Stability thresholds and calculation techniques for fast entangling gates on trapped ions

0.91 MB·

by  C D B Bentley

Stability thresholds and calculation techniques for fast entangling gates on
trapped ions
C D B Bentley,1,∗ R L Taylor,1 A R R Carvalho,1,2 and J J Hope1
1Department of Quantum Science, Research School of Physics and Engineering,
The Australian National University, Canberra, Australia
2ARC Centre for Quantum Computation and Communication Technology,
The Australian National University, Canberra, Australia
(Dated: January 14, 2016)
Fast entangling gates have been proposed for trapped ions that are orders of magnitude faster
than current implementations. We present here a detailed analysis of the challenges involved in
6 performing a successful fast gate. We show that the RWA is a stable approximation with respect
1 to pulse numbers: the timescale on which we can neglect terms rotating at the atomic frequency is
0 negligiblyaffectedbythenumberofpulsesinthefastgate. Incontrast,weshowthatthelaserpulse
2 instabilitydoesgiverisetoapulse-numberdependenteffect; thefastgateinfidelityiscompounded
withthenumberofappliedimperfectpulses. Usingadimensionalreductionmethodpresentedhere,
n
we find bounds on the pulse stability required to achieve two-qubit gate fidelity thresholds.
a
J

3
I. INTRODUCTION GZC and FRAG gates are independent of the ini-
1
tial motional states, errors in the gate are enhanced

] Atwo-qubitentanglinggateisanessentialcompo- accordingtothemeanvibrationalmodeoccupation,
h nent of any quantum information processing (QIP) as shown in [4, 6]. Certain gate error sources have
p
system [1]. Fast gates for trapped ions using con- been considered: the effects of trap anharmonicity
-
t trolled large momentum kicks offer a significantly on both schemes [2, 4, 6], dissipation effects on the
n
faster operation timescale than traditional gates re- GZCscheme[4]andlasercontrolerrorsintheFRAG
a
u quiring spectral resolution of sidebands [2–7]. This schemesuchasinsufficientlaserrepetitionratesand
q in turn leads to simpler gate adaption for long ion pulse timing or direction errors [5]. Only a prelimi-
[ crystals or more complex geometries [3, 8–10], with nary analysis of pulsed fast gate errors due to pulse

1 relatively invariant schemes required for sufficiently area imperfections has been performed, despite the
v fast gates [6, 11]. There has been recent progress conclusionthatsucherrorsaresignificant[2,5]. Fur-
0 towards the implementation of pulsed fast gates in thermore, a phase stability analysis is still required.
1 the production of the required high repetition-rate Laser phase dependence in each momentum kick
1 pulsed lasers [12] and their application to perform comprising the fast gates arises when the rotating
3 a single-qubit gate [13], as well as spin-motion en- wave approximation (RWA) is no longer valid due
0
tanglement [14]. In this paper we outline challenges to short pulse durations relative to the atomic tran-
.
1 for performing a complete fast gate protocol, and sition frequency. This leads to imperfect popula-
0 present both the techniques for quantifying gate fi- tion transfer between internal states. Short pulse
6 delity subject to imperfections, as well as the re- durationsarenecessarysuchthatthetotalmotional
1
quired thresholds in laser stability and pulse times evolution during the pulses is negligible. Thus the
:
v to perform high-fidelity gates. combined duration from all of the applied pulses is
Xi For implementation of a fast gate, and certainly required to be much shorter than the trap period,
for considering their application to large-scale al- which is on the order of 1 µs. Few pulses or very
r
a gorithms, detailed analysis of the stability require- short pulse durations seem preferable, however in-
ments for the trap and control are critical. Er- creasing the number of pulse pairs in a fast gate im-
ror correction can be applied between gate oper- proves the gate speed, fidelity and scaling with the
ations, however individual gate operations are re- numberofions. TheRWAprovidesthelowerbound
quired to meet high-fidelity thresholds [15–17]. The forpulsedurationsforfastgates,andexaminingthe
scheme proposed by Garc´ıa-Ripoll, Zoller and Cirac dependence of this lower bound on the number of
(GZCscheme)[2]andtheFastRobustAntisymmet- pulses in a gate is essential for applications of fast
ricGate(FRAGscheme)[5]wereshowntohavevery gates to QIP.
