Actual computational time-cost of the Quantum Fourier Transform in a quantum computer using nuclear spins

4 Pages·0.11 MB·English

by  A. Saito

Actual computational time-cost of the Quantum Fourier Transform in a quantum
computer using nuclear spins
∗
A. Saito , K. Kioi, Y. Akagi, N. Hashizume, K. Ohta
Advanced Technology Research Laboratories, Sharp Corporation, Tenri-shi, 632 Japan
quantum computers being considered are constructed
There have been many proposed methods for the prac-
from an array of n quantum registers labeled by j(1
tical implementation of quantum computing. Now quantum ≤
j n) in one dimension. They calculate n qubit data
computationhasreachedtheturningpointfrombeingacon- wh≤ich correspondsto N =2n states. We defined the fol-
ceptualsystemtobecomingaphysicalone. Inthispaper,we
lowing matrices. The suffixes on each matrix represent
discuss a practical elementary gate and the actual computa-
the quantum registers on which it operates. In the def-
0 tionaltime-costoftheQFTintwophysicalimplementations,
initions of C (θ) and D (θ), the first and the second
0 namely the bulk spin resonance computer and the Spin Res- j,k j,k
suffixes represent the target bit and the controlled bit
0 onance Transistor. We show that almost all universal gates
2 require different times for operation. The actual time-cost of respectively.
n the QFT is O(n2n) for large n. This differs drastically from 1 1 1 cosθ/2 sinθ/2
Ja thereportedcostO(n2)basedonidealquantumcomputation. Hj = √2(cid:18)1 −1(cid:19), Ryj(θ)=(cid:18)−sinθ/2 cosθ/2(cid:19)

1
3 1 0 0 0
1 Recent technological development has stimulated pro- Rzj(α)=(cid:18)eiα0/2 e−i0α/2 (cid:19), Cj,k(θ)= 00 10 01 00 
v posals for many quantum computers: Bulk Spin Res-  0 0 0 eiθ 
3 onance(BSR) [1–3], trapped ions [4], cavityQED [5],  
1
Josephson junctions [6], coupled quantum dots [7] and
1
theSpinResonanceTransistor(SRT)[8,9]. BSRhasbeen eiθ 0 0 0
1
0 implemented experimentally in some organic molecules eiδ 0  0 e−iθ 0 0 
0 by the use of conventional nuclear magnetic reso- Φj(δ)=(cid:18) 0 eiδ (cid:19),Dj,k(θ)= 0 0 e−iθ 0
 
0 nance(NMR)equipment[1–3]. theSRTisattractivefrom  0 0 0 eiθ 
/ the viewpoint of it’s integration ability and it’s compat-  
h
p ibility with silicon technology. Firstly, we discuss the practical elementary gate in
- Complexity analysis classifies quantum algorithms ac- these implementations. The practical elementary gate is
nt cordingtoafunctionthatdescribeshowacomputational defined as a quantum gate that can be achieved directly
a cost incurred in solving a problem scales up as larger byphysicalphenomenaintheimplementationbeingcon-
u problems are considered [10]. The computational cost of sidered.
q a quantum algorithm has usually been estimated as the BSRconsistsofquantumregistersthatarethenuclear
v: sum of the universal gates required in such ideal mathe- spinsofeachatominanorganicmolecule[1–3]. External
i matical models as the Quantum Turing Machine(QTM) RF-pulsesandmagneticfieldscontrolthequantumstates
X
andthe quantumcircuit. Thecomputationalcomplexity and the quantum correlations in the following way.
