Chemical Process Considerations in Physical Realizations of Quantum Computing Ion Traps

7/27/2019 Chemical Process Considerations in Physical Realizations of Quantum Computing Ion Traps

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CHEMICAL PROCESS CONSIDERATIONS IN PHYSICAL

REALIZATIONS OF QUANTUM COMPUTING ION TRAPS

Arjan Singh Puniani

University of California, Berkeley

Department of Chemical and Biomolecular Engineering

Spring 2013

Abstract

Numericalsimulation of quantum systems is computationally difficult because the quan-

tity of information required to specify a physical configuration scales exponentially with the

number of particles [2]. This letter will briefly survey the basic design parameters experi-

mentalists subscribe to when architecting the quantum simulator, a brief exposition on the

governing Hamiltonian along with the physical dynamics that enable efficient quantum

simulation, and major pitfalls. Special emphasis will be on the process engineering me-

chanics in surface-electrode radio frequency (rf) ion traps for trapping and controlling ions.

Most speculative conjectures in physically implementing the quantum computers (QC)

qubit representation is through photons, including quantum dots and cavity QED devices.

Recruiting atomic and nuclear states in lieu of the photon motivates ion trapping experi-

mentation, which continue to challenge researchers due to the difficulty in observing and

controlling otherwise robust quantum state arrays. Ion traps, which confine ions to verysmall regions of space, represent qubits as hyperfine states of the atom, and exquisite con-

trol was observed for the lowest level vibrational modes (better known as phonons) by suit-

able electric and magnetic fields. As individually trapped ions serve as leading candidates

for physically-realizable quantum simulators, an increasing emphasis on surface traps is

observed in the literature [3]-[5], [10] due to amenability of the scalable architecture with

standard fabrication techniques (including photolithography, wire bonding, metal evapo-

ration, and many more). Of particular relevance to micro-fabrication in this letter is an em-

phasis upon junctions and backside loading holes incorporated into the surface geometry

of the trap.

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Feynman was first to suggest the use of a

well-known quantum system to experi-

mentally simulate complex states of matter in

a 1982 paper [1]. The peculiar nature of quan-

tum mechanics (probabilities encoded with

elements from C) insists the information re-quired to specify a particular system of quan-

tum states grow exponentially as the number

of particles simulated increases.

Regardless of specific implementation (be

it cavity QED devices, nonlinear optical me-

dia, molecular NMR, etc.), a QC requires a

robustly-defined quantum bit (qubit). Clas-

sical information theory defines the bit as a

system capable of assuming either one or an-

other distinct stateeach representing either

(exclusively) 1 or 0. The fundamental build-

ing blocks of computation both classically and

quantum mechanically are logical operators

(gates). Classical gate operations are lim-

ited to the identity, 1, which is succinctly ex-

pressed as (0 0, 1 1), and logical NOT:(0 1, 1 0) [5]. A qubit, in direct contrast,is a two-level quantum system, constructed

in a two-dimensional complex Hilbert space.

The quantum mechanical correspondence be-

tween the classical notion of 0 or 1 is de-scribed with the familiar ket notation in the

quantum mechanical workspace:

0Classical Bit |0Quantum Bit

1

0

,

1Classical Bit |1Quantum Bit

0

1

The principle of superposition, often the

first postulate when describing QM, suggeststhe qubit representation assume the form:

| =|0+|1

where , C represent probability ampli-tudes, and constrained by physical reality

(unitarity of probabilities):

||2 +||2 = 1Ultimately, the qubit spans an entire frontier

of solutions in the complex Hilbert space, pa-

rameterized continuously by,. Up to an ar-bitrary global phase factor of ei, the generic

qubit can be represented in an Eulerian fash-

ion:

| = cos 2|0+ei sin

2|1

=

cos 2

ei sin 2

, [0,], [0,2)

(1)

The number of quantum states completely

dwarves the possibilities assumed by the bits

realized in todays Turing machinesprecisely

2. This macroscopic bias to mutually exclu-

sively assign one state or the other to a clas-

sical bit fuels the peculiar correspondence be-

tween quantum and everyday systems. Con-

sider the fair coin. This can certainly repre-

sent a bit, provided a mapping (e.g. {Heads 1,Tails 0}) is made, since the number ofstates is binary. Quantum mechanics is (in a

sense) unencumbered by this limitation dueto the superposition principle, granting an in-

finite spectrum of possibilities. While it may

be tempting to conclude a single qubit could

encode an infinite amount of information,

the reality is that an infinite number of bits

would be required to characterize and .

