the Hamil-tonian as follows,

H =

"!ra

†a+X

j=1,2

✓⌦

2+ g0

◆�zj

#�"X

j=1,2

g0�zj +

X

j=1,2

g�xj (a

† + a)

#. (3.21)

25

In the interaction picture taking as H0 the first part of the Hamiltonian inthe Eq. 3.21, the operators �j and a become:

��j (t) = ��

j (0)e�i(⌦+2g0)t,

a(t) = a(0)e�i!r

t. (3.22)

Hence,

HI(t) = �X

j=1,2

g0�zj �

X

j=1,2

g(a†�+j e

+i(!r

+⌦+2g0)t + a�+j e

�i(!r

�⌦�2g0)t

+a†��j e

+i(!r

�⌦�2g0)t + a��j e

�i(!r

+⌦+2g0)t). (3.23)

In the same way as before, we apply the rotating wave approximation andthe most oscillating terms are neglected taking into account that !r+⌦+2g0 �g and |!r � ⌦� 2g0| ⌧ !r + ⌦+ 2g0. The Hamiltonian becomes:

HI(t) = �X

j=1,2

g0�zj �

X

j=1,2

g(a�+j e

�i(!r

�⌦�2g0)t + a†��j e

+i(!r

�⌦�2g0)t). (3.24)

We consider now a second order e↵ective Hamiltonian obtained from thegeneral time-dependent expression,

HI(t) =X

j

[A†je

i�j

t + Aje�i�

j

t], (3.25)

where A is a time-independent function of system operator that can be rewrit-ten in an e↵ective time-independent form as the following:

He↵ =X

j

[A†j, Aj]

�j

, (3.26)

if |�j| � gj, 8j and |�j ±�k| � gk, 8j 6= k.

We apply this to the last term of our Hamiltonian: A† = ga(�+1 + �+

2 ) and� = ⌦+ 2g0 � !r. This part of the Hamiltonian results:

he↵ =g2

�[a†a(�z

1 + �z2) + |e1ihe1| + |e2ihe2| + (�+

1 ��2 + �+

2 ��1 )]

=g2

�[a†a(�z

1 + �z2) + |e1ihe1| + |e2ihe2| +

1

2(�x

1�x2 + �y

1�y2)]. (3.27)

26

Adding a constant � g2

2� we can rewrite the total Hamiltonian as:

HIe↵ = �g0(�

z1 + �z

2) +g2

�[a†a(�z

1 + �z2) +

1

2(�z

1 + �z2) +

1

2(�x

1�x2 + �y

1�y2)]

=g2

�(�z

1 + �z2)a

†a+1

2(�2g0 +

g2

�)(�z

1 + �z2) +

g2

2�(�x

1�x2 + �y

1�y2).

(3.28)

It is straightforward to show that in case there is no photon in the cavityand choosing g0 =

g2

2� the Hamiltonian finally becomes:

HIe↵ = Hxy

12 ⌘ g2

2�(�x

1�x2 + �y

1�y2), (3.29)

where the coupling constant typically takes the value g2/2� = 0.05 GHz.

The unitary evolution is:

U(t) = exp

�i

g2

2�(�x

1�x2 + �y

1�y2)t

�

=

0

BB@

1 0 0 0

0 cos 2 g2

2�t �i sin 2 g2

2�t 0

0 �i sin 2 g2

2�t cos 2 g2

2�t 00 0 0 1

1

CCA . (3.30)

In case g2

2�t = �⇡/4, thepiSWAP gate is achieved:

UpiSWAP =

0

BB@

1 0 0 00 0 i 00 i 0 00 0 0 1

1

CCA . (3.31)

27

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28

Chapter 4

Quantum simulators

In 1982, Feynman conjectured that quantum computers might be able to sim-ulate quantum systems in a more e�cient way than classical computers [6]. Infact, a computer made of classical bits, either 0 or 1, can only solve problems bymethods that use polynomial amounts of computational resources. These kindof problems are commonly named tractable. On the other hand, the methodsthat use exponential amounts of computational resources are the intractableones.

The state of one particle of spin 1/2 is defined by two complex numbers,but considering N spins the complexity of the systems grows exponentially.In this case, we would need 2N numbers to define the state of the system and2N ⇥ 2N for the evolution. Even the most powerful computers cannot do thiskind of tasks if the number of particles is higher than, say, 40 spins.

