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P : 0 0eiϕ . (102)
4 ′ ′
→
1 0
When again 0 0 and 1 1 , then the corresponding unitary evolution
≡ ′ ≡ 0 ≡ ′ ≡ 1
(cid:18) (cid:19) (cid:18) (cid:19)
matrix which transforms any coherent superposition of 0 and 1 into a superposition of 0 and 1
′ ′
is given by
TM 21Z(α,β,ω,ϕ)= −ie−i(β+ω 2) −e e− −i( iα α+ cϕ o) ss 2inω 2 e −i sϕ inc 2osω 2 . (103)
ω ω
(cid:18) (cid:19)
ThecorrespondencebetweenTbs(T(ω),α,β,ϕ)withTMZ(α,β ,ω ,ϕ)inequations(94)(103)
21 21 ′ ′ ′ ′
can be verified by comparing the elements of these matrices. The resulting four equations can be
used to eliminate the four unknown parameters ω = 2ω, β = β ω, α = α π/2, β = β ω
′ ′ ′ ′
− − −
and ϕ =ϕ π/2; i.e.,
′
−
π π
Tbs(ω,α,β,ϕ)=TMZ(α ,β ω,2ω,ϕ ) . (104)
21 21 − 2 − − 2
Both elementary quantum interference devices are universal in the sense that every unitary
quantum evolution operator in twodimensional Hilbert space can be brought into a one-to-one
correspondence to Tbs and TMZ; with corresponding values of T,α,β,ϕ or α,ω,β,ϕ. This can
21 21
be easily seen by a similar calculation as before; i.e., by comparing equations (94) (103) with the
“canonical”formofaunitarymatrix,whichistheproductofaU(1)=e iβ andoftheunimodular
−
unitary matrix SU(2) [72]
eiα cosω e iϕ sinω
T(ω,α,ϕ)= − − , (105)
eiϕ sinω e iα cosω
−
(cid:18) (cid:19)
where π β,ω π, π α,ϕ π. Let
− ≤ ≤ − 2 ≤ ≤ 2
T(ω,α,β,ϕ)=e iβT(ω,α,ϕ) . (106)
−
23
A proper identification of the parameters α,β,ω,ϕ yields
π π π π
T(ω,α,β,ϕ)=Tbs(ω , α ϕ ,β+α+ ,ϕ α+ ) . (107)
21 − 2 − − − 2 2 − 2
Let us examine the realization of a few primitive logical “gates” corresponding to (unitary)
unary operations on qbits. The “identity” element I is defined by 0 0, 1 1 and can be
→ →
realized by
π π π π 1 0
I=Tbs( , , , )=TMZ( π,π, π,0)= . (108)
21 −2 −2 2 2 21 − − 0 1
(cid:18) (cid:19)
The “not” element is defined by 0 1, 1 0 and can be realized by
→ →
π π 0 1
not=Tbs(0,0,0,0)=TMZ( ,0,0, )= . (109)
21 21 − 2 − 2 1 0
(cid:18) (cid:19)
The next element, “√not” is a truly quantum mechanical; i.e., nonclassical, one, since it
converts a classical bit into a coherent superposition of 0 and 1. √not is defined by 0 0+1,
→
1 0+1 and can be realized by
→−
π π π π 3π π 1 1 1
√not=Tbs( , , , )=TMZ( π, , ,0)= − . (110)
21 −4 −2 2 2 21 − 4 −2 √2 1 1
(cid:18) (cid:19)
Note that √not √not=not diag(1, 1)=not(mod1). The relative phases in the output ports
· · −
showing up in diag(1, 1) can be avoided by defining
−
π π π π π π 1 1+i 1 i
√not′ =Tbs( ,0, ,0)=TMZ( , , , )= − . (111)
21 − 4 4 21 − 2 2 − 2 − 2 2 1 i 1+i
(cid:18) − (cid:19)
With this definition, √not′√not′ =not.
It is very important that the elementary quantum interference device realizes an arbitrary
quantumtimeevolutionofatwodimensionalsystem. Theperformanceofthequantuminterference
device is determined by four parameters, corresponding to the phases α,β,ϕ,ω.
References
[1] D. Z. Albert, Phys. Lett. 98A, 249 (1983).
[2] L. E. Ballentine, Quantum Mechanics (Prentice Hall, Englewood Cliffs, 1989); for a short
expose, see also L. E. Ballentine, Rev. Mod. Phys. 42, 358 (1970).
[3] A. Barenco, C. H. Bennett, R. Cleve, D. P. DiVincenzo, N. Margolus, P. Shor, T. Sleator,
J. Smolin and H. Weinfurter, Elementary gates for quantum computation, e-print quant-
ph/9503016(URL: http://xxx.lanl.gov/abs/quant-ph/9503016).
[4] J. S. Bell, Speakable and Unspeakable in Quantum Mechanics (Cambridge University Press,
Cambridge, 1987).
[5] P. Benioff, J. Stat. Phys. 29, 515 (1982); Phys. Rev. Lett. 48, 1581 (1982).
[6] P. Benioff, Annals New York Academy of Sciences 480, 475 (1986).
[7] C. H. Bennett, Logical Reversibility of Computation, IBM J. Res. Dev. 17, 525-532 (1973);
reprinted in: Maxwell’s Demon, ed. by H. S. Leff and A. F. Rex (Princeton University Press,
1990), pp. 197-204.
[8] C.H.Bennett,G.Brassard,S.BreidbartandS.Wiesner,Quantumcryptography,orunforgable
subway tokens,in Advancesin Cryptography: ProceedingsofCrypto ’82(PlenumPress,New
York, 1982), pp. 78-82.
[9] C. H. Bennett, The Thermodynamics of Computation–A Review, Int. J. Theor. Phys. 21,