high fidelity and robustness in [6], and we focus on Significantinfidelityalsoarisesfromimperfectap-
these schemes for our error analysis. While perfect plied pulses. Ideal pulses keep the internal qubit
states invariant throughout the phase gate and re-
storetheinitialmotionalstateattheendofthegate,
however imperfect pulses cause internal state trans-
∗ [email protected] fers as well as occupation of a range of motional
2
levels after the gate. Random errors in the pulse trap frequency, Ω (cid:29) ν. In this regime, multiple
duration were considered in [2] for a four-kick se- numberstatesofeachsharedmotionalmodeareex-
quence, and in [5] a worst-case error bound was cal- cited by pairs of counter-propagating laser pulses,
culatedusingperturbationtheoryforsmallerrorsin as shown in Figure 1. These π-pulse pairs provide
the pulse area and low numbers of pulse pairs. The momentum kicks such that a closed trajectory in
perturbation technique was used for just four pulse phase-space is described for the centre of a coherent
pairs, and fails well before 100 pulse pairs with er- state, as in Figure 1 (h). The area enclosed in each
ror on the order of 1% in the pulse area [5]. It was mode’s phase-space determines a conditional phase
concludedthatimperfectpulseareawilllimitthefi- applied to different two-qubit computational states.
delity of fast gates; a more complete analysis of the The evolution of an ideal fast gate can be de-
required pulse stability is necessary for gate imple- scribedasalternatingdisplacementandrotationop-
mentation. erators in phase space for each motional mode. A
It is possible to model the full dynamics of a displacement operator for mode p is described by
gate without the RWA or with pulse area imper-
fections, and thereby directly calculate the gate fi- Dˆp(α)=exp[αa†p−α∗ap], (1)
delity. However, the Hamiltonian operator required
for this calculation has a dimension given by the for a displacement of α, where ap is the mode anni-
square of the full state vector dimension, which in- hilationoperator. UndertheRWA,pairsofcounter-
cludes both the internal qubit states and the vibra- propagatingπ-pulsesgiverisetomodedisplacement
tional mode states for each shared mode. In the operators as follows [6]:
ideal-pulsecase, thecomplexityisvastlyreducedby
simplifying the requirements for performing a high- Ukick =e−2izk(x1σ1z+x2σ2z) (2)
fidelity gate to three control conditions [2]. In this =ΠL Dˆ (−2iz(b(p)σz+b(p)σz)η ), (3)
paper,wepresentasimplificationmethodforimper- p=1 p 1 1 2 2 p
fect gates that permits fidelity calculation for large when there is negligible motional evolution between
momentumkicksintrapswithmanyionsandacor- the two pulses. Here z is the direction of the first
responding number of shared motional modes. Our pulseinthepair,kisthelaserwavenumber,x isthe
i
method is presented in Section II following a review position operator for ion i and σz is the usual Pauli
i
offastgates. InSectionIII,weapplythisfidelitycal- Z operator acting on ion i. There are L motional
culationtechniquetoexploretheeffectofpulsenum- modes corresponding to L ions in the crystal, and
ber on the phase dependence of the gate with short b(p) is the ion-mode coupling coefficient between ion
pulsedurations. Thisprovidesaminimumpulsedu- i
i and mode p. The Lamb-Dicke parameter η for
p
rationboundforhigh-fidelityfastgatescomposedof
mode p is given by
varying numbers of pulses. In Section IV, we apply
our method to consider imperfect pulse areas com- (cid:115)
(cid:126)
prehensively. We introduce the errors in the atom- η =k , (4)
p 2Mν
lightevolutionunitaries,andconstructtheimperfect p
gate evolution operators to directly compute the fi-
for ion mass M and mode frequency ν .
delity. This gives us an accurate measure of fidelity p
Two main causes of imperfect momentum kicks
for large numbers of pulses. Finally, we present our
to the ions come from counter-propagating pulses
conclusions in Section V.
applied with area not equal to π, or from breaking
the RWA through short pulse durations. These im-
perfections are shown in Figure 1, which illustrates
II. QUANTIFYING FAST GATE ERRORS
their effect at each stage in the fast gate evolution
process.