Indeed, a physical limitation caps our infor-

mation extraction capabilities, as a measure-

ment made on an arbitrary polarization state

n of a qubit along an arbitrary axis n yields

precisely 1 bit worth of information: eithern is +1 or -1. Completely specifying both

and requires infinitely many measurements

on identically-prepared statistical ensembles

of single-qubit states. Measurements collapse

the wave-function of fleeting superpositions

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into a single state with probability modulus of

the respective eigenvalue.

Pausing for qualitative reflection, exper-

iments suggest a delicate balance must be

sought here, as the demands for the qubits

operational fitness are rather discriminatory.Quantum mechanics (which QCs obey) is es-

sentially applied probability, and physical ob-

servables may manifest in systems unitarily

only from ensembles of identically-prepared

initial configurations. Maintaining the super-

position of states allows QCs to circumvent the

exponential complexity faced by Turing ma-

chines, as computational paths are traversed

simultaneously across the superpositionin

parallel, as opposed to serially. Fundamen-

tally, the building a good quantum computer

is a constraint problem: a system internally

isolated enough to discourage qubits coupling

with the environment, but accessible enough

externallyto facilitate manipulation.

Ultimately, from surveys published else-

where [6] [7] [8], intractability of a QC is

avoided even with the physical limitations in-

herent to tasking a two-level quantum system

as the systems qubit. In the coarsest general-

ization of these results, three primary designparameters appear to have entered consensus:

1. if a two-level quantum system can be

prepared consistently in a well-defined

state, then this particular two-level sys-

tem qualifies as a robust qubit (in accor-

dance with the literature, the initialized

state constituting the statistical ensem-

ble is known as the fiducial qubit) state;

2. if states can be transformed into otherstates unitarily; and

3. the measurement bases is computa-

tional in nature; for example, a par-

ticularly good system is {|0, |1}, be-cause qubit polarization can be mea-

sured along the z-axis. In this particular

case, we can use a Pauli matrix:

z =

1 0

0 1

,

which we know is satisfactory, since |0and |1 are eigenstates ofz. Continu-ing along this train of thought, suppose a

qubit state was best described with Eqn.

(1). The realization of either 1 or 0

manifests as a measurement yielding ei-

ther z = +1 or z = 1. The probabilityassociated with each measurement is:

p0 = |0||2 = cos2

2,

p1 = |1||2 = sin2 2

(2)

While these points seem to solely bela-

bor the qualitative aspects, rigorous traction

in the underlying mathematical structure in-

spires confidence in the impending quantum

simulator. A quantum computation, at its very

essential core, can be reduced to a sequence

of unitary transforms that obey the statisti-

cal theories underlying quantum mechanics.However, an efficient implementation of these

unitary transforms requires clever and novel

engineering techniques and physical insight.

Nakaharas treatise expounds this point lucidly

[9]; thus, summarized only cursorily below:

Quantum Gates: A Mathematical

Treatment

The motivating force behind this section is to

focus on the more challenging elements of themathematical framework governing QCs. The

efficient implementation of the unitary trans-

forms is an edifice built upon a strong founda-

tion of classical computing algorithmic pedi-

gree, which ultimately assist in structuring the

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quantum computation via finitely-many ba-

sic transformation functions. One particularly

satisfying result is the rigorous proof establish-

ing equivalency between an arbitrary unitary

matrix and an approximation treatment of this

unitary matrix via a product of simpler uni-tary matrices. These atomic elements of the

unitary matrix describing the computation are

known as quantum gates, and analogize the

wires and transistors that define modern com-

puters, which are simply physical manifesta-

tions of recursive and primitive functions. A

powerful proof, sketched below, guides the de-

sign basis for the ion trapping QC.