A quantum computer is made of quantum bits. These are not only 0 or1 but also a superposition of them. As the state of a spin can be up anddown at the same time, a quantum computer can e↵ect tasks making use ofquantum properties such as linear superposition or entanglement. Therefore,given that the architecture of quantum computers is more complex than theclassical ones, they are expected to solve problems that are unfeasible at themoment [12].

Unfortunately, the di�culty to control a great number of qubits impedesto create a universal quantum computer. Nevertheless, it is possible to ma-nipulate a certain number of qubits and use them to mimic other quantumsystems. This process is called quantum simulation.

Given that there are many di↵erent kinds of quantum simulators, one can

29

establish a basic set of criteria that must be fulfilled [65]. We show it below:

• Quantum system: A quantum simulator should contain quantum sys-tems, e.g., qubits, bosons, or fermions confined in a region of space. Thesystem must have many degrees of freedom.

• Initialization: It is necessary to prepare the system in a certain purestate. Even though it may also be interesting to study the dynamics ofa mixed state, knowing which one is the state at t = 0 is the first stepto control the system.

• Hamiltonian engineering: One should be able to control the system mak-ing use of a set of interactions with external fields or between the particlesof the system. Turning on and o↵ these interactions, it should be possibleto generate the dynamics of another quantum system.

• Detection: It should be feasible to perform individual and/or collectivemeasurements on the system. Once the evolution has finished, this isrequired to determine the state of the system, or the result of the mea-surement of an observable over the system.

• Verification: It is crucial to know whether the simulation is correct.In case a classical machine could not reproduce the dynamics of thesimulated system, one should work with consistency arguments to beconfident on the correctness of the simulation. For example, it is expectedthat the dynamics of two di↵erent quantum systems should be the sameif the Hilbert spaces and their interactions are similar.

4.1 Analog and digital methods

Two di↵erent kinds of quantum simulators can be distinguished: analog anddigital. Those systems whose dynamics is the same of the mimicked systems arecalled analog quantum simulators. Just adjusting a few parameters, namely,the coupling constants or the frequencies that appear in the Hamiltonian of thequantum simulators, one can obtain the same Hamiltonian of the simulatedsystem. This provides a direct feasible quantum simulation. However, onlya few systems share the same dynamics, and as a result, one needs to findmethods in order to simulate systems whose dynamics are di↵erent from theones of controllable systems, making it possible to realize universal quantum

30

simulations. Nevertheless, this kind of simulations have an additional di�culty,namely, they require a more complex gate sequence. In the following we reviewthe digital methods.

In physics, most of the time, we deal with Hamiltonians that can be writtenas a sum of local terms,

H =NX

k=1

Hk, (4.1)

where generally each Hk acts on a small Hilbert space and H describes nquantum subsystems. In principle, if H takes the form of Eq. (4.1), with apolynomial growth N(n) in the number of sum terms, its dynamics can beimplemented e�ciently using the Trotter expansion or equivalent methods [1,9]. This is the reason why quantum simulators are e�cient. Notice, however,that implementing an arbitrary Hamiltonian evolution that is not associatedwith Eq. (4.1), that is, an arbitrary unitary gate upon n qubits, requiresan exponential number of single and two-qubit gates in n. Accordingly, itwould be ine�cient even for a quantum computer. Remarkably, most physicalprocesses can be reduced to interactions of the form of Eq. (4.1), such that aquantum computer or simulator is in principle e�cient for reproducing mostof the natural processes.

The method that Lloyd envisioned to simulate interactions of type Eq.(4.1) is the so-called Trotter formula [1, 9], that can be expressed in the form:

e�iHt = liml!1

(e�iH1t/le�iH2t/l · · · e�iHN

t/l)l. (4.2)

To approximate e�iHt to arbitrary precision, all that one has to do is todivide the time into l time intervals of length t/l and to apply sequentiallythe evolution operator of each local term for each time interval, repeating thesequence l times. The error can be made as small as desired just by increasingl. In the end, there will be other competing processes (e.g., switching times),such that there will be a limit to the number of local e�iH

k

t/l gates that canbe realized. Accordingly, one has to optimize l to get the best result possible.

Eq. (4.2) is just the lowest order approximation, emerging directly fromthe Baker-Campbell-Hausdor↵ formula. For H = A+B, it reads

e�i(A+B)�t = e�iA�te�iB�t +O(�t2), (4.3)

such that by decreasing �t (basically, �t = t/l, such that this amounts toincreasing l), the error is reduced quadratically.

