Scaling of Decoherence Effects in Quantum Computers

19 Pages·0.23 MB·

by  B. J. Dalton

Scaling of Decoherence Effects in Quantum
Computers
3
0
B. J. DALTON
0
†‡
2 Centre for Atom Optics and Ultrafast Spectroscopy,
†
n Swinburne University, Hawthorn, Victoria 3122,
a Australia
J
Department of Physics, University of Queensland,
7
‡
1 St Lucia, Queensland 4072, Australia

3 February 1, 2008
v
1
7
Abstract
0
9 The scaling of decoherence rates with qubit number N is studied for
0
a simplemodel of aquantumcomputerin thesituation where Nis large.
2
The two state qubits are localised around well-separated positions via
0
trapping potentials and vibrational centre of mass motion of the qubits
/
h occurs. Coherent one and two qubit gating processes are controlled by
p external classical fields and facilitated by a cavity mode ancilla. Deco-
- herence due to qubit coupling to a bath of spontaneous modes, cavity
t
n decay modes and to the vibrational modes is treated. A non-Markovian
a treatment of the short time behaviour of the fidelity is presented, and
u
expressionsforthecharacteristicdecoherencetimescalesobtainedforthe
q
casewherethequbit/cavitymodeancilla isinapurestateandthebaths
:
v are in thermal states. Specific results are given for the case where the
i cavity mode is in the vacuum state and gating processes are absent and
X
the qubitsare in (a) theHadamard state (b) the GHZ state.
r
a
1 Introduction
⊂
Quantumcomputersareexpectedtohaveakeyadvantageoverclassicalcomput-
ers in terms of computational complexity, an advantage based on the idealised
features of parallelism, entanglement and unitary quantum computation pro-
cesses. For certainalgorithms [1, 2], the number ofcomputationalsteps needed
usingquantumcomputersshouldincreasemuchmoreslowlywithinputnumber
sizethanforclassicalcomputers,enablingcomputationsthatwereinfeasibleus-
ing classical computers to be carried out. In the case of the quantum search
algorithm for example, the number of steps increases with the square root of
the number of items, rather thanlinearly as in classicalsearching[2]. However,
the physical system whose quantum states define the N qubit system and the
1
quantum devices involved in the gating processes both interact with the envi-
ronment. Suchinteractionschangethedensityoperatordescribingthequantum
computer state from a pure to a mixed state, with coherences between states
evolvedfromdifferentinputstatesbeingpartiallyorcompletelydestroyed. This
process of decoherence is the enemy of quantum computation, though methods
such as quantum error correction [3, 4], decoherence free sub-spaces [5, 6, 7, 8]
and dynamical suppression of decoherence [9, 10] could be used to minimize
its effects. Utilizing the computational complexity advantage of quantum com-
puters requires suitably large qubit numbers (ca 105 qubits may be needed for
searching or factoring algorithms with error correction [11] for input numbers
of practical importance), so the scaling with the number of qubits of the de-
coherence rates in comparison to the coherent processing rates is important in
determining the size limits of useful quantum computers [12, 13].
This paperextends previouswork[14,15,16,17,18,19,20]treatingscaling
effects. Decoherence effects are studied for a simple model of anN qubit quan-
tum computer, involving N two state qubit systems (see figure 1). The qubits
are localised around well-separated positions via trapping potentials, and the
centre of mass (CM) vibrational motions of the qubits are treated. Coherent
one and two qubit gating processes are controlled by time dependent localised
classical electromagnetic (EM) fields and magnetic fields that address specific
qubits. The two qubit gating processesare facilitated by a cavity mode ancilla,
which permits state interchange between qubits. The magnetic fields are used
to bring specific qubits into resonancewith the classicalEMfields orthe cavity
mode. The two state qubits are coupled to a bath of EM field spontaneous
emission (SE) modes, and the cavity mode is coupled to a bath of cavity decay
modes. For large N the numerous vibrational modes of the qubits also act as
a reservoir, coupled to the qubits, the cavity mode and the SE modes. The
system-environmentcouplinginteractionsincludeelectricdipolecouplingofthe
qubitstotheSEmodes,quasi-modecouplingofthecavitymodetocavitydecay
EM field modes and Lamb-Dicke coupling of qubits to CM vibrational modes
and each of: (a) the SE modes (b) the cavity mode (c) the classical EM gating
fields (d) the classical magnetic gating field. The system-environment coupling
interactions considered include both σ phase destroying and σ population
Z X
destroying terms, though the latter are more important in this model. The
modelissimilarto thoseofthe ion-trap[21], neutralatom[22]andcavityQED
[23, 24, 25] varieties.