A few refreshers on pre-requisite mathe-

matics follow: it is well known that for irra-

tional number , the set {exp (2i n) |n Z}is dense on the unit circle. This is equivalent

to the statement: the 1 1-matrix exp(2i)is universal for the set of all unitary 1 1-matrices. This is more intuitively understood

as an assertion that any complex number on

the unit circle can be expressed as some prod-

uct with exp (2i)the quantity exp(2i) is

a universal gate on the unit circle. Unitary ma-

trices U are d d of simple form if one of itsblock-diagonal forms is a 2 2-matrix of rota-tions:

U2 =

cos sinsin cos

,

This serves as a motivation for approximat-

ing unitary matrices as products of the simple-

formed matrices.

Unitary Transform Theorem. For every arbi-

trary and unitary 2 2-matrix, U, with > 0,there exists a classical algorithm computable

in the time polynomial in log1 unitary matri-

ces of the simple form U1,U2 such that:

||UU1()U2()|| < .Proof. (i) for the case of d = 1, designateU1() = ei(+

/2), which implies that ||U1

U1()|| < . Now, for the case ofd= 2,

U2 =

cos sinsin cos

,

we would take that:

U2() =

cos(+/2) sin(+/2)sin(+/2) cos(+}/2).

Hence, ||UU1()|| is

=

/2 /2/2

/2.

<

This statement is holds true for any unitaryU

in C2.

This establishes the universality of the QC

provided a quantum Turing machine is real-

ized from the ion trap implementation.

Returning to the classical NOT gate, we

now turn our attention to the realizability

of a quantum analogue to this Turing ma-

chine workhorse. Theoretically speaking,

this quantum NOT gate houses some process

which performs the following map on quan-

tum states: |0 |1 and |1 |0. Its tempt-ing to patch ad-hoc corrections to the classi-

cal NOT gate and tease a correspondence (Cf.

()3n in grand canonical ensembles prevalentin statistical mechanics). The first pitfall to this

nave correspondence is the absence of an an-

alytics prescription for the new solution fron-

tier the qubit function spans; that is, how does

one adjust the qubit in response changes to |0and |1? One solution for a N-ary array of ionsis to designate the ground state of the n-th ion

as:

|gn |0nand the excited state as:

|en |1n.

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Hence, the macroscopic superposition of

quantum registers represented by the ion trap

is:

| =2N1x

=0

cx|x

x

={0,1}N

cx|x,

where

|x = |xN1N1 . . . |x00,which will be extremely useful when describ-

ing the theory behind laser cooling. Let H0denote the Hamiltonian of the system without

laser cooling. If the laser configuration solely

affects the n-th ion and the frequency chosen

is n= x such that the ions equilibrium posi-tion coincides with a node of a stationary laser

wave, the corresponding Hamiltonian in the

interaction picture is:

Hn,q =N

2

|eqng|aei+|gneq|aei

where a, aare (respectively) creation and an-

nihilation operators for phonons, is the

Rabi frequency, is the laser phase, and is

the Lamb-Dicke parameter (explained below).

Without modifying the CM motion of ions but

tuning the laser frequency to the internal tran-

sition ( = 0), an ion of polarization q = 0 atan equilibrium position in the standing waves

antinode is described by the Hamiltonian:

Hn= (/2)|e0ng|ei+|gne0|ei

For the typical interaction time of laser pulse

k, t = k/, the unitary evolution operatoris:

Uk

n

()=

expi k

2|e0ng|e

i

+|g

n

e0

|ei

Thus establishing the correspondence:

|gn cos(k/2)|gn i ei sin(k/2)|e0n,

|e0n cos(k/2)|e0n i ei sin(k/2)|gn

Surface-Electrode Ion Traps

Introduction

The primary ingredients of the ion trap quan-

tum computer were laid out by Cirac and

Zoller in a landmark 1995 paper [4]. Many of

the most common physical-implementations

of the QC represent qubits use photons. It

is possible via ion trapping to represent a ro-

bust qubit through atomic and nuclear states.

A single nuclear spin is a better qubit than,

say, a coin, since the superposition of quan-

tum states (aligned or not aligned with an ap-

plied magnetization, H) resists decoherence

better than other qubit representations. In

fact, this systems coupling to the environment

is so small, QC scientists find it challenging to

tease out the orientation of these nuclei, sim-

ply because these particular qubits are exceed-

ingly superior in retaining its quantum prop-

erties, than, say a harmonic oscillator QC. As

the qubits are virtually inaccessible, it is dif-

ficult to build a useful computing system thatrequires the participation and manipulation of

the quantum transistors without being able

to control them very well. Spins are measured

to establish the on-off correspondence with

the traditional bit.