31

There are higher-order expansions with errors that decrease faster withl, at the price of introducing more gates per Trotter step; the next orderapproximation is [1]:

e�i(A+B)�t = e�iA�t/2e�iB�te�iA�t/2 +O(�t3), (4.4)

with an error that decreases cubically with l. Depending on the implemen-tation and the problem, sometimes it will be better to use a higher-orderexpansion, or a lower-order one. But in all of them, by increasing the numberof Trotter steps l, the accuracy can be made as high as desired. One expectsthat it would be advantageous to make l very large, but this is not the casein general. Sometimes, to reduce the error per Trotter step can be a betterapproach. For this purpose, one may employ higher-order integrators, the socalled Lie-Suzuki-Trotter formula. For details we refer to Ref. [66].

In our case, we will make use of complex interactions in order to simu-late the desired Hamiltonian. In order to estimate the error we consider thefollowing expression:

e�iHt = (e�iH1t/le�iH2t/l · · · e�iHN

t/l)l +X

i>j

[Hi, Hj]t2/2l +

1X

k=3

E(k), (4.5)

where the higher order error terms E(k) are bounded by

||E(k)||sup l||Ht/l||ksup/k! (4.6)

and here ||A||sup is defined as the supremum, or the maximum expectationvalue, of the operator A over the states that play a role in the simulation.Hence, the total error in approximating e�iHt = (e�iH1t/l · · · e�iH

N

t/l)l is lessthan ||l(eiHt/l � 1 � iHt/l)||sup, that can be made as small as one wants bytaking l large enough.

To date several experiments have been performed using Trotter expansion.These include the realization of a quantum simulator for quantum chemistryin a photonic setup [19], as well as a recent work implementing a universaldigital quantum simulator in trapped ions [21].

It is noteworthy to mention that, in Eq. (4.1), it is not needed that theHilbert space of each Hk be small, it is enough that its dynamics can beimplemented in polynomial time. The size of the Hilbert space of Hk couldbe large with the condition that the number of available gates required toimplement it is polynomial. In the general case, the dimension of the total

32

Hilbert space associated with H can be large (say, for more than 40 qubits, itis beyond classical computation).

In this Thesis, we will analyze a digital quantum simulation of spin chainsin superconducting quantum circuits. As it is shown in Chapters 2 and 3,spin systems and superconducting circuits do not share the same Hamiltonian.Hence, we need to employ a gate decomposition to achieve the same behavior.We study in detail this proposal in Chapter 5.

33

Pagina en blanco

34

Chapter 5

Simulating spin models insuperconducting circuits

Until now we made a brief review of spin models, superconducting circuits andquantum simulators, and in this chapter we explain the main results of thisThesis: how to simulate a many-body spin dynamics making use of digitalmethods in superconducting circuits.

Circuit Quantum Electrodynamics is a very promising quantum technologyto realize a quantum computer due to its good scalability features. Actually,single and two-qubit gates with high fidelity [32], preparation of complex entan-gled states [33] and basic protocols for quantum error correction [34] have beenperformed in this system. When making proposals for quantum simulationsin superconducting circuits, it is advisable that they are realistic according tocurrent or near future technology. Accordingly, we make here our proposalstaking into account state-of-the-art experimental parameters.

In this work, we present the first proposal of a digital quantum simulationin cQED [67]. So far, digital methods have been employed in other quantumtechnologies such as trapped ions [25], but never before employing supercon-ducting qubits. Although analog quantum simulations have been already con-sidered in this plaform [20], the techniques proposed by Lloyd [9] provide theopportunity to generate interactions that could not be achieved in other ways.

Below we investigate the possibility of implementing digital quantum sim-ulations of spin Hamiltonians in a superconducting setup consisting of severaltransmon qubits coupled to a coplanar waveguide resonator. Indeed, we willshow that a variety of spin dynamics can be retrieved provided elementary

35

single and two-qubit gates.

We investigate the possible models, simulation times, and fidelities withcurrent cQED technology, showing the current simulating power of supercon-ducting qubits in terms of digital simulations. We estimate coherence timesand gate errors needed in order to provide simulation of relevant dynamics.In that way, one could estimate which resources are required to implementin a realistic setup a universal quantum simulator of spin dynamics, capableof simulating arbitrary many-qubit spin Hamiltonians, bounded by realisticsimulation times.

In the following we present the studied physical models in circuit QED.