The object is to study decoherence effects in quantum computers for the
situationwherethenumberofqubitsN becomeslarge. Decoherenceeffectswill
be specifiedby the fidelity, which measureshowclose is the actualbehaviourof
thedensityoperatorforthequbitssystem(includingtheancilla)toitsidealised
behaviour due to coherent gating processes only. The initial quantum state for
the quantum computer will be assumed to be pure and the reservoirstates will
be assumed to be thermal. Previous work [14, 15, 16, 17, 18, 19, 20] indicates
that the decoherence time t can decrease inverselywith N (qubits decohering
D
independently) or N2 (qubits decohering collectively), so it can become very
shortforthecasewhereN islargeandmaybecomecomparabletothereservoir
2
correlation time t . In the case of atomic spontaneous emission relaxation, t
C C
10−17 s, and t 10−8s for one qubit could become t 10−18s for 105
D D
∼ ∼ ∼
qubits in the case of collective decoherence. However,one aspect of the present
study is whether the qubits decohere independently or not, so we do not arbi-
trarily assume either case. As maintaining coherence between all states of the
quantum computer is important for its operation, we are interested in the be-
haviour of all density matrix elements, and not just those for a smaller density
matrix describing a single qubit, even though the latter may have long deco-
herence times and satisfy Markovian equations. Treating the density operator
forthe fullN qubitsystemviathe standardBorn-Markoffmasterequation(see
for example, [26, 27]) requires that all the interaction picture density matrix
elements do not change significantly during the reservoir correlation times. As
the latter are just the time scales over which two-time correlation functions of
reservoir operators involved in the system-reservoir interaction decay to zero,
we can see that the reservoir correlation times are independent of the number
of qubits. Since the generaldecrease in decoherence time with increasing qubit
numbersindicates that some density matrix elements mustchangeovershorter
and shorter time scales, then using Markovian master equations for studying
decoherence therefore becomes questionable if the number of qubits becomes
large enoughIn the present paper we do not assume that Markovianbehaviour
will necessarily occur, and use methods that do not rely on the standardBorn-
Markoffmaster equation. If the behaviour does turn out to be Markovianthen
our approach is still correct. However as we will see, in certain cases the be-
haviour is not consistent with the Markoff approximation, so the necessity for
our choice of non-Markovian methods is justified a posteriori. Several authors
[28, 29, 30, 31] have also emphasisedthe importance of studying the short time
regime for macroscopic and mesoscopic systems (such as quantum computers
with large qubit numbers) using non-Markovian methods. One such approach
[29,30]involvesusingeigenstatesofthesystemoperatorsinvolvedinthesystem-
reservoirinteraction(pointer basis),anothertreatment[32]developsshort-time
expressions for the linear entropy (or impurity). Here the approach used is
similar to that in [33] and uses the Liouville-von Neumann equations to ob-
tain short-time expansions for the fidelity. This leads to expressions for the
characteristic decoherence time scales for the general case with gating EM and
magnetic fields present, and at non-zero temperature, and with the qubit and
ancilla system in an arbitrary pure state. The treatment enables situations
where independent or collective decoherence occur to be distinguished. In this
initial paper, the decoherence time scales will be evaluated for specific initial
quantumcomputerstatessuchas: (a)theequalsuperpositionofallqubitstates
(Hadamardstate)and(b)thegeneralisedGHZstates,withthecavitymodean-
cilla in the vacuum state and gating processes ignored. The effects of gating
processesandtherelationshipoftheresultstothoseobtainableforlongertimes
using Markovianmaster equation methods will be treated later.