We present the fast gate mechanism and sum-
The free motion of the ions, and their motional
marise the GZC and FRAG gate schemes, followed
modes, corresponds to rotation operators in each
by a general fidelity calculation method for two
mode p’s phase space:
trapped ions as well as a two ion gate in a longer
ion crystal. Up,mot =e−iνpδtka†pap, (5)
where δt is the time between the kth and (k+1)th
k
A. Gate dynamics and limitations momentum kicks.
The displacements and free rotations are deter-
Fast gates operate in the strong-coupling regime, mined according to particular pulse schemes. These
where the laser coupling is much greater than the schemes satisfy the required gate conditions: (1)
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(cid:6)(cid:7)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:14)(cid:15)(cid:16)(cid:17)(cid:16)(cid:13) (cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:14)(cid:15)(cid:16)(cid:17)(cid:18)(cid:19)(cid:20)(cid:19)(cid:16) (cid:6)(cid:7)(cid:8)(cid:9)(cid:10)(cid:11)(cid:12)(cid:13)(cid:14)(cid:15)(cid:16)(cid:17)(cid:16)(cid:13)
(cid:1) (cid:1)=(cid:1)(cid:2)(cid:3)(cid:4)(cid:5) (cid:2) (cid:1)=(cid:6)(cid:7)(cid:8)(cid:9)(cid:2)(cid:3)(cid:4)(cid:5) (cid:3) (cid:1)=(cid:1)(cid:2)(cid:2)=(cid:9)(cid:10)(cid:11)(cid:2)(cid:3)=(cid:12)(cid:4)/(cid:9)
FIG. 1. (h): Centre-of-mass (COM) phase-space trajectory for the centre of a coherent state during the gate
operation. The sides of the trajectory correspond to momentum kicks. The angle of each vertex corresponds to
free evolution between kicks, and marks the gate evolution point for the other subfigures, from a→g. (a)→(g):
Population occupying COM mode number states, for both ions in the excited state, at different points during the
GZC gate operation, with n = 1 (14 total pulse pairs). Blue circles represent an ideal gate, satisfying the rotating
wave approximation (RWA) and with no pulse area imperfections given by (1−ξ). A gate with systematic pulse
imperfections (ξ = 0.95) is also considered (yellow squares), as well as a gate where the RWA is invalid (green
diamonds) such that the pulse duration and laser phase must be defined (τ = 5 fs, φ = 3π/5). After the non-ideal
gates, population is lost to other internal states, and some of the population is imperfectly restored to the initial
COM state, |2(cid:105) , the second excited number state. Values were chosen to illustrate the effects of these errors. The
c
point in the gate operation described by subfigures (a)→(g) corresponds to a→g marked in subfigure (h).
conditional phase evolution according to the two- TheintegerndeterminesthegatetimeT ,which
G
qubit gate described by scales optimally with the total number of pulses in
the scheme N as T ∝N−2/3 [4, 6].
Ugate =eiπ4σ1zσ2z (6) The FRAGpand GGZC scphemes consist of 10n and
14n pulses respectively. The FRAG scheme has a
for a gate applied to ions 1 and 2, and (2) no mo-
state-averaged fidelity, as defined in [6], of 0.96 for
tional dependence, such that the initial motional
n = 1, and 0.995 for n = 2, while for higher n the
state is restored following the gate operation. The infidelity is below 10−8. We neglect the low-fidelity
GZCandFRAGschemesarecharacterizedbypulse
n = 1 case of the FRAG scheme, which obscures
pairszappliedattimest,interspersedwithfreeevo-
thestabilityanalysis. TheGZCscheme,withhigher
lution. For the FRAG scheme [5]:
totalnumbersofpulsesforeachn,achievesinfidelity
on the order of 10−5 for n=1, and infidelity below
z = (−n, 2n, −2n, 2n, −2n, n) 10−8 for higher n.
t = (−τ , −τ , −τ , τ , τ , τ ).
1 2 3 3 2 1 The scaling of errors with the number of pulses
is examined for both the FRAG and GZC schemes.
At time −τ , n counter-propagating pulse pairs
1 Weexploretheeffectsoferrorsonschemeswithlow
are applied along the trap axis (aligned with the z
pulse-numbers using the n = 1 GZC scheme due to
axis) to provide a 2n(cid:126)k momentum kick in the −z
its high fidelity. While more robust for lower num-
direction.
bers of pulses, the GZC scheme is slower than the
TheGZCscheme[2,4]ischaracterisedasfollows:
FRAG scheme for n ≥ 2, as shown in [5]. The ef-
fects of finite laser repetition rates on these schemes
z = (−2n, 3n, −2n, 2n, −3n, 2n) were explored in [5, 6], where it is shown that for
t = (−τ1, −τ2, −τ3, τ3, τ2, τ1). repetition rates of around 300 MHz, even a gate
4
with perfect π-pulses has non-negligible infidelity. stretch modes respectively. The motional inner
Faster repetition rates have robust fidelities, partic- product is stricter than the computational fidelity
ularly for the two-qubit case. Errors due to imper- of [6], with a stronger motional restoration require-
fectpulseareasorfrombreakingtheRWAaffectany ment that population must be restored to the ini-
schemeregardlessofrepetitionrate;inthispaperwe tial number state for each mode at the end of the
consider these errors independently by assuming an gate operation. This is a convenient choice for our
infinite repetition rate. Furthermore, this approx- number basis, and directly considers effective heat-
imation allows a clear analysis of the relationship ing caused by the gate to be infidelity.