5.1 XYZ Hamiltonian (2-qubit case)

Here we show the implementation of the Heisenberg spin model in circuit QED.For two qubits, the Heisenberg model reads:

H = ~Jxx�x1�

x2 + ~Jyy�y

1�y2 + ~Jzz�z

1�z2. (5.1)

Following the Trotter method, one can see that while this complete Hamil-tonian is hard to implement directly with superconducting qubits, implement-ing some of these terms separately is feasible. For instance, it is well knownhow to generate an XY Hamiltonian through the known exchange interactionfor spins (see Chapter 3),

Hxy12 ⌘ g2

2�(�x

1�x2 + �y

1�y2) ⌘ ~J (�x

1�x2 + �y

1�y2) . (5.2)

Assuming we have su�cient controllability over the interaction time and thepossibility to make global rotations—even without single-qubit addressing—,

Rx(⇡/4)�yRx(⇡/4)

�1 = exp(�i�x⇡/4)�y exp(i�x⇡/4) = �z,

Ry(⇡/4)�xRy(⇡/4)

�1 = exp(�i�y⇡/4)�x exp(i�y⇡/4) = ��z; (5.3)

we can transform the XY Hamiltonian as it is shown below:

Hxz12 = Rx(⇡/4)H

xy12R

†x(⇡/4) = ~J (�x

1�x2 + �z

1�z2) ,

Hzy12 = Ry(⇡/4)H

xy12R

†y(⇡/4) = ~J (�z

1�z2 + �y

1�y2) . (5.4)

36

One can now implement the desired XYZ Hamiltonian according to theprotocol:

Step 1: The qubits interact for a time �t according to the XY Hamiltonian.

Step 2: Apply single-qubit rotation Rx to both qubits.


Step 4: Apply inverse single-qubit rotation Rx to both qubits.

Step 5: Apply single-qubit rotation Ry to both qubits.


Step 7: Apply inverse single-qubit rotation Ry to both qubits.

HxyHxy

RxHxy

Ry

Rx Ry

R†y

R†y

R†x

R†x

Figure 5.1: Protocol for Heisenberg model with two qubits.

The unitary evolution would be:

U = exp (�iHxy12 �t) exp (�iHxz

12 �t) exp (�iHzy12 �t)

= exp (�i2J�t(�x1�

x2 + �y

1�y2 + �z

1�z2)) . (5.5)

Notice that in this case just one Trotter step is needed to achieve a simula-tion without errors (the only source of error would be the gate errors), becausethe three interactions XX+YY, XX+ZZ and YY+ZZ commute among them-selves. As given by Eq. (4.2), the sequential application of this 6-step protocolallows us to extend the simulation of the complete XYZ spin Hamiltonian foran arbitrary interaction time.

It is worth mentioning that in case the three operations did not commute,then the error coming from the Trotter formula would be approximately pro-portional to the sum of the commutators of these Hamiltonians. In addition,as the error is also proportional to 1/l where l is the number of Trotter steps,the more Trotter steps we do, the smaller the error. In contrast, the number of

37

gates we need to apply to accomplish the protocol is proportional to the num-ber of Trotter steps, and these gates introduce an error due to the imperfectimplementation in experimental set-ups. Therefore, in general we will needto study which the optimal number of states is for each problem taking intoaccount errors coming both from the Trotter expansion and from the imperfectgates.

In this case, our protocol makes use of eight single-qubit gates and threetwo-qubit gates. As there is no error coming from the Trotter expansion, theprotocol can be done in a single step. These gates have a 99% and 95% fidelity,and 10ns and 20ns execution times respectively [62, 68]. This will give a totalfidelity of the protocol around ⇠ 77% and a total execution time of 0.10µs,which is well below the T1 and T2 in current experiments.

In Fig. 5.2 we plot the fidelity of a state evolution with the unitary (5.5)in one Trotter step for the case with gate errors with respect to the ideal caseduring a phase given by ✓ = Jt = ⇡/2. While the black (dashed) line shows theevolution with two-qubit gate errors, the red (solid) one includes errors bothin single and two-qubit gates (see section 5.4 for the detailed error analysis).The dynamics associated to this phase is non-trivial, i.e., the evolved state issignificantly modified.

0 0.31 0.63 0.94 1.26 1.570.8

0.85

0.9

0.95

1

�

F

Figure 5.2: Fidelity of the evolved states in the Heisenberg model for twoqubits during a phase ✓ = ⇡/2 of the case with gate errors with respect to theideal case. The black (dashed) line shows the evolution considering an errorin two-qubit gates while the red (solid) one considers errors both in single andtwo-qubit gates. The dynamics associated to this phase is non-trivial, i.e., thesurvival probability of the initial state is significantly modified.

38

This is the first proposal for an implementation of XYZ interactions incircuit QED. Additionally, this is a first step approach to digital quantumsimulation techniques in circuit QED, paving the way to more complex proto-cols.