Spin is fundamental property of elemen-

tary particles. A particle with nonzero spin be-

haves as if a current loop were run equatorially

on the surfaceand is only ever either inte-

ger or half-integer. However, the spin statesof an atom are very small relative to the en-

ergy scales corresponding to, say, the kinetic

energy experienced by atoms at room temper-

ature. However, one possible solution to this

obstacle is laser cooling.

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Brief Primer on Laser Cooling

One of the most clever solutions to observing

nuclear spin states as opposed to thermalized

kinetic noise was laser cooling ions.

Technically speaking, cooling must be suf-ficient to satisfy kBT z. While the truepotential is non-quadratic for large displace-

ments in any direction away from the trap cen-

ter, the one-dimensional harmonic approx-

imation is very much applicable and valid.

When higher order vibrational modes have

harmonic oscillations with higher energy in-

tensity than the center of mass mode, unique

behavior is observed, as ions are cooled to

their motional ground state. From the de-

sired goal ofkBT z, where T is the tem-perature reflecting the kinetic energy of the

ions, laser cooling relies on the fact that pho-

tons carry not only energy but momentum

(p = h/). Akin to the Doppler shift, wherethe whistles pitch of an approaching train is

higher than that of a departing train, an atom

moving towards a laser beam has a higher

pitchhigher-energy transition frequencies

than those found on atoms moving away from

the laser beam. If the laser is tuned such thatit is absorbed only by approaching atoms, then

the atoms slow down because the photons kick

them in theopposite direction. To cool beyond

the limit sideband cooling is applied to reach

the aspirational kBT.

Consider our representation of the ground

and excited states of the atom, |g and |e.Suppose the two-level system is separated by

energy0. One important assumption to fo-cus on here is the rate atwhich

|e

decays to

|g

via one-photon electric dipole transitions; weassume this is . Laser radiation at frequency

is applied to the two-level system, with con-

stituent photons of energy and momen-tum, k. Since k is the wave vector associ-ated with the monochromatic radiation wave-

length,, the correspondence holds:

k= 2

= c

Now suppose the ion travels towards the

laser with velocity, v. The frequency will be

Doppler-shifted to laser frequency, (1+ v/c).If the laser is detuned to below the transition

frequency, 0, such that the detune factor, ,

is precisely equal to Doppler factor:

=0 =v/c= kvzthen the ion will absorb a photon of en-

ergy , transitioning from |g to |e. Con-served momentum asserts that the traveling

ion experiences a momentum reduction ofk.Even when properly tuned (momentum com-

ponents are aligned along the translation axis),

the cooling limit for the applied laser is kBT /2, where is the radiative width of the tran-sition employed to cool the ion. In order to at-

tain the desired kBT z, sideband coolingis required.

Another important criterion to consider is

maintenance of the ion oscillations width be-

ing small relative to the wavelength of the in-

cident light. This is known as the Lamb-Dicke

criterionwith parameter:= 2z0/,

where = wavelength and z0 =

/2N M isthe characteristic length scale of the spacing

between ions in the trap. In order to be use-

ful for quantum computation, 1 at the veryleast, but ideally should be 1.

Geometry and Physics

An early apparatus of the ion trap consistedof four cylindrical electrodes, with its end seg-

ments biased to a different voltage than the

middleresulting in z-axially confined ions

conforming to the potential:

dc =U0[z2 (x2 +y2)]/2 (3)

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where U0 is the electrode endpoint biasing

voltage and is a geometric factor. How-

ever, this violates Earnshaws theorem, which

posits that charge cannot be confined in three-

dimensions by static potentials (the static field

configuration is a saddle-point, with a des-ignated trajectory for instability). Thus, the

potential must be generated by a rapidly-

oscillating voltage; in this case, such a poten-

tial is the radio frequency (rf) variant:

rf= (V0 cosTt+Ur)(1+ (x2 y2)/R2)/2 (4)where R is the new geometrical factor. As the

electrodes are capacitively coupled, the RF po-

tential is constant across each cylinder, and

the combination ofdc

and rf

over T

ulti-

mately results in a harmonic potential over x,

y, and z. When we account for the Coulombic

repulsion that arise from the ions, the govern-

ing Hamiltonian for N ions in a trap is:

H=N

i=1

M

2

2xx

2i +2yy2i +2zz2i +

|pi|2M2

+N

i=1

j>i

e2

40|rirj|

(5)

with M equal to the ion mass. Typical con-ditions assert that x,y x; hence, thetrapped ions assume a linear arrangement

along thezaxis (as shown in Figure 4).