between these errors and number of applied pulses. The ideal gate operation of equation (6), with
Themethodsinthispapercanbeappliedusingpar- duration T , applies a state-dependent phase while
G
ticular, finite repetition rates to model the errors in preserving the internal and motional states:
an experiment more precisely.
(cid:16)
To model the effect of imperfect pulses or an in- U |φ (cid:105)= eiπ/4a |gg(cid:105)+e−iπ/4a |ge(cid:105)
id i 00 01
valid RWA, we expand the appropriate unitary op-
(cid:17)
erator for the applied gate, U , in the number ba- + e−iπ/4a |eg(cid:105)+eiπ/4a |ee(cid:105)
re 10 11
sis. We can then observe the phase-space evolution
during the gate process, and calculate the fidelity ⊗e−iTG(νcnc+νrnr)|ncnr(cid:105), (12)
of the gate. While coherent states are preserved by
where the motional component is global phase cor-
the momentum kicks and rotations, the momentum
responding to free evolution for each mode.
kicksdeformaninitialnumberstatetospreadacross
Therealgateisamorecomplexoperationonboth
manymodes. Attheendofahigh-fidelitygate,how-
thecomputationalandmotionalstates, andwecon-
ever, this spread resolves back into the initial num-
sider the error to first order in the small gate im-
ber state, as shown in Figure 1(a)-(g).
perfection. Since the ideal gate does not transform
the basis states but just applies a phase, a real gate
approximating the ideal operation has only small
B. Fidelity calculation: dimensional reduction
population transfer between internal states. The
gate schemes are designed to restore the motional
To assess the impact of particular errors, we use
states for preserved internal states; only a fraction
the state-averaged fidelity as a measure of the gate
of the motional state population (to second order
performance. The fidelity of a pure state |ψ(cid:105) with
in the error) will be restored for altered internal
respect to a density matrix σ is given by the state
states with changed state-dependent displacement
overlap [18]:
operators. Thesetermswithchangedinternalstates
thusprovideasecond-ordercorrectiontothefidelity,
F =(cid:104)ψ|σ|ψ(cid:105). (7)
which we neglect here.
For an initial state |φ (cid:105), the final state following the Similarly, an ideal counter-propagating pair of
i
ideal gate operation U is given by pulses acting on two ions with the same internal
id
stateaffectsonlytheCOMmode. Animperfectpair
|ψ(cid:105)=Uid|φi(cid:105), (8) of pulses may alter the stretch mode to some small
degree; a first-order error term. This perturbation
and the final density matrix following the real, im-
to the stretch mode also has only a small effect on
perfect, operation U is given by
re the fidelity; only a fraction of the perturbation is
σ =U |φ (cid:105)(cid:104)φ |U† . (9) expected to be restored to its initial motional state
re i i re
andthissecond-ordercontributionisneglected. Ions
The state-averaged fidelity is thus withoppositeinternalstatesareassumedtohavean
(cid:90) invariant COM mode, with the gate acting on the
F = |(cid:104)φ |U†U |φ (cid:105)|2, (10) stretch mode.