In Section 2 the general theory used will be outlined. Results for the deco-
herencetimescalesarepresentedinSection3andSection4containsasummary
of the paper.
3
2 Theory
2.1 Hamiltonian
The total Hamiltonian is written as the sum:
H =H +H +H +V +V (1)
S C B S I
The Hamiltonian for the qubit system and cavity mode ancilla is given in
termsofqubitatomicspinoperatorsσi andcavitymodeannihilation,creation
Z
operators b,b† by:
1
H = ~ω σi +~ω b†b (2)
S 2 0 Z b
Xi
Here the qubits all have transition frequency ω and ω is the cavity mode
0 b
frequency. Thecavitymodefacilitatesstatetransferbetweendifferentqubits in
two qubit gating process. In the case of a large number of charged qubits, the
CM vibrationalmodes are too closelyspacedto enable the in-phase vibrational
mode tobe usedinthis role,asinthe caseofion-trapquantumcomputers[21].
The array of qubits confined in their trapping potentials is rather analogous to
alowdensitysolid. Also,becausethecavitymodeenergyquantumislarge,this
ancillaisnotsubjecttothermaleffectsasarethelowfrequencyCMvibrational
modes.
The Hamiltonian for the centre of mass vibrational motion of the qubit
system is given in the harmonic approximation as:
1 1
H = p2 + Vαβδr δr (3)
C 2m iα 2 ij iα jβ
Xiα iXjαβ
= ~ν A† A (4)
K K K
XK
In these equations the qubit CM momentum operators are p and the centre
iα
of mass of each qubit vibrates with displacements δr about well-separated
iα
positions r defined by trapping potentials (α,β = x,y,z). The vibrational
i0
mode K has angularfrequencyν andphononannihilation,creationoperators
K
A ,A† . ThevibrationalHamiltonianH appliestothegeneralcasewherethe
K K C
qubitsvibratecollectively,suchasintheion-trapcasewherethechargedqubits
interactvialong-rangeCoulombpotentials. Ifthequbitsvibrateindependently,
suchasforneutralqubitswithdipole-dipolecouplingsignored,thenthecoupling
matrix is given by Vαβ = δ δ V, and the vibrational modes K are specified
ij ij αβ
iα. These all have the same frequency ν.
The Hamiltonian for the bath of spontaneous emission and cavity decay
modes is of the form:
H = ~ω a†a + ~ξ b†b (5)
B k k k k k k
Xk Xk
4
Here the SE mode k has angular frequency ω and annihilation, creation oper-
k
atorsa ,a†, and the cavitydecay mode k has angularfrequencyξ andannihi-
k k k
lation, creation operators b ,b†. These modes are described in the quasi-mode
k k
picture [34].
The coherent coupling (or gating) processes for the qubit system are de-
scribed by the Hamiltonian:
1
V = (~Ω σi +Hc)+ ~(∆ ∆ )σi
S i X 2 i1− i0 Z
Xi Xi
+ (~g σi b+Hc) (6)
i X
Xi
In order, the terms are the coupling of the qubits to the classical EM field, to
the magnetic field and to the cavity mode. Each qubit can be addressed by
localised classical EM fields E and magnetic fields B to facilitate one and two
qubit gating processes. The two qubit gating process involves the creation and
annihilation of a single photon in the cavity mode, which acts as an ancilla.
The quantities Ω ,∆ and g are coupling constants. The coupling constants
i ia i
Ω ,∆ are both spatially localised and time dependent in accordance with the
i ia
sequence of gating processes involved for a particular algorithm.