The purpose of the trap apparatus de-

scribed above is to allow sufficient cooling

of the atoms to attain some semblance of a

stable initial computation state. Specifically,

we require that the vibrational state gener-

ates zero phonons. In addition to a natu-

ral starting state, a zero-phonon initialization

allows for the storage and manipulation ofpreviously computed resultsessentially, you

can preload intermediate calculated results to

build more complex solutions.

Formally speaking, the internal atomic

states relevant to a trapped ion are the addi-

tive sum (F) of the electron (S) and nuclear

spins (I): F = S+ I. An entire discipline is de-voted to this art: the addition of angular mo-

mentum. However, only a cursory survey of

the study is presented. Recall how a lasers

photon and an ion exchange units of angu-lar momenta. The atoms angular momentum,

however, is not limited to linear translation;

on the contrary: orbital, electron and nuclear

spin can absorb and dissipate angular mo-

menta. How this distribution occurs depends

on the energy state. Consider two fermions

(with spin-1/2) selected as a computational

basis. Explicitly, the state possibilities are:

|00

,|01

,|10

,|11

While this may span our state space, an

equivocally-comprehensive basis establishes a

correspondence with the states above:

|0, 0J =|01 |10

2

|1,1J = |00

|1, 0J =|01+ |10

2

|1, 1J = |11

(6)

What distinguishes (6) from equivalent de-

scriptions is the fact that these are eigenstates

of the total angular momentum operator:

J2 = j2x+j2y+j2z

where j2x = (X1 +X2)/2 and so on. Each of thestates,

|j, mj

J are eigenstates ofJ

2with eigen-

value j(j+1), and, simultaneously, eigenstatesofjz with eigenvalue mj. These states are nat-

ural, since thez-axis is usually the orientation

by which magnetic fields are aligned with the

magnetic moment in the Hamiltonian Bz(proportional to mj).

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Fabrication

Many research groups [12] are hoping to

fabricate surface-electrode radio frequency

ion traps for trapping and manipulating cal-

cium ions. Confinement forces are gener-ated through time-dependent potentials that

trap low kinetic energy running across elec-

trodes. A variety of techniques can be gleaned

from the relevant literature, with several im-

plementation schemes ranging from gold on

quartz to traditional printed circuit boards

to aluminum deposited upon silicon oxide.

The traps described in this letter are com-

parable in scale, and engineering techniques

that minimize the line of sight (LOS) between

the trapped ions and the exposed dielectrics

is emphasized to reflect the prevailing re-

search direction [10] in reducing stray electric

charges. To realize this particular design con-

sideration, the top metal layer of the traps

consisting of electrodes, leads, and necessar-

ily grounded regionsoverhang the support-

ing oxide pillars by 5m. This overhang is con-

trollable thanks to strategically-placed vertical

etch stops around pillars; the resultant over-

hang allows for vertical deposition of metalalong the top of the, say, aluminum, electrode

layer without shorting dc control or rf elec-

trodes. In addition, electrodes require a thick,

insulating dielectric film that may survive 4.2K

operating conditions without suffering sub-

stantial deformation for quantum computa-

tion. Internal stresses are exacerbated by ex-

treme temperature swings during the cooling

sequence. However, during cooling, the film

and substrate of ions shrink at different rates

due to unequal coefficients of thermal expan-sion. Because the substrate and oxide strips

run along the same length, the thermal expan-

sion differential will create an uncompensated

bending moment. Specifically, the film caus-

ing upward bending on the oxide introduces

tensile forces, whereas downward bending in-

troduces compressive forces. To reduce ox-

ide buckling, the chemical process objective is

minimizing material stresses. In a 2006 letter

[11], Seidelin alludes to an auxiliary study con-

ducted to determine the optimal parametersfor plasma-enhanced chemical vapor deposi-

tion (PECVD) yielding the lowest oxide stress.