i id re i
φi For the basis state
integrating uniformly over the unit hypersphere de- a |gg(cid:105)⊗|n n (cid:105)≡a |ggn n (cid:105), (13)
00 c r 00 c r
scribedbytheinitialstatewitharbitrarycoefficients
a : the fidelity inner product element is thus
jk
|φi(cid:105)=(a00|gg(cid:105)+a01|ge(cid:105)+a10|eg(cid:105)+a11|ee(cid:105)) |a00|2(cid:104)ggnc|Ui†dUre|ggnc(cid:105)
⊗|ncnr(cid:105). (11) =|a00|2e−iπ/4(cid:104)ggnc|Ure|ggnc(cid:105), (14)
The initial motional state is the number product wherethestretchmodeisallowedtoevolvefreelyby
state |n (cid:105)⊗|n (cid:105) for the centre-of-mass (COM) and the ideal and real unitaries, and thus cancels from
c r
5
the inner product. Only the population retained in during the gate operation. This truncation occurs
the computational ground state of both ions is re- for higher phonon numbers with larger numbers of
tained in the fidelity term. The unitaries act sym- applied pulses in a gate corresponding to larger mo-
metrically on |ee(cid:105), and the same symmetry between mentum kicks. We truncate our number basis at 50
|eg(cid:105) and |ge(cid:105) allows us to simplify our full fidelity statesforn=1foreachgate,70statesforn=2and
expression: n = 5, and 130 states for n = 10. The single-mode
analysis thus reduces the dimensions for calculating
(cid:90) (cid:12) thestatevectorevolutionbyafactorofaround100,
F = (cid:12)(|a |2+|a |2)eiνcTGnc−iπ/4
(cid:12) 00 11 depending on the number of applied pulses. The
φi state vector’s dimension is four times the dimension
×(cid:104)ggn |U |ggn (cid:105)+(|a |2+|a |2)eiνrTGnr+iπ/4
c re c 01 10 of the number basis, due to the two basis internal
×(cid:104)gen |U |gen (cid:105)|2. (15) states for each ion.
r re r
For each internal state of the two qubits, the effect
of the real gate on just a single motional mode con- C. Extending fidelity calculation to larger
tributes to the fidelity expression. We thus expand traps
U for each mode independently, and accordingly
re
cancel the motional phase term in the ideal unitary Performingfastgateswithinalargequantumpro-
from free evolution of the other mode. cessorrequiresanalysisofgateimperfectionsintraps
Ageneralpositionoperatordecompositionfortwo with larger numbers of ions. The number of mo-
trapped ions is described by tionalmodesisequaltothenumberoftrappedions.
Thepositiondecompositioncanbegeneralisedfrom
kx =b(c)η (a +a†)+b(r)η (a +a†), (16) equation (16) for arbitrary numbers of ions, and we
i i c c c i r r r
againapplytheapproximationofseparablemotional
wherethesubscriptcdescribestheCOMmode,and modes in a harmonic trap.
subscriptrdescribesthestretchmode. Thecoupling The ideal and real unitaries can be written as a
operators for two trapped ions are separable product of the operation on each mode:
b(c) =(√1 ,√1 ) (17) (cid:89)L
2 2 Uid = Uid,p (21)
1 1 p=1
b(r) =(−√ ,√ ), (18)
2 2 (cid:89)L
U = U , (22)
re re,p
with the jth vector element representing the √cou- p=1
pling for ion j, and mode frequencies of ν and 3ν
for the COM and stretch modes respectively. where L is the number of modes and
The real gate unitary U is applied to both
modes, howevertoconsiderfirrest-ordererrorswecan Uid,p =eiφpσ1zσ2z, (23)
treat it as separable for each mode. This allows us
up to global phase. This simplifies the fidelity F
to apply a single-mode expansion of kx for each in-
i expression to require only the product of the mode-
ternal state of the qubits. For the states |gg(cid:105) and
dependent fidelities F up to first order in the error
p
|ee(cid:105), the terms in U contributing to the fidelity
re term:
equation (15) are given by
(cid:90)
F (cid:39) |Π F |2, (24)
kx =b(c)η (a +a†), (19) p p
i i c c c φi
F =(cid:104)φ |U† U |φ (cid:105). (25)
while for |ge(cid:105) and |eg(cid:105), p i id,p re,p i
We can calculate the contributions U of each
kx =b(r)η (a +a†). (20) re,p
i i r r r modetotherealgateunitary,usinganappropriately
truncated single-mode number basis for each calcu-
Our separable representation of U reduces the
re lation. For a high-fidelity gate scheme, the ideal
dimension of the state vector by a factor given by
phase contribution for each mode is given by [6]
the number of required motional basis states. The
Hamiltonian is reduced in dimension by the square N m−1
ofthisfactor. Weusethenumberbasistomodelthe φ =8η2σzσzb(p)b(p) (cid:88) (cid:88) z z sin(ν (t −t )),
p p 1 2 1 2 m k p m k
stateevolution,andtruncatethebasissuchthatneg-
m=2 k=1
ligible population occupies the maximal basis states (26)
6
wherez isthenumberofpulsepairsappliedattime is:
k
t determinedbythegatescheme. Forahigh-fidelity
gkate, (cid:80)pφp ≈ π4. Upulse,1 =exp(cid:20)−2iτπ (cid:18)(cid:90) tf σ1+e−i(kx1−2ωatt+φ)dt
Our method to estimate fidelity in this fashion is ti
to first calculate the real unitaries for each mode (cid:90) tf
+ σ−ei(kx1−2ωatt+φ)dt+τσ+ei(kx1+φ)
U and the ideal phases φ . The individual 1 1
re,p p ti
mode expansion of the gate unitary ensures a low- (cid:17)(cid:105)
dimensional state vector, as required for computa- +τσ1−e−i(kx1+φ) , (28)
tion.