ar is effective in estimating the essential performance of an The nuclear spin under a strong magnetic field
algorithmtofactoroutthe variationsinperformanceex- B = (0,0,B0) is described by the Hamiltonian H =
perienced by different makes of computers with different γ¯hB0Ijz, where γ is the gyro-magnetic ratio for the
−
amountsofRandomAccessMemory(RAM),swapspace, spin, and Ijz is the z component of the jth nuclear
and processor speeds. The above cost is proportional to spin. The time evolution of the system is exp(iHt/¯h) =
anactualtime-costinthephysicalimplementationwhere exp( iγ¯hB0tSjz) and this then corresponds to the rota-
−
allquantumoperationscanbeachievedinthesametime. tiononthez-axisRzj(θ =γ¯hB0t). TheRF-pulseenables
However,if the implementation being consideredtakes a the rotation on the other axis [11]. The time evolution
different time for each quantum gate, there is a possibil- showsthat the rotationangleθ can be controlledby two
itythattheactualtime-costwillhaveadifferentbehavior factors: theintensityB0,andthedurationt,oftheexter-
from the ideal cost. A hardware dependent time-cost is nal magnetic field. Consequently, there are two control
importanttoexperimentalistswhoresearchintopractical modes. In the intensity control mode, each single qubit
implementations. In this paper, we focus on the actual rotation takes the same time. In the duration control
computational time-cost of the Quantum Fourier Trans- mode however, each single qubit phase rotation takes a
form(QFT) intwophysicalimplementations thatutilise different time, which is proportional to the rotation an-
nuclear spins: BSR and the SRT. gle.
For the following discussion, we assumed that the The exchange interaction between the j-th and the k-
th registers is described by the Hamiltonian H =
exch.
1
J I I , where J is the exchange coupling constant. for execution. As discussed above, almost all univer-
jk jz kz jk
The time evolution of the system is described by sal gates take a different time in these implementations.
It suggests the possibility that the actual time-cost of
exp(iJjktIjzIkz/¯h)=Dj,k(θ =Jjkt/2h¯). (1) the quantum algorithm is different from the ideal cost,
though the complexity is not affected in the QFT case.
Sequence(1) shows that the exchange interaction can Next we estimate the actual time-cost of the QFT
control the phase rotation angle. Any external fields, thatcouldbeachievedbytheabovepracticalelementary
however,cannot controlthe interactiondirectly, because gates, by considering the time resolutions of the control-
atomsinamoleculealwaysinteractwitheachother. The ling external fields in these implementations.
refocusing technique can control the angle θ effectively The QFT is the transform with base N = 2n, corre-
[11]. Only the time duration between the refocusing sponding to the n qubit defined by
pulses determines the angle θ. The operation D (θ),
j,k
therefore, takes a time that is proportional to the rota- 1 N−1 2πicx
tionangleθ. AnactualoperationsuchasD iseffective x e N c . (3)
j,k | i→ √q | i
only for adjacent registers because the exchange interac- Xc=0
tion J between non-adjacent registers is very small.
j,k Shor proposed the algorithm factoring a composite inte-
The SRT is composed of quantum registers which are
ger by the QFT [13]. The QFT can be constructed by
the nuclear spins of arrayed phosphorus ions in silicon,
the sequence [13] in the order (from right to left)
with a globally static magnetic field B and an AC mag-
netic field BAC [8]. The implementation consists of two H0C0,1(θ1)C0,2(θ2) C0,n−1(θn−1)H1 Hn−3
··· ···
gates on the surface: the A-gate above each ion and the
Cn−3,n−2(θ1)Cn−3,n−1(θ2)Hn−2Cn−2,n−1(θ1)Hn−1 (4)
J-gate between adjacent ions.
The A-gate controls the strength of hyperfine inter- followed by a bit reversal transformation, where θ
j
≡
actions and the resonance frequency of the nuclear spin π/2j. The n qubit QFT requires n(n 1)/2 controlled
−
beneath it. A globally applied magnetic field BAC flips phase-shifter (Cj,k(θk−j)s and n Hadamard transforma-
nuclearspinsresonantwiththe fieldbythe sameprocess tion H s, and then n(n+1)/2 O(n2) [13]. This esti-
j
≃
thatoccursinBSR.Inthiscase,onlythedurationofthe mation is based on the complexity analysis method. It
resonance determines the rotation angle. It implies that coincides with an actual time-cost in the case where all
each single qubit phase rotation always takes a different gates required in the sequence(4) are practical elemen-
time. tary ones.
The electron wave function extends over a large dis- The controlled phase rotation Cj,k(θk−j) can be
tance and makes an effective electron-mediated coupling achieved by the sequence
for two nuclear spins sharing it in semiconductors. The
J-gatecontrolstheoverlapoftheelectronwavefunctions Cj,k(θk−j)=Rzk( θk−j+1)Φk(θk−j+2)Rzj( θk−j+1)
− −
bounded to two adjacent phosphorus atoms, and hence XOR(j,k)Rzj(θk−j+1)XOR(j,k). (5)
the electron-mediated exchange coupling J = J(t) in
j,k
the time evolution(1) directly. It therefore controls the The intensity control mode can make all operations
phase rotation angle in the operation Dj,k. The opera- (Cj,k(θk−j),Hj) take almost the same time. Conse-
tion D in this implementation also operates only for quently,theactualtime-costcoincideswiththeidealcost
j,k
adjacent registers. in this mode.