The results are summarized below.

Typical micro-fabricated ion traps [12] use

PECVD to deposit oxides on Si wafers at 250C.Preliminary analyses showed that variations in

pressure, rf power, and stoichiometric ratios

of reactant gases impacted oxide stress non-

trivially. The parameters were varied to yield

the lowest stress oxide film, which was mea-

sured with a profilometer and optical stress

gauge. Oxide film veracity was established

with a refraction index measurement. In gen-

eral, pressure is inversely related to the oxide

stressindeed, the machine limit of 1500torr

yielded a 100MPa reduction in oxide stress.

As for rf power, a positive relationship exists,

with increasing rf power resulting in higher

oxide stress. Finally, to adjust stoichiome-

try, the ratio of nitrous oxide (N2O) to 2%silane gas was manipulated to establish a cor-

respondence. An explanation for this particu-

lar factor dependence lay in Si density within

the oxide film. Films with higher densities

are more stress-resistant. This is best illus-

trated with Figure 1s depiction of a disloca-

tion. An additional 1/2-plane of atoms from

the top layer introduces compressive stresses

at the top-half of the lattice, while the bottom-

half experiences tensile forces (tension). Of

course, the actual oxide stresses are muchmore complex than a single dislocation, but

this analysis serves as an analytical spring-

board in which we may extrapolate the conse-

quent high-stress dislocation array associated

with a low-density system. Additional consid-

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erations include dc rails inside rf rails to fa-

cilitate additional principle axis rotation and

compensation. The ventral surface of the chip

is evaporated with gold slightly off-normal

to the surface in order to coat the exposed

vertical edges of the Si substrate and elec-trode supportultimately preventing charge

buildup by pulling the ventral chip surface to

ground. Cross-sectional areas of the fabricated

ion trap from different perspectives are shown

in Figure 5.

Finally, a note about the rf capacity. The

ion traps rf capacity is determined after ap-

plying a 30MHz (2W) rf drive through a Q =150 resonator. Typically, the capacitance be-

tween the rf driver and ground is 10pF. Wedge-

bonded gold ribbons connect I/O electrodes

on the electrical plane to the package bond

pads and pulled taut to minimize supra-planar

projections on the the ion trap surface. The re-

sultant electrode array is shown in Figure 6.

Analysis

Similar to the case of a mass on a spring

behaving like a quantum system provided

environmental coupling is negligible, theseelectromagnetically-confined ions become

quantized when sufficiently isolated. For sim-

ple harmonic oscillators, it is well known that

energy levels are equally-spaced and in units

ofz. When N atoms are trapped in such amanner, the energy eigenstates represent the

vibrational modes of the entire linear chain of

ions moving together as if a single entity with

mass N M. Each z quantum of vibrationalenergy is known as a phonon. The phonon de-

scription of the trapped ions are valid only un-der particular conditions. For example, cou-

pling of the trapped ion system to the exter-

nal environment introduces thermalization,

randomizing the system and causing parti-

cles to behave classically. Nearby electric and

magnetic fields may excite ions into randomly

transitioning between energy eigenstates

thus introducing the universal nuisance of

noise. Imperfect voltage sources everywhere

introduce finite resistance, and this introduces

systemic Johnson noise, which causes fluctua-tions on time-scales trapped ions are sensitive

to. Shoddy patchwork of service fields E andB drive ion motion randomly in one direction,

and as randomness permeates theions shared

system, classical statistical averages become

more accurate tools than quantum mechan-

ics when describing quantum systems. Re-

searchers can control noise sources quite well

given todays technology, but must exercise on

the side of caution and avoid excessively heat-

ing and de-phasing the trapped ions. In the

harmonic approximation, the trapped ions ex-

hibit a discriminatory palette when determin-

ing which fluctuations to remain sensitive to

(drawn primarily towards high spectral den-

sity around z).