5.2 XYZ Hamiltonian (3-qubit case)

Let us now consider a 1D chain of three spins. For the nearest-neighbor inter-action the Hamiltonian is:

H = ~Jxx(�x1�

x2 + �x

2�x3 ) + ~Jyy(�y

1�y2 + �y

2�y3) + ~Jzz(�z

1�z2 + �z

2�z3). (5.6)

One can follow a similar approach for the implementation of the Hamilto-nian, as shown in Fig. 5.3:

Step 1: The qubits 1 and 2 interact for a time �t according to the XY Hamilto-nian.


Step 3: Apply single-qubit rotation Rx to the three qubits.



Step 6: Apply inverse single-qubit rotation Rx to the three qubits.

Step 7: Apply single-qubit rotation Ry to the three qubits.



Step 10: Apply inverse single-qubit rotation Ry to the three qubits.

39

Hxy

Hxy

Hxy

Hxy

Hxy

Hxy

R†y

R†y

R†y

R†x

R†x

R†x

Ry

Ry

RyRx

Rx

Rx

Figure 5.3: Protocol for digital quantum simulation of Heisenberg model withthree superconducting qubits.

The unitary evolution is:

U = exp (�iHxy12 �t) exp (�iHxy

23 �t) exp (�iHxz12 �t)

⇥ exp (�iHxz23 �t) exp (�iHyz

12 �t) exp (�iHyz23 �t) (5.7)

First, we calculate which the error coming from the Trotter formula is, Eq.(4.5):

X

i>j

[Hi, Hj]t2/2l =

~2g22l

2it2[��x1�

y2�

z3 � �x

1�z2�

y3 + 3�y

1�x2�

z3

��y1�

z2�

x3 + 3�z

1�x2�

y3 + �z

1�y2�

x3 ]. (5.8)

In this case, the terms of the Hamiltonian do not commute, so there isan error proportional to their commutators. One can simulate a non-trivialdynamics of a state considering perfect gates. In case the dynamics is not thesame as the one with the ideal Hamiltonian, it means there is digital error, orthe sum of the commutators is non-zero.

One the one hand, this error is proportional to (1/l) as is explained above,so the more Trotter steps we consider, the lower the digital error. On theother hand, the more Trotter steps we take into account, the more gates mustbe executed and the higher the single-gate error we get from the imperfectgates. Thus, here we analyze which is the optimal number of Trotter steps byconsidering both contributions to the error.

Given that in each Trotter step we apply 4 single-qubit gates (at di↵erenttimes) and 6 two-qubit gates we expect it takes about 0.16µs. The transmonand Xmon qubits have a decoherence time much longer than this, so one mayin principle apply from 100 to 1000 Trotter steps, depending on the physicalsystem.

40

0 0.31 0.630

0.02

0.04

0.06

0 0.31 0.630

0.1

0.2

� �

1�

F

a) b)

Figure 5.4: Fidelity loss for a Trotter implementation of the Heisenberg modelfor 3 qubits during a phase ✓ = ⇡/4. Curved lines show the error due to theTrotter expansion while the horizontal lines show the error due to imperfectgates. a) Error in each Trotter step: 10�2. Red (solid): l=3, black (dashed):l=5. b) Error in each Trotter step: 5⇥10�2. Red (solid): l=2, black (dashed):l=3. When the curves are lower than the corresponding horizontal lines, themain source of error is due to the accumulated single-gate imperfections. Whenthe curves become higher than the corresponding horizontal lines, the domi-nating error is the digital one.

Figure 5.4 shows the errors during a certain dynamics. The horizontal linesshow the error of the imperfect gates multiplied by the number of Trotter steps,i.e., the total accumulated error due to individual gate imperfection, while thepoints show the decrease of the error with increasing number of Trotter steps.In this way, one can straightforwardly check during the evolution which themain source of error is. The errors due to imperfect gates that we consider arestill lower than the current experimental ones such that in this case technologymust improve to make near one fidelities (see section 5.4 for the detailed erroranalysis).

41

5.3 XX Hamiltonian with frustration (3-qubit

example)

Let us consider a 3-spin Ising model with periodic boundary conditions. Thismodel includes frustration for the case of antiferromagnetic interactions, and,when scaled to many spins, it is ine�ciently solvable in a classical computer,while e�cient for a quantum simulator. The coupling is the same for the threequbits, J > 0, so one gets a Hamiltonian with frustration that can be writtenas:

H = ~JX

i<j

�xi �

xj . (5.9)

In order to implement this Hamiltonian we need to apply a ⇡/2 rotationin one of the qubits, to cancel the YY part,

Rx(⇡/2)�yR†

x(⇡/2) = exp(�i�x⇡/2)�y exp(i�x⇡/2) = ��y.