for a pulse of duration τ =t −t , where
f i
(cid:90) tf
σ+e−i(kx1−2ωatt+φ)dt
1
ti
−iσ+ (cid:16) (cid:17)
III. BREAKING THE RWA = 1 e−i(kx1−2ωattf+φ)−e−i(kx1−2ωatti+φ) .
2ω
at
(29)
Pulses with duration on the order of the atomic
transition period 2π/ω render the RWA invalid, Fortwoions,theinteractionHamiltonianisH(cid:48)+H(cid:48)
at 1 2
and cause infidelity in fast gate schemes which rely for ions 1 and 2, using H(cid:48) from equation (27). The
i
ontheRWA.Inthissectionweapplyourfidelitycal- pulseunitaryoperatorUpulse,1 issimilarlyextended,
culation method to explore the tradeoff in pulse du- and combined with the motional free evolution uni-
ration between performing large numbers of pulses tary,equation(5),toconstructtherealgateunitary
in a short time for fast, high-fidelity gates, while Ure. This allows us to explore the validity of the
staying in the regime where the RWA holds. Gates RWA for different pulse lengths by solving for the
significantly faster than the trap evolution period phase dependence and fidelity.
(∼1µs)requirelargenumbersofpulses,whichmust Figure 2 shows the effect of short pulse duration
thus have very short durations. We demonstrate ongatefidelity. Forshortpulseduration,thefidelity
that the valid RWA regime is altered little by the decreases as more pulses are applied for the FRAG
number of applied pulses in a gate. and GZC gates. The mean infidelity is plotted for
varying phase φ, and error bars mark a standard
We perform fast gates using short pulses of vary-
deviationininfidelityduetophasedependence. The
ing duration without performing the RWA to inves-
FRAG gate with n = 2 has fidelity of 0.988, and
tigate the regime where the approximation holds.
approachesthisvaluewithastandarddeviationless
The gate should also be independent of the optical
than 10−3 for pulse lengths τ > 40 fs. The GZC
phase φ [19], and we quantify the pulse lengths re-
schemeandtheFRAGschemeforn=5havefidelity
quired for phase-independence. In the interaction
above 0.999 and standard deviation less than 10−4
frame with respect to the internal states of a sin-
for τ ≥60 fs.
gle ion, ion 1 marked by subscripts, the atom-light
Pulsesmuchlongerthantheatomictransitionpe-
interaction Hamiltonian is
riod are accurately described under the RWA, and
the number of pulses in the gate does not signifi-
(cid:126)Ω cantlyalterthisthreshold. Forgateswithincreasing
H(cid:48) = (σ+e−i(kx1−(ωL+ωat)t+φ)+σ+ei(kx1−δt+φ) speed or scalability with the number of ions, large
1 2 1 1
numbers of pulses must be performed much faster
+σ1−e−i(kx1−δt+φ)+σ1−ei(kx1−(ωL+ωat)t+φ)), (27) than the trap motional frequency, ν/(2π)(cid:39)1 MHz,
or even much faster than 10 ns for momentum ap-
plication schemes exciting short-lived atomic lev-
where δ = (ω − ω ). Typical atomic frequency
L at els [20]. This provides five orders of magnitude be-
transitions are on the order of ω ∼ 2π×1015 Hz,
at tweenasafepulseduration∼100fsandthelifetime
and the fast rotating terms can be neglected fol-
of typical short-lived levels, such as P in 40Ca+.
lowing the RWA. Pulse durations are typically as- 3/2
sumed to be much longer than the rotation period,
τ(π×1015)(cid:29)1.