Physically speaking, the ions are observed

to hover 80m above the top electrode layer,

which is consistent with several simulations

[10]-[12]. The uncooled ion lifetime lasts from3-5min, and is a function of DAC cable shield-

ing; though this lifetime can reach the order

of hours in meticulously-prepared UHC cham-

bers with non-invasive control voltage sets for

storage and shuttling. Though the literature

reports a veritable spectrum of rf drive fre-

quencies, multi-atom trapping is best accom-

plished with higher rf drives. In particular, the

rf voltages were delivered through a cavity res-

onator ofQ

100 and amplitude ranging from

50-200V. The change in radial and axial secu-lar frequencies when stepping through the dc

voltage at a fixed rf voltage are used to deter-

mine geometric potential factors for the con-

trol electrodes. Of course, the applied dc offset

applied to the rf electrodes actually changes

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Figure 1: Dislocation illustration for Si lattice comprising the oxide layer in ion trap scheme.

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Figure 2: SEM image depicting 5m overhang from supporting oxide pillar. The 7m gap be-

tween electrodes is also visible [12]

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Figure 3: Schematics of the most enduring, classical ion traps. The modern quadropole ar-

rangement is a modified Paul trap.

Figure 4: Geometric abstraction of the linear quadropole trap.

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Figure 5: Cross-sectional area of (a) the overhead view of the ion trap and (b) the ion trap itself.

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Figure 6: Example electrode layout of surface-electrode ion trap.

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the radial secular frequencies and the princi-

pla axis rotation; hence, a correction via nu-

merical fitting of the functional dependence

between rf and rf voltage offset must be made

to the operating parameters.

Summary of Specific Ion Trap

QC Requirements

Representation of the Qubit. Hyper-

fine (nuclear spin state) of an atom; also:

lowest phonon levels of trapped atoms

(vibrational modes)

Unitary Evolution. Arbitrary transfor-

mations constructed from applied laser

pulses, which interact with atoms via the

Jaynes-Cumming mechanism. A shared

phonon state links qubits.

Preparation of Initial State. Atoms

cooled via trapping and optical pumping

to motional ground state and hyperfine

ground state

Measurement Readout. Measurements

made of hyperfine states.

Shortcomings. Phonon lifetimes are

short-lived and ions require continuous

effort for motional ground state prepara-

tion.

Closing Remarks

Performing Doppler and sideband cooling on

trapped ions results in quantum statistical en-semble initiation with 95% fidelity. Readout

of this computational measurement is accom-

plished primarily by collecting fluorescence

data from circularly polarized light undergo-

ing the cycling transition. This is how one

verifies truth-table fidelity of the implemen-

tation to a controlled-NOT gate, but in order

to become computationally useful, the life-

times of the trapped ions stable states must

be extended via the short-lived motional state

to exceed decoherence times that couple thesystem to the environment negatively. Scal-

ing to larger numbers of ions is conceptually

viable, but a scalable-in-principle technique

for trap fabrication must be established first.

The surface geometry is most amenable to

micro-fabrication, but several challenges per-

sist in, for example, low trap depth, and several

difficulties surrounding the working shuttling

junction. Overall, a scalable quantum com-

puter using ion traps will require creative new

thinking in process design and chemical engi-

neering.

References

1. Feynman, R.P. Simulating physics with

computers. Int. J. Theor. Phys. 21, 467-

488 (1982).

2. Aaronson, S. Why Quantum Chemistry

is Hard. Nature Phys. 10, 707-708 (2009).

3. Cai, J. A. et alLarge-scale quantum sim-

ulator on a diamond surface at room

temperature. Nature Phys. 9, 168-173

(2013).

4. Cirac, J.I., Zoller P. Quantum Computa-

tions with Cold Trapped Ions. Phys. Rev.

Lett. 74, 4091 (1995).

5. Benenti, G., Casati, G., Strini, G. Princi-

ples of Quantum Computation and In-

formation, Volume I: Basic Concepts.

World Scientific, Singapore(2004).

6. Monz, T. et al. Phys. Rev. Lett. 106,

130506 (2011).

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Figure 8: How spins of various fermionic origin contribute to atomic energy levels.

7. Maurer, P.C. et al. Science. 336, 1283-

1286 (2012).

8. Plaff, W. et al. Nature Phys. 9, 29-33

(2013).