(5.10)

Hence, the Hamiltonian transforms as the following:

Hx,�y12 = Rx(⇡/2)H

xy12R

†x(⇡/2) = ~J (�x

1�x2 � �y

1�y2) . (5.11)

The protocol, as shown in Fig. 5.5, consists of the following steps:


Step 2: Apply single-qubit rotation Rx to the qubit 1.


Step 4: Apply inverse single-qubit rotation Rx to the qubit 1.




42

Hxy Hxy Hxy Hxy

Hxy Hxy

R†x Rx R†

x Rx

R†x Rx

Figure 5.5: Protocol for digital quantum simulation of a frustrated Ising modelwith three superconducting qubits.






The unitary evolution is given by:

U = exp (�iHxy12 �t) exp

��iHx,�y

12 �t�exp (�iHxy

13 �t)

⇥ exp��iHx,�y


23 �t) exp��iHx,�y

23 �t�

(5.12)

As the terms of the Ising Hamiltonian naturally commute, there is no errorfrom the Trotter expansion. Hence, proceeding similarly as with the two-qubitHeisenberg model, we only need to apply one Trotter step to simulate the Isingmodel. Moreover, the error due to imperfect gates is minimal, since only onestep is needed. We obtain a fidelity of the protocol around ⇠ 64%. The timeit takes to execute all the gates is of 0.18µs.

Figure 5.6 shows the fidelity of a state evolved with this method taking careof the errors coming from the imperfect gates (see section 5.4 for the detailederror analysis). Given that in this case there is not a big di↵erence betweenthe fidelities taking into account errors from the single and the two-qubit gateswe only show the one that takes into account both at the same time. This isdue to the fact that there is no error coming from the single-qubit gates att = 0.

43

0 0.31 0.63 0.94 1.26 1.570.75

0.8

0.85

0.9

0.95

1

�

F

Figure 5.6: Fidelity of the evolved state in the frustrated Ising model for threequbits during a phase ✓ = ⇡/2 considering both single- and two-qubit gateerrors. The dynamics associated to this phase is non-trivial, i.e., the survivalprobability of the initial state is significantly modified.

We have also studied the Ising model with a transverse magnetic field. Inthis case the Hamiltonian is,

H = ~JX

i>j

�xi �

xj +B

X

i

�yi . (5.13)

Now the terms of the Hamiltonian do not commute, so we need to ap-ply more than one Trotter step to achieve adequate fidelities. The unitaryevolution in this case is given by,

U = exp (�iHxy12 �t) exp

��iHx,�y


13 �t)

⇥ exp��iHx,�y


23 �t) exp��iHx,�y

23 �t�

⇥ exp (�iB�y1�t) exp (�iB�y

2�t) exp (�iB�y3�t)

= e�i2J�t(�x

1�x

2+�x

1�x

3+�x

2�x

3 )e�iB�t(�y

1+�y

2+�y

3 ) (5.14)

Figure 5.7 shows which the optimal number of Trotter steps for this simu-lation is, considering a certain error for each step due to the imperfect gates(see section 5.4 for the detailed error analysis). Even though the errors perTrotter step are still not achievable, this analysis makes an extrapolation fornear future technology improvements.

44

0 0.31 0.630

0.02

0.04

0.06

0 0.31 0.630

0.1

0.2

� �

1�

F

0 0.31 0.6300.01

0.03

0.05

0.07

0 0.31 0.630

0.1

0.2a) b)

Figure 5.7: Fidelity loss for a Trotter implementation of the Ising model for 3qubits with a transverse magnetic field during a phase ✓ = ⇡/4. The curvesshow the error due to the Trotter expansion while the horizontal lines showthe accumulated error due to imperfect gates. a) Error in a Trotter step:10�2. Red (solid): l=3, black (dashed): l=5. b) Error in a Trotter step:5 ⇥ 10�2. Red (solid): l=2, black (dashed): l=3.When the curves are lowerthan the corresponding horizontal lines, the main source of error is due to theaccumulated single-gate imperfections. When the curves become higher thanthe corresponding horizontal lines, the dominating error is the digital one.

45

5.4 Error analysis

Quantum technologies have been improved along many years, but at the mo-ment they are still not perfect. Current technology allows to do quantumsimulations with high fidelities but the errors coming from the imperfect exe-cutions of quantum gates should be taken into account when making proposals.Below we make an error analysis based on realistic gate imperfections in cur-rent experiments, which have been used in sections 5.1, 5.2 and 5.3.