IV. IMPERFECT PULSES
We focus on resonant transitions where δ = 0 for
simplicity, such that ω = ω . Assuming constant Significant errors also arise from imperfect π-
L at
Ω and a perfect π-pulse, such that Ω = π/τ, the pulses, which construct the momentum kicks fun-
unitary operator from equation (27) for a single ion damental to fast gates. π-pulses with arbitrarily
7
(a) � ◆◆●■●■ (b) � ●■●■
◆ ■
����� ●■◆◆●■ ����� ●●■
■ ■
������ ����� ● ◆■ ��●������� ������ ����� ● ●■ ● ● ● ● ● ��������
-��� ����� ● ◆●■ ◆●■ ◆●■ ◆■ �� -��� ����� ■ ■ ■ ●■ ��
◆
●■ ◆● ■
■ ■
��-� ��-�
� �� �� �� �� ��� � �� �� �� �� ���
�������������(��) �������������(��)
FIG. 2. Infidelity following the (a) GZC and (b) FRAG gate operations with different numbers of pulses, governed
by n. The effect of changing the duration of the pulses composing the gate is shown. The initial motional state is
|2(cid:105) |2(cid:105) , the second excited number state for each mode. We determine the mean and the standard deviation (error
c r
bars) by varying the phase φ for a given pulse duration τ.
high fidelity can be constructed using composite ing to the pulse applied to a single ion follows:
pulses [21, 22]; laser repetition rates must be suffi-
ciently high to accommodate the pulse components Upulse =e−i2ξπ(σ+ei(kx+φ)+σ−e−i(kx+φ)). (31)
inthisapproach. Inthissection,weconsidertheim-
pact of infidelity in the π rotations on the full gate Reversing the pulse direction changes the sign of k
fidelity. Imperfect π-pulses cause imperfect state intheevolutionoperator. Thepulserotationfidelity
transfer,errantmomentumkicksandacquiredphase canbefoundforidealandrealpulseunitariesU(cid:48)
pulse
infidelity. and U respectively:
pulse
While different methods for performing π-pulses
hpauvlseevraortyaitniognrofibduelsittnyesfsortoanlyasmeretflhuocdtuhaatisonas,fixtehde R. Fid.=Minψi(cid:12)(cid:12)(cid:104)ψi|(Up(cid:48)ulse)†Upulse|ψi(cid:105)(cid:12)(cid:12)2 (32)
(1−ξ)2π2
relation to the full gate fidelity. We consider here (cid:39)1− , (33)
the simplest case of square pulses to calculate the 4
relation between rotation fidelity and gate fidelity.
up to third order in (1−ξ)π/2.
To model the imperfect gate process, we assume a
Assumingthatthesamelaserproduceseachpulse,
suitable pulse length for the RWA, with δ =0:
and that phase drift is minimal during the gate du-
(cid:126)Ω ration (< 1 µs), φ is fixed. We fix ξ to be constant
HR(cid:48)WA = 2 (σ+ei(kx+φ)+σ−e−i(kx+φ)). (30) during a gate operation to find the systematic error
effects.
ForΩconstantintime,aπ-pulsesatisfiesΩτ =π, The unitary for a counter-propagating pulse pair,
for a pulse duration τ. An approximate π-pulse sat- with first pulse direction z, can be expressed in the
isfies Ωτ =ξπ, with ξ (cid:39)1. The unitary correspond- computational basis {e,g}:
(cid:18)e−izkx(cos(kx)cos(πξ)+izsin(kx)) cos(kx)sin(πξ)(−icos(φ)+sin(φ))(cid:19)
U (z,ξ)= , (34)
pair cos(kx)sin(πξ)(−icos(φ)−sin(φ)) eizkx(cos(kx)cos(πξ)−izsin(kx))
such that ξ =1 gives states, and represents the angle of rotation on the
(cid:18)e−2izkx 0 (cid:19) Bloch sphere. It does not affect the magnitude of
U (z,1)=− , (35) rotation which provides the error, and we set φ=0
pair 0 e2ikx
for simplicity.
withtheexpectedstate-dependentmomentumkicks
and no φ-dependence. The φ-dependence for im-
perfect pulses is in the terms of equation (34) cor- Themotionalandinternaloperatorscommutefor
responding to population transfer between internal separate ions, and the unitary for a two-ion imper-
8
increaseswithn,theerrorsineachpulsecausecom-
(a)���
pounding gate infidelities, shown in Figure 4. For
����������� ������◆◆◆◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ��������●������������� bmofoattthhicethimnecoFrteiRoanAseaGslsiantnatdtheGefoZmlCleoawgnaintagensad,Fgsatigatuenrdaeas4rtdhshedoenwvuismadtbiroeanr-
������� ������■■■ ■ ■ ■ ■ ■ ■ ■ ■ ◆■ �� oroftpautilosnesiinnficdreealisteys.leFssortheaanch1s0c−h5emise,rewqiuthirend(cid:46)for10a,
���●●● ● ● ● ● ● ● ● ● gate fidelity above 0.99, or rotation infidelity less
������ ������ ������ ������ ������ ������
than 10−4 is required for a gate fidelity above 0.9.