Theinteractionofthequbitsystemandancillawiththebathsandthequbits
centre of mass vibrational degrees of freedom is given by:
V = (~giσi a +Hc)+ (~b†(u b +w b†)+Hc)
I k X k k k k k
Xik Xk
+ (~ni σi a [A +A† ]+Hc)
kK X k K K
XikK
+ (~pi σi b[A +A† ]+Hc)
K X K K
XiK
+ (~Θi σi [A +A† ]+Hc)
K X K K
XiK
1
+ ~(mi (1 σi )+mi (1+σi ))[A +A† ] (7)
2 0K − Z 1K Z K K
XiK
The first two terms are the electric dipole coupling of the qubits to SE modes
and the quasi-mode coupling of the cavity mode to the cavity decay modes.
These terms involve coupling constants gi,u and w . The remaining terms
k k k
are Lamb-Dicke couplings with the various fields, allowing for the vibrational
motion of the qubits CM around their equilibrium positions r . The third
i0
term allows for coupling to the SE modes, the fourth to the cavity mode, the
fifth to the classical EM field and the sixth to the magnetic field. These terms
involve coupling constants ni ,pi ,Θi and mi . Since this paper studies
kK K K 0K
short time scale effects, the standard rotating wave approximation (RWA) has
not been made. Other possible interactions, such as those due to Roentgen
currents [35, 36] for neutral and charged qubit systems, and those due to ionic
5
currents [35, 36] for chargedqubit systems, were also investigated but found to
be relatively small in comparison to the interactions listed above.
2.1.1 Expressions
Detailed definitions or commutation rules for the qubit, cavity mode, CM vi-
brational modes and bath modes operators are as follows:
σi = (1 0 + 0 1) σi = i(1 0 0 1)
X | ih | | ih | i Y − | ih |−| ih | i
σi = (1 1 0 0) (8)
Z | ih |−| ih | i
[b,b†] = 1 [A ,A†]=δ
K L KL
[a ,a†] = δ [b ,b†]=δ (9)
k l kl k k kl
ThequbitsarerepresentedbyPaulispinoperators,whereasbosonicannihilation
andcreationoperatorsapply for the cavitymode (quantized involume V ), the
b
CM vibrational modes, the SE modes and the cavity decay modes (quantized
in volume V).
The CM displacements δr (α=x,y,z) are related to the vibrational nor-
iα
mal coordinates via a unitary matrix S in the form:
~
δr = S (A +A† ), (10)
iα iα;Kr2mν K K
XK K
where the matrix S is determined from the eigenvalue equations
VαβS =mν2 S . (11)
ij jβ;K K iα;K
Xjβ
These equations also give the vibrational frequencies ν . For the case where
K
1
the qubits vibrate independently Siα;K =δiα;K and ν =(V/m)2.
6
Explicit expressions for the coupling constants are:
ω
Ω = i c (d e )α expi(k r ω t) (12)
i − Xc r2ǫ0~V 10· c c c· i0− c
∆ = [m B]/~ a=0,1 (13)
ia aa
− ·
ω
g = i b (d e )exp(ik r ) (14)
i − r2ǫ0~Vb 10· b b· i0
ω
gi = i k (d e )exp(ik r ) (15)
k − r2ǫ0~V 10· k · i0
ω ~
ni = k (d e )(k S )
kK r2ǫ0~V r2mνK 10· k · iK ×
exp(ik r ) (16)
i0
× ·
ω ~
pi = b (d e )(k S )
K r2ǫ0~Vbr2mνK 10· b b· iK ×
exp(ik r ) (17)
b i0
× ·
ω ~
Θi = c (d e )(k S )
K Xc r2ǫ0~V r2mνK 10· c c· iK ×
α expi(k r ω t) (18)
c c i0 c
× · −
∂(m B) ~
miaK = − ( ∂Raa· )0r2mν Siβ;K /~ a=0,1. (19)
Xβ iβ K
Here d is the electric dipole matrix element, m (a = 0,1) are the magnetic
10 aa
dipole matrix elements for each qubit. The polarisation unit vectors for the
classicalfield,thecavitymode,theSEmodeorcavitydecaymodekaree ,e ,e
c b k
respectively, and k ,k ,k give the wave vectors. The classical EM gating field
c b
couldbe thoughtofasbeingrepresentedbyamulti-modecoherentstate,where
α isthecoherentstateamplitudeforthemodec. Allthemodeshaveessentially
c
the same polarisation vector, wave vector and frequency, though the frequency
andwavevectorbandwidthsaresufficientlywidetoproducetherequiredspatio-
temporal localisation. The equivalent classical electric field at the equilibrium
positionofthe iqubitis givenbyE=i ~ωc e α expi(k r ω t)+cc.