In the literature one can find which the decoherence times for the gates are.Making use of this information it is possible to calculate how much decoherenceis introduced by the gate. Let us consider a general pure state of a qubit:

| i = ↵|ei + �ei�|gi (5.15)

Then, the time-dependent density matrix of the qubit, ⇢(t), is a↵ected by thedecoherence in the following way:

⇢(t) =

✓↵2e�t/T1 ↵�e�i�e�t/T2

↵�ei�e�t/T2 1 � ↵2e�t/T1

◆, (5.16)

where T1 and T2 are the relaxation and dephasing times which are related as

1

T2=

1

2T1+

1

T�

, (5.17)

T� being the pure dephasing time. Neglecting the contribution of this one, thedensity matrix can be written in terms of the relaxation time:

⇢(t) =

✓↵2e�t/T1 ↵�e�i�e�t/2T1

↵�ei�e�t/2T1 1 � ↵2e�t/T1

◆. (5.18)

Straightforwardly, we calculate the fidelity of ⇢(t) with the pure state | i:

h |⇢(t)| i = �2 + e�t/T1 [↵4 + ↵2�2(2e+t/2T1 � 1)]. (5.19)

Now, we integrate over the possible states in order to obtain the fidelityaverage. For this, we use the condition of normalization to rename ↵ and � ascos ✓ and sin ✓ respectively.

Fav =

Rh |⇢(t)| iR

h | i, (5.20)

46

whereZ 2⇡

0

Z ⇡

0

h |⇢(t)| id✓ sin ✓d� = 2⇡2

15(10 + e�t/T1 + 4e�t/2T1), (5.21)

Z 2⇡

0

Z ⇡

0

h | id✓ sin ✓d� = 4⇡. (5.22)

Consequently, the average fidelity over all possible states is:

Fav =1

15(10 + e�t/T1 + 4e�t/2T1). (5.23)

Now we consider the typical decoherence times for superconducting qubits,and also the execution times for single-qubit gates. Introducing these valuesin the expression calculated above, one gets a fidelity larger than 99%. Thisresult agrees with the experimental fidelities for single-qubit rotations. It ispossible to do a similar approach for multiqubit gates.

We consider now the error that one makes in applying a sequence of Ngates each one with its own error ✏i versus the ideal faultless gates. The idealevolution under the single perfect gate is given by the unitary operator U ,while the real gate is described by U . The di↵erence can be written

Ui � Ui = ✏iOi (5.24)

where kOik = 1.

Applying sequentially these gates one has, for ✏i ⌧ 1,

U =NY

i

Ui =NY

i

(Ui � ✏iOi) ⇡ U �NX

j=1

Pj

NY

i=1

Ui, (5.25)

where Pj acts on a sequence of UNUN�1 · · ·U1 as

Pj[UNUN�1 · · ·U1] = [UNUN�1 · · · ✏jOj · · ·U1] (5.26)

The norm of the error is bounded by

kU � Uk .NX

i=1

✏ikOik =NX

i=1

✏i, (5.27)

and this gives an error on the total unitary which is bounded under this ap-proximation by the sum of the single errors on each gate. The higher order

47

terms in the error go like ✓Nk

◆ kY

i=1

✏i, (5.28)

and are negligible in the limit of small single gate errors ✏i.

In order to make a model of the imperfect execution of the gates for thecomputational simulations, we have characterized the unitary transformationsdisrupting the Pauli matrices:

�i = �i + ✏A. (5.29)

The parameter ✏ is introduced by hand for establishing an estimate of the errorwhereas A is a 2 ⇥ 2 random Hermitian matrix with kAk ⇠ 1. In order toobtain this we have considered a random weighted sum of the identity and thePauli matrices:

A = C1 + C2�x + C3�

y + C4�z, (5.30)

where Ci 2 [�1, 1] 8i = 1, ..., 4.

To correctly estimate the size of the error ✏, we choose a random state andapply both a perfect rotation and an imperfect one, calculating the fidelitybetween them. Doing this for a large number of states, we modify ✏ until theaverage fidelity is the one we want to simulate. Let us show an example:

We consider two ⇡/2 rotations along OX for a qubit in a state | i, oneperfectly implemented and the other with error:

U | i = exp(�i�x⇡/2)| i,U | i = exp[�i(�x + ✏A)⇡/2]| i. (5.31)

We calculate the average fidelity between these states for N initial states:

Fav =1

N

NX

i

|h i|U †Ui| ii|2. (5.32)

Notice that the imperfect unitary is di↵erent each time we employ it becausethe matrix A is random.