�-����������������
Both schemes are similarly affected by pulse error
(b)��������◆●■◆●■◆●■ ◆●■ ◆●■ ◆■ ◆■ comUspionugnsdqiunagrweiptuhlspeusl,sewhnuermebtehre. pulse area is pro-
������������ ������������ ● ● ◆●■ ◆■ ◆■ ◆■ ��������●■�������������� preoqrutiiorenmalentots.ξ,Swysetecmanatfiicnpdutlhseeapruealseerarorera(1st−abξ)iliotny
���� ● ◆ � the order of 0.4% is permissible for fidelity better
���� ● than 0.9 and n (cid:46) 10 for each scheme. Pulse area
���� ● error (1−ξ) ≤ 0.2% is required for a fidelity above
������ ������ ������ ������ ������ ������
�-���������������� 0.98. Figure 5 demonstrates the impact of system-
aticpulse-areaerrorsontheinternalstateandmode
occupation following a GZC gate with n = 1; pop-
FIG. 3. A GZC gate with n=1 is applied with varying
rotationfidelityforindividualpulsesanddifferentinitial ulation is lost to other internal states with variable
motional occupation. (a) The mean and standard devi- motional mode occupation.
ation (error bars) in the occupation of motional states
following the gate are shown. (b) Gate fidelity is shown
as a function of pulse rotation fidelity.
V. CONCLUSIONS
The duration of fast gates directly impacts their
fect π-pulse is given by
fidelity and scalability with the number of trapped
ions. We have presented a technique for calculat-
U (z)
2pulse ing gate fidelities to first order in the error for large
=e−i2ξπ(σ1+eizkx1+σ1−e−izkx1+σ2+eizkx2+σ2−e−izkx2). numbers of applied pulses. Applying this technique
to two ions, we have demonstrated that pulse er-
(36)
rors cause compounding infidelity with the number
Using this unitary we construct the evolution from of pulses composing the gate. Gate duration scales
pulse pairs, which we intersperse with the motional with the number of pulses, so this pulse fidelity re-
free evolution unitaries to build up our gate oper- quirementisofgreatimportanceforusingfastgates
ations. The necessary pulse times for gates with for scalable QIP. Pulse infidelity less than 10−5 is
varying numbers of pulses are found according to required for gate fidelity above 0.99 with up to 140
the applied scheme [2, 6]. pulse pairs in the FRAG and GZC gate schemes.
We have also shown that different numbers of ap-
Figure 3 shows the effect of the initial motional
pliedpulsesdonotsignificantlyalterthevalidRWA
state on final mode occupation and gate fidelity for
regime: pulsedurationsmuchlongerthantheatomic
aGZCgatewithn=1. Increasinginfidelityinindi-
transition period are required. Experimental imple-
vidual π-pulses, or rotation infidelity, damages the
mentation of a fast gate, which requires fast and
full gate fidelity and increases both the mean and
robust π-pulses, will be a significant step towards
standarddeviationofthemodeoccupationafterthe
large-scale QIP with ions.
gate. The initial motional state before the gate is
applied affects the magnitude of the gate infidelity.
There is not a clear relationship between initial mo-
tional state and infidelity; however each initial state VI. ACKNOWLEDGMENTS
isharmedbypulseerrors. Rotationinfidelityaround
3×10−4 is required for gate fidelity better than 0.9, This work was supported by the Australian Re-
or rotation infidelity around 10−5 for a gate fidelity search Council Centre of Excellence for Quan-
above 0.99. tum Computation and Communication Technol-
Higher numbers of perfect π-pulses provide faster ogy (Project number CE110001027) (ARRC),
gate times, more stability, and improved scalabil- Australian Research Council Future Fellowship
ity. However, as the number of pulses in the gate (FT120100291)(JJH)aswellasDP130101613(JJH,
9
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FIG. 4. A GZC (a,b) and FRAG (c,d) fast gate are applied to |ee(cid:105)|1(cid:105) |1(cid:105) with varying n and pulse rotation error.
c r
(a,c): The mean and standard deviation (error bars) in the occupation of the COM mode are shown following the
gate applied to the |ee(cid:105) internal state. (b,d): Gate fidelity is shown as a function of pulse rotation infidelity.
ARRC).
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1. The fraction ξ of a perfect square π-pulse performed
determinestherestorationoftheinternalstateandCOM
motional mode to the initial state.