cq2ǫ0V c c c· i0− c
P
2.2 Dynamics
The total density operator W for the full system of qubits, qubits CM motion,
cavity mode and baths satisfies the Liouville-von Neumann equation:
∂W
i~ =[H,W] (20)
∂t
Initial conditions are assumed of an uncorrelated state for qubits and ancilla,
bath and CM vibrational motion given by:.
W(0)=ρ (0)ρ (0)ρ (0) (21)
S B C
7
and for the bath and CM in thermal states the average value of V is zero.
I
Tr V ρ (0)ρ (0)=Tr ρ (0)ρ (0)V =0 (22)
BC I B C BC B C I
In the present paper the initial state for the qubits and ancilla will be assumed
in a pure state ψ , so that ρ (0)= ψ ψ .
| Si S | Sih S|
The dynamicsofthe quantumcomputer isdescribedbythe reduceddensity
operator ρ for the qubits and ancilla subsystem defined by:
S
ρ =Tr W (23)
S BC
The exactevolutionallowingfor the coherentcouplingandthe interactionwith
the bath and the CM vibrationalmodes is embodied in the time dependence of
ρ . Noassumptionismadeherethatρ satisfiesaMarkovianmasterequation,
S S
thoughitmaydosofortimescaleswhicharelargecomparedtobathcorrelation
times.
The coherent evolution of the qubits and ancilla subsystem is described by
the reduced density operator ρ , which is chosen to coincide with the exact
S0
reduced density operator at time zero, and is then allowed to evolve under the
coherent coupling term V only. Hence:
S
∂ρ
i~ S0 = [H +V ,ρ ] (24)
∂t S S S0
ρ (0) = ρ (0) (25)
S0 S
Thus the interaction between the qubits/ancilla and the baths and CM motion
isconsideredtobeswitchedoffhere,henceρ evolvesaccordingtoaLiouville-
S0
von Neumann equation..
2.3 Decoherence
Decoherence effects are specified by the fidelity, defined as:
F =Tr (ρ ρ ) (26)
S S0 S
The fidelity specifies how close the actual evolution of qubits and ancilla is to
the idealisedcoherentevolution. The time dependence of the fidelity is entirely
duetothedecoherenceeffectscausedbythebathandCMinteractionswiththe
qubits and cavity mode ancilla.
As in [33] the short-time behaviour of the fidelity with the qubits in a pure
statecanbe expressedasapowerseriesinthetime elapsedandexplicitexpres-
8
sions obtained for the first few time constants involved. Thus:
t t2
F(t) = 1 ( ) ( )+.. (27)
− τ − 2τ2
1 2
~
= 0 (28)
τ
1
~2
= Tr ψ V (0)2 ψ ρ (0)ρ (0)
2τ2 BCh S| I | Si B C
2
Tr ψ V (0)ψ 2ρ (0)ρ (0) (29)
− BCh S| I | Si B C
∆V (0)2 (30)
I S BC
≡ hh i i
The initial value of the fidelity is unity. The times τ ,τ ,..specify characteris-
1 2
tic decoherence times for the qubit and ancilla system, their inverses defining
decoherencerates. ForthermalbathandCMstates,onlyτ isinvolvedinspec-
2
ifyingshorttimedecoherence. Forthe timeconstantτ ,the expressioninvolves
2
the average of the square of the fluctuation ∆V (0) = V (0) V (0) of the
I I I S
−h i
zero time interaction operator V (0). The squared fluctuation operator is first
I
averaged over the initial qubit and ancilla pure state ψ , then the result is
| Si
averagedover the bath and CM initial state ρ (0)ρ (0).