Varying ✏ by hand, one can find which is the correct number that weshould introduce to obtain the typical fidelity of a single-qubit gate. For 99%fidelity, the error size that we should introduce for a phase of ✓ = ⇡/2 is about✏ ⇠ 0.012, whereas for ✓ = ⇡/4 it is about ✏ ⇠ 0.18. As can be seen, the

48

smaller the phase, the larger the introduced error should be to get the samefidelity.

This method can also be applied for two-qubit gates. In this case, althoughwe use 4 ⇥ 4 matrices we use the same model. For example, the gate �x

1 ⌦ �x2

during a phase of ✓ = ⇡/2 with error can be modeled as below:

U = exp(�i(�x1 + ✏A1) ⌦ (�x

2 + ✏A2)⇡/2). (5.33)

Similarly to the single-qubit case, we obtain the average fidelity for severalstates, and varying ✏ we finally achieve a 95% fidelity introducing an error sizeof ✏ ⇠ 0.076.

By this method, we are able to simulate the imperfect implementation ofthe gates on the computer. In this way, we can accomplish numerical calcula-tions and make realistic proposals for nearly future experiments.

A more general error model would involve considering completely posi-tive maps instead of random unitary matrix errors. However, this implies anapproach for the numerical simulation di↵erent from the standard Trotter evo-lution, with the consequent increased di�culty. Our method will be accuratefor simulating many kinds of experimental errors, e.g., standard magnetic fieldor charge fluctuations, and a master equation approach is in general not neededgiven that the qubit decoherence times T1 and T2 are much longer than theprotocol time.

49

Pagina en blanco

50

Chapter 6

Conclusions

In this Thesis, we propose a digital quantum simulation of spin systems insuperconducting qubits. To achieve this goal, we introduce the sections: spinsystems, circuit Quantum Electrodynamics, and quantum simulators. Finally,we provide with the last section where our results are explained in detail.

In the following, we show the conclusions of each section:

• We have analyzed the contribution of magnetic dipoles to the electro-magnetic field by realizing a multipole expansion. We have reviewedthe discovery of the spin in the experiment of Stern and Gerlach and theproperties of this magnitude. In addition, we have studied two basic spinsystems: the Ising model and the Heisenberg model, in which both clas-sical and quantum scenarios have been presented, studying the cases oflattices in one and two dimensions. We have also introduced the conceptof frustration in these models.

• We have studied circuit Quantum Electrodynamics analyzing the foun-dations, e.g., the Josephson e↵ect. In the same way, we have shownhow to make a superconducting qubit by using a nonlinear element, aJosephson junction, and a capacitor, constituting a superconducting is-land with the correct selection of external parameters. Furthermore, wehave analyzed the transmission line resonator used to manipulate thesuperconducting qubits within it and, besides, to realize measurementson the system. We have examined the complete Hamiltonian of cQED,we have also obtained the detuned Jaynes-Cummings Hamiltonian andthe exchange gate from it. We have also introduced the DiVincenzo cri-teria, which list the requirements that a quantum system must fulfill to

51

implement universal quantum computation, confirming that cQED is anincreasinlgy important quantum technology.

• We have presented a short review of quantum simulators, where we haveintroduced Feynman’s idea: the simulations of a quantum system makinguse of another quantum system may be, in principle, more e�cient thanthe classical simulations. We have listed a basic set of criteria thatthe simulating system must satisfy. Additionally, we have introducedboth analog and digital methods in quantum simulations, giving theadvantages and disadvantages of each one. We have also analyzed theTrotter expansion in digital quantum simulations.

• We have studied the quantum simulation of spin chains in superconduct-ing circuits using digital quantum simulation protocols. We have consid-ered Heisenberg and frustrated Ising models for up to three qubits, andwe have shown the feasibility of the simulation of the dynamics relatedto these models with current superconducting state-of-the-art technol-ogy. We have done an exhaustive analysis of the error coming from theimperfect implementation of single and two-qubit gates in cQED, andalso the error coming from the digital methods used in the simulation.

Summarizing, cQED is an excellent quantum technology for quantum sim-ulations. We have demonstrated that it can reproduce with good fidelitiesthe behavior of spin interactions by using digital methods. In the near fu-ture, these protocols may be extended to many-qubit spin models and morecomplex interactions, paving the way towards universal quantum simulationof spin dynamics in circuit QED setups.

52

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