9. Nakahara, M. and Ohmi, T. Quantum

Computing: From Linear Algebra to

Physical Realizations. CRC Press | A Tay-lor & Francis Book, Boca Raton(2008).

10. Nielsen M.A. and Chuang, I.L. Quan-

tum Computation and Quantum Infor-

mation. Cambridge University Press,

Cambridge, UK(2010).

11. Seidelin, S. et al. A micro-fabricated

surface-electrode ion trap for scalable

quantum information processing. Phys.

Rev. Lett. 96, 253003 (2006).

12. Wallace, M. Low Stress Oxides for use in

Micro-fabricated Ion Traps for Quantum

Computation. 2010 NNIN REU Research

Accomplishments

Appendix

Below, several basic quantum mechanical con-

cepts are detailed below:

Spin

A particle is said to have spin if it has a mag-

netic moment that appears to suggest the par-ticle itself is a composite particle with a current

loop running equatorially across the surface.

This interpretation fails to account for par-

ticles exhibiting nonzero spin, including the

electronwhich are elementary particles and

cannot be reduced materially furtheras well

as the quark (substituent of nucleons), which

does not appear to generate spin from orbital

motion. The final property that defies expla-

nation is the fact that particle spins are only

ever integers or half-integers.Particles with integer-valued spins are

known as bosons, and include familiar species,

such as photons. As photons are massless, the

possible spin states are 1 and not 0. Thesedichotomous states correspond to the familiar

17


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orthogonal polarization states light assumes.

On the other hand, half-integer spins corre-

spond to fermionselectrons, protons, and

neutrons, for example. Fermions are usu-

ally spin-1/2 particles, and many experiments

show fermions are either +1/2 (spin up) or -1/2 (spin down). The energy eigenstates of

atoms inextricably depend upon the coupling

of spins resulting in fermions occupying a rela-

tively small, concentrated system (the atom).

For example, 49C has a nuclear spin of 3/2. Be-

cause the spin is a magnetic moment, spins

will interact with a magnetic field, B. Elec-

trons, spin-1/2 particles of spin S, possesses

energy geS B. Whereas the nuclei, spin-3/2particles with spin I, have energy gnI

B (Cf.

Figure 8 for illustrative fan-out).

Transition Amplitudes

A common misconception in quantum

mechanics is the time evolution of kets

specifically, states and bases. However, it is im-

portant to distinguish between the time evo-

lution of state kets and the time evolution of

base kets. Starting from the simple eigenvalue

equation, it is natural to ask what the time evo-lution of this relation is:

A|a = a|aIn the Shrdinger picture, A is stationary, so

whatever base kets obtained that correspond

to t = 0 solutions must also be static. Unlikestate kets, the base kets actually dont change

in the Shrdinger representation. However, in

the Heisenberg picture, the eigenvalue equa-

tion assumes a different form:

A(H)(t) =UA(0)UWhen the Shrdinger eigenvalue equation is

evaluated at t = 0, the resultant Heisenbergequivalence is:

(UA(0)U)[U|a] = (a)[U|a]

and when the Heisenberg operator is substi-

tuted:

A(H)(U|a) = a(U|a)The Heisenberg-picture base kets, |a, tH,evolve through time as:

|a, tH =U|a

with U implying solution of the improperly-

signed Shrdinger equation:

i t

|a, tH = H|a, tH

Now, taking queue from Sakurai, suppose we

initialize a system, t= 0, with physical observ-able Aand corresponding eigenket,

|a

. If one

were interested in computing the probability

amplitude of the system at time t correspond-

ing to physical observable B with eigenvalue

b, the quantity that links the two is knownas the transition amplitude. The definition

is sufficiently flexible to accommodate both

A, B different and A, B same. Starting with

the Shrdinger picture, the state ket at time t

requires U|a, though the specific base kets,|a, |b, are stationary. Thus, the transitionamplitude is generically:

Base Bra| (|State Ket)

And, specifically, for the scenario described

above, the corresponding transition amplitude

is:

b| (U|a)And, from the Heisenberg-picture, the base

kets evolve oppositely:

(b|U) |aSince the Shrdinger and Heisenberg pictures

are equivalent, the composite transition am-

plitude from above is:

b|U(t, 0)|a

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Chemical Process Considerations in Physical Realizations of Quantum Computing Ion Traps