B C
3 Results
3.1 General case
The evaluation of τ for the general case with gating EM and magnetic fields
2
both present, and at non-zero temperature, and with the qubit and ancilla
system in an arbitrary pure state, leads to expressions for τ that involve:
2
(a) Expectation values for the state ψ of one qubit operators σi, two
qubit operators σiσj (i=j), (α,β =X,Z|)S.i α
α β 6
(b) Expectation values of cavity operators b,b†,bb†,b†b,b2,b†2.
(c) Expectation values of products of these qubit and cavity operators.
(d) Sums over bath modes and centre of mass vibrational modes of prod-
ucts of pairs of the coupling constants gi,w ,ni ,pi ,Θi ,mi , the products
k k kK K K aK
being weightedby factorsinvolving the thermally averagedphotonandphonon
quantum numbers n ,m ,N for the SE modes, cavity decay modes and vi-
k k K
brational modes. The thermally averaged quantum numbers are given by the
Planck function.
The general case will be studied in further work.
3.2 Case of spontaneous emission for stationary qubits,
no cavity mode
Inthiscasethecavitymodeisignored,thequbitsareassumedstationarysothat
CMvibrationalmodesareexcluded(asareallLamb-Dickeinteractions),andthe
only coupling constantsincluded aregi, givenby g expik r . The qubit pure
k k · i0
9
stateis φ . Thisspecialsituationhasbeenstudiedin[33],wheretemperature
| Qi
effects and criteria for the qubits decohering independently or collectively were
examined.
For this case the decoherence time τ is given by:
2
1
= ( σi σj σi σj ) gi gj∗(2n +1) (31)
2τ2 h X Xi−h Xih Xi k k k
2 Xij Xk
~ω
= ∆σi ∆σj g 2cosk d coth k (32)
h X Xi | k| · ij 2k T
Xij Xk B
where .. = φ ..φ ,∆Ω = Ω Ω and d = r r . This result is the
h i h Q| | Qi −h i ij i0 − j0
same as in [33].
As in[33], if∆k is the bandwidth andk the meanof a normalisedGaussian
modelforthefunction g 2coth ~ωk ,thentheconditionforindependentdeco-
| k| 2kBT
herence(whereonlytermswithi=j contributetoτ )requiresthat∆kd 1
2 ij
≫
foranypairofqubits(d = d ). Collectivedecoherence(wheretermsfromall
ij ij
| |
pairs i= j of qubits contribute to τ ) requires that ∆kd 1 and kd 1.
2 ij ij
6 ≪ ≪
Further discussion about independent and collective decoherence is given be-
low for the case of zero temperature, but based on evaluating the sum over SE
modes k for a three dimensional model.
3.3 Case of no gating processes, zero temperature and
cavity in vacuum state
The case where no gating is taking place, the ancilla is back in its original no
photon state 0 and the temperature is zero, provides a simple case of the
A
| i
general results, enabling the dependence on the qubit state to be examined.
Here ψ = φ 0 , where the state of the qubit system is φ . This case is
| Si | Qi| iA | Qi
ofsomephysicalinterest,since it correspondsto states producedafter idealised
coherentgatingprocesseshaveoccurred. Toorapidadecoherenceforsuchstates
would be of concern for the implementation of quantum computers.
3.3.1 Case of correlated qubit states
1
= ( σi σj σi σj )G
2τ2 h X Xi−h Xih Xi ij
2 Xij
+ σi σj K +L (33)
h X Xi ij
Xij
G = gi gj∗+ ni nj∗ (34)
ij k k kK kK
Xk XkK
K = pi pj∗ (35)
ij K K
XK
L = w w∗ (36)
k k
Xk
10