In both implementations, all single qubit phase rota- The duration control mode, however, makes each
tionsandcontrolledphaserotationsarepracticalelemen- operation Cj,k(θk−j) require the time τk−j in propor-
tary gates. They can make the quantum XOR in the tion to the phase rotation angle θk−j. The operation
sequence below, ordered from right to left [1] C0,n−1(θn−1)rotatestheminimumphaseangleθn−1 and
takes the minimum time τn−1 of all rotations in the
√ iXOR(j,k) QFT(4). The range of required phase rotations in the
− π π π π π QFT(4) increases with 2n. For example, the time ratio
=Ryj(−2)Rzk(−2)Rzj(−2)Dj,k(4)Ryj(2). (2) τ0/τn−1 approximates to 2100 1030 in the 100 qubit
≃
QFT.
The suffixes j and k represent the target bit and the Ingeneral,thetimeresolutiont controllingtheexter-
R
controlled bit respectively. The sequence(2) shows that nalfieldisdeterminedbytheresponsetimeofthesystem,
the quantum XOR depends on single qubit gates in this the delay ofthe electronic signalandso on. We canonly
technique, and is then concerned with the time required execute in the physical implementations that satisfy the
for phase rotation. relationship:
Barenco and co-workers showed that other universal
gates can be constructed by all single qubit gates and τ0 >τ1 > >τn−1 tR. (6)
··· ≥
thequantumXOR[12]. Eachn( 1)qubitgaterequired
≥ It is important for the actual time-cost estimation to
adifferentnumberofthem[12],andthenadifferenttime
determine how we set up the unit time t . The unit
unit
2
timet shouldbealsogreaterthanthetimeresolution using adjacent ones so here we estimate the actual time-
unit
t . The QFT’s in these implementations have various cost required for constructing such non-adjacent gates
R
actual time-costs from tunit =τ0 to tunit =τn−1. fromadjacentones. Wemustconstructanynon-adjacent
If we adopt the maximum rotation time τ0 as the unit gates by use of the adjacent swap technique. The swap
time t , the actual time-cost is O(n), since S is an operation for exchanging data between two
unit j,k
quantum registers simply. It is achieved via the se-
n−2 n−1 τk−j =n−2 n−1 θk−j =n−2 n−1 2j−k nquonen-acdejaScje,nkt=twoXqOubRit(kg,ajt)eXUOR(isj,akc)hXieOveRd(bky,ja).djacTehnet
Xj=0k=Xj+1 τ0 Xj=0k=Xj+1 θ0 Xj=0k=Xj+1 swapsandadjacentUl,l+1 throju,kghthefollowingk j 1
=n+21−n 2 O(n). (7) sequences. − −
− ≃
On the other hand, the condition tunit = τn−1 makes Uj,k =Sk,k−1Sk−1,k−2···Sj+3,j+2Sj+2,j+1Uj,j+1
the actual time-cost O(n2n), since Sj+1,j+2Sj+2,j+3 Sk−2,k−1Sk−1,k (9)
···
n−2 n−1 τk−j =n−2 n−1 θk−j =SSlk,,lk−−11···SSj+l+12,j,lS+l1+S1j,,lj++21···SSkl−−11,,klU(jl,l<+1l <k)
Xj=0k=Xj+1τn−1 Xj=0k=Xj+1 θn−1 =Sj,j+1·S·j·+1,j+2Sj+2,j+3 ···Sk−2,k−1Uk−1,k
···
=n−2 n−1 2n−1+j−k Sk−1,k−2···Sj+3,j+2Sj+2,j+1Sj+1,j. (10)
Xj=0k=Xj+1 Figure 1 represents the sequence(9). These sequences
=(n 2)2n−1+1 O(n2n). (8) show that the non-adjacent operationUj,k requires both
− ≃ the time for the adjacent operation Uj,j+1 and another
In this way, the actual time-cost varies from O(n) to time in proportionto k j for swapstransferring data.
