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probabilities also have to satisfy (cid:104)0|Γ (|0(cid:105)(cid:104)0|)|0(cid:105) = 1 computationally secure against attacks using quantum
z(cid:48)
and (cid:104)+|Γ (|+(cid:105)(cid:104)+|)|+(cid:105) = 1/2, which leads contradic- computers.
z(cid:48)
tion to the previous results. Acknowledgements: The authors thank M. R¨otteler,
Thus,ifthereisanalgorithmA,wecandecidewhether M.Ukita,Y.KawamotoandD.Markhamforusefulcom-
an obfuscated quantum gate sequence z(cid:48)(C ), obtained ments. ThisworkispartlysupportedbySpecialCoordi-
U
by using sufficiently-random shuffling algorithm for a nation Funds for Promoting Science and Technology.
[1] B. Barak, O. Goldreich, R. Impagliazzo, S. Rudich, A. shift with phase 2π/p , where p is a prime number and
i i
Sahai, S. Vadhan, K. Yang, Advances in Cryptogra- p (cid:54)=p ,(i(cid:54)=j). Considering two operations P =⊗n P
i j n i=1 i
phy – CRYPTO’01, Lecture notes in Computer Science, and H = H⊗n, one can show that U = eiθI iff
n
Springer-Verlag, 2001. [H ,U] = [P ,U] = 0. Thus, define U = UP U†P† and
n n 1 n n
[2] S. Aaronson, Ten Semi-Grand Chal- U = UH U†H for a given unitary operation U, and
2 n n
lenges for Quantum Computing Theory replace z(C ) with z(C ) and z(C ).
U U1 U2
http://www.scottaaronson.com/writings/qchallenge.html. [6] Y. Tanaka, quant-ph/0903.0675v1.
[3] P. Arrighi and L. Salvail, Int. J. of Quantum Information [7] A.Y.Kitaev,A.H.Shen,andM.N.VyalyiClassical and
4,883(2006);A.Broadbent,J.FitzsimonsandE.Kashefi, QuantumComputation,GraduateStudiesinMathematics
quant-ph/0807.4154. 47, Am. Math. Soc., Providence, Rhode Island, (2002).
[4] M.A.NielsenandI.L.Chuang,Phys.Rev.Lett.79,321 [8] M.Hayashi,QuantumInformation,Springer-Verlag,2006.
(1997).
[5] Let H and P be a Hadamard operation and a phase
i
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GLL/07-24
Duality and Recycling Computing in Quantum Computers
Gui Lu Long and Yang Liu
1 Key Laboratory of Atomic and Molecular NanoSciencs and Department of Physics,
Tsinghua University, Beijing 100084, P. R. China
2 Tsinghua National Laboratory for Information Science and Technology, Beijing 100084, P. R. China
(Dated: October30, 2018)
Quantumcomputerpossessesquantumparallelismand Thegeneralquantumgate p U ,ordualitygate,isno
i i i
offers great computing power over classical computer longerunitary. Asaresult,PthefinalwavefunctioninEq.
[1,2]. Asiswell-know,amovingquantumobjectpassing (3)is only partofa wavefunction. The powerofduality
7002
through a double-slit exhibits particle wave duality. A computer depends sensitively on the result of measure-
quantum computer is static and lacks this duality prop- ment ofpartof a wavefunction. Three possibilities have
erty. The recently proposed duality computer has ex- been suggested [3]: 1) one will get a result immediately
guA ploited this particle wave duality property, and it may asifawholewavefunctionweremeasured;2)onewillget
offer additional computing power [3]. Simply put it, a a result but with a longer time; 3) one sometimes gets
dualitycomputerisamovingquantumcomputerpassing a result and sometimes one does not. In the first two
through a double-slit. A duality computer offers the ca- scenarios, a duality computer could solve NP-complete
51
pabilitytoperformseparateoperationsonthesub-waves problems in polynomial time [3, 4]. The mathematical
coming out of the different slits, in the so-called dual- description of duality computer with the first two sce-
ity parallelism. Here we show that an n-dubit duality narios has been given recently [5, 6]. In this work, we
]hp-tnauq[
computer can be modeled by an (n+1)-qubit quantum assume case 3, which comes out from the measurement
computer. In a duality mode, computing operations are postulate of quantum mechanics naturally.
not necessarily unitary. A n-qubit quantum computer A symmetric 2-routes duality computer has p1 =p2 =
canbe usedasann-bitreversibleclassicalcomputerand 1/2. Thecompletewavefunctionofadualitycomputeris
is energy efficient. Our result further enables a (n+1)- ψ = ϕ κ where ϕ is the internalstate and κ is the
| i | i| i | i | i
qubit quantumcomputer to runclassicalalgorithmsin a centerofmasstranslationalmotionwavefunction. When
O(2n)-bitclassicalcomputer. Thedualitymodeprovides a QWD operation is performed, it changes the state to
a naturallink betweenclassicalcomputing andquantum
1v6891.8070:viXra computing. Here we also propose a recycling computing 2 1
modeinwhichaquantumcomputerwillcontinuetocom- ψ ′ = ϕ κ i . (4)
| i ⊕2| i| i
pute until the result is obtained. These two modes pro- Xi=1
vide new tool for algorithm design. A search algorithm
One then performs different gate operations on the sub-
for the unsorted database search problem is designed.
waves,
In a duality computer, there exist two new computing
gates in addition to the usual universal gates for quan- 2
1
tum computers [3]. One such gate is the quantum wave ψ = U ϕ κ . (5)
′′ i i
divider (QWD), and a double-slit is an example of such | i 2 Xi=1⊕ | i| i
gate. Suppose there is a complex Hilbert space H, a
QWD reproduces copies of the wave function with at- Then a QWC operation is performed and changes the
tenuated coefficient in m direct summed Hilbert spaces state to
m m
H = H , namely
⊕ Pi=1⊕ i
|ψ =(p1U1+p2U2) |ϕ i|κ i. (6)
if
D (ψ )= p ψ , (1)
p i i
| i ⊕ | i
i=X1,m A final measurement is performed on ψ f to read-out
| i
the outcome. Under the 3rd assumption about partial
where p = 1 and p is the strength of the sub-wave
i i i measurement, the wave function should not renormal-
inthe iP-thslit, m is the numberofslits. Another duality
ized, and a result to a conditional measurement is ob-
gateisthe reverseofQWD,the quantumwavecombiner