| − |
O(n2n), and it depends on which of these is adopted as Considering the construction (9 10), the QFT(4) re-
theunittimet . Theformertime-costhowever,isnot quires more (n 1)n(2n 1)/6 ∼O(n3) adjacent swaps
unit − − ∼
valid for any n in the following way. for data transfer besides the adjacent controlled phase
Theformercondition,tunit =τ0,meansthatallphases rotations.
arealwaysrotatedbytheexternalfieldwithconstantin- There is, however, a way to reduce some swaps.
tensity B for various data of magnitude n. In this case, From sequences(4) and (10), we can obtain the actual
the minimum time τn−1 decreasesexponentially with in- quantum circuit shown by Fig.2. In the figure, swaps
creasing n. We cannot rotate the phase θj to satisfy the Sj,j+1···Sk−1,kSk,k−1···Sj+1,j inside each box can be
condition τ < t . There exists the upper bound n , reduced to the identity operationI. The n qubit QFT
j R b
satisfying the relationship(6) for the intensity B under requires(n 1)(n 2) O(n2) swaps as a result. It has
− − ∼
consideration. The estimated time-cost(7) is valid for the same polynomialorderasthe idealcost. Inthis way,
any n satisfying n n . It is, however,not valid for any therearesomecaseswheretheswapscanbereduceddue
b
≤
n which is greater than n . to the symmetry of the algorithm under consideration.
b
The latter condition tunit = τn−1 means that the in- We canconclude thatthe actualtime-costofthe QFT
tensity B decreases exponentially with n satisfying the isdominatedbythephaserotationsandthenareO(n2n)
relation(6). In this case, we can rotate all phase angles in the range n>n .
b
θ (0 j n 1)intheQFT(4)accurately. Aparticular The range of required phase rotation angles increases
j
≤ ≤ −
condition tunit = τn−1 = tR always achieves all phase exponentially with n in the QFT if the accuracy is pre-
rotationsinthe minimumtotaltime,andthenyieldsthe served. In the implementations that are considering, we
best computing performance for each value of n. The need to obtain it by controlling the duration or the in-
actual time-cost always obeys eq.(8) for any n. tensity of the external field. The duration control mode
In this way, the actual time-cost varies from O(n) to increasestheactualtime-costdrasticallywith2n. Thein-
O(n2n)foranyn( n ),andfollowsonlythelatterforall tensity control mode in BSR takes least time and seems
b
≤
other values of n. These costs are estimated making the to be the most efficient case for the QFT. This mode,
assumption that the QFT is always executed accurately. however,leadstoanotherburdenontheequipment. The
An approximate QFT, (AQFT) can reduce the arbi- required range of intensity for the applied field increases
trary numbers of the controlledphase shift gates by sac- with 2n. If the minimum phase rotation is implemented
rificing the accuracy [14]. We can select the AQFT with by the field B = 10−3 T, the maximum phase rotation
an actual time-cost between O(n) and O(n2n), by con- forthe100qubitQFTrequiresafieldintensityB 1027
∼
sidering the required accuracy for any n. T, which is far beyond the current feasible intensity for
We havediscussedthe actualtime-costfromthe view- the magnetic field.
point of the phase rotations in the QFT(4). Almost Our results lead to concerns about the feasibility of
Cj,k(θk−j) operations occur for non-adjacentregisters in factoringhugenumberswithpolynomialordertime-costs
the QFT. We need to construct such non-adjacent gates in the implementations we are considering.
3
We have shown that almost all universal gates do not FIG.2. Figures(a),(b)and(c)inorderaretheactualquan-
expend the same time in BSR and the SRT. This causes tumcircuits of the5qubitQFT. This quantumcircuit is de-
rivedfromsequences(4)and(10). Theswapsinsidetheboxes
theactualtime-costoftheQFTtobedrasticallydifferent
are reducible.
from the ideal one. The ideal cost of the QFT is not
effective in the range n > n for practical cases of these
b
implementations if the accuracy is preserved.
We believe that both ideal and actual discussions are
importantforthedevelopmentofquantumcomputersci-
ence. Theformerdiscussionstimulatesstudyofthechar-
acteristic of the algorithm itself, and the latter is also
important for the current situation in quantum compu- ∗
e-mail address: [email protected]
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