Datasets:
wesley7137
/
quantum

Dataset card Data Studio Files
xet
 Community
text
stringlengths
0
8.13M
space. Thus, the standard basis of Cartesian coordinates can be chosen for a representation of t
and f; i.e.,
1 0
t and f . (12)
≡ 0 ≡ 1
(cid:18) (cid:19) (cid:18) (cid:19)
The evolutionrepresentingdiagonalization(effectively, agentA’stask)canbeexpressedbythe
unitary operator D by
Dt=f and Df =t . (13)
Thus, D acts essentially as a not-gate. In the above state basis, D can be represented as follows:
0 1
D = . (14)
1 0
(cid:18) (cid:19)
D will be called diagonalization operator,despite the fact that the only nonvanishing components
are off-diagonal.
As has been pointed out earlier, quantum information theory allows a coherent superposition
h =αt+βf of the cbit states t and f. D acts on cbits. It has a fixed point at the qbit state
α,β
t+f 1 1
h∗ :=h = . (15)
√1 2, √1 2 √2 ≡ √2 1
(cid:18) (cid:19)
h does not give rise to inconsistencies [88]. If agent A hands over the fixed point state h to the
∗ ∗
diagonalizationoperatorD,thesamestateh isrecovered. Stateddifferently,aslongastheoutput
∗
of the “halting algorithm” to input A(A) is h , diagonalizationdoes not change it. Hence, even if
∗
the (classically)“paradoxical”constructionofdiagonalizationismaintained,quantumtheorydoes
not giverise to a paradox,because the quantum rangeof solutions is largerthan the classicalone.
Therefore, standard proofs of the recursive unsolvability of the halting problem do not apply if
agent A is allowed a qbit.
Another, less abstract, application for quantum information theory is the handling of incon-
sistent information in databases. Thereby, two contradicting cbits of information t and f are
resolved by the qbit h = (t+f)/√2. Throughout the rest of the computation the coherence is
∗
maintained. After the processing,the result is obtained by an irreversiblemeasurement. The pro-
cessing of qbits, however, would require an exponential space overhead on classical computers in
cbit base [40]. Thus, in order to remain tractable,the correspondingqbits should be implemented
on truly quantum universal computers.
It shouldbe noted, however,thatthe fixedpoint qbit “solution”to the abovehalting problem,
as far as problem solving is concerned, is of not much practical help. In particular, if one is
interestedinthe “classical”answerwhether ornotA(A) halts,thenone ultimately hasto perform
anirreversiblemeasurementonthefixedpointstate. Thiscausesastatereductionintotheclassical
statescorrespondingtotandf. Anysinglemeasurementwillyieldanindeterministicresult. There
is a 50:50 chance that the fixed point state will be either in t or f, since P (h ) = P (h ) = 1.
t ∗ f ∗ 2
Thereby, classical undecidability is recovered. Stated pointedly: With regards to the question of
whether or not a computer halts, the “solution” h is equivalent to the throwing of a fair coin.
∗
Therefore, the advance of quantum recursion theory over classical recursion theory is not so
much classical problem solving but the consistent representation of statements which would give
rise to classical paradoxes.
4.6.3 Proper quantum diagonalization
The above argument used the continuity of qbit states as compared to the two cbit states for a
constructionof fixed points of the diagonalizationoperator. One could proceed a step further and
allownonclassicaldiagonalization procedures. Suchastep,albeitoperationalizable,hasnoclassical
operational equivalent, and thus no classical interpretation.
Consider the entire range of twodimensional unitary transformations [72]
eiα cosω e iϕ sinω
U(2)(ω,α,β,ϕ)=e −iβ − − , (16)
eiϕ sinω e iα cosω
−
(cid:18) (cid:19)
12
where π β,ω π, π α,ϕ π, to act on the qbit. A typical example of a nonclassical
− ≤ ≤ − 2 ≤ ≤ 2
operation on a qbit is the “square root of not” gate (√not√not=D)
1 1+i 1 i
√not= − . (17)
2 1 i 1+i
(cid:18) − (cid:19)
Not all these unitary transformations have eigenvectors associated with eigenvalues 1 and thus
fixed points. Indeed, it is not difficult to see that only unitary transformations of the form
cosω2+eiλsinω2 1+eiλ e i(α+ϕ) sin(2ω)
[U(2)(ω,α,β,ϕ)] 1diag(1,eiλ)U(2)(ω,α,β,ϕ)= − 2 −
− −1+ 2eiλ ei(α+ϕ) sin(2ω) eiλcosω2+sinω2 !
(18)
have fixed points.
Applying nonclassical operations on qbits with no fixed points
D = [U(2)(ω,α,β,ϕ)] 1diag(eiµ,eiλ)U(2)(ω,α,β,ϕ)
′ −
eiµcos(ω)2+eiλsin(ω)2 e−i(α+p) eiλ eiµ sin(2ω)
= 2 − (19)
ei(α 2+p) eiλ −eiµ sin(2ω) eiλcos( (cid:0)ω)2+eiµ (cid:1)sin(ω)2 !
with µ,λ = nπ, n N gives r(cid:0)ise to eigen(cid:1)vectors which are not fixed points, but which acquire
0
6 ∈
nonvanishing phases µ,λ in the generalized diagonalization process.
5 Quantum algorithmic information
Quantum algorithmic information theory can be developed in analogy to algorithmic information
theory [28, 24, 63]. Before proceeding, though, one decisive strategic decision concerning the
physicalcharacteroftheprogramhastobemade. Thisamountstoarestrictiontopurelyclassical
prefix-free programs.
The reason for classical programs, as well as for the requirement of instant decodability,
is the desired convergence of the Kraft sum over the exponentially weighted program length
exp(p logk) 1, where p stands for the length of p and k is the base of the code (for
p | | ≤ | |
binary code, k = 2). If arbitrary qbits were allowed as program code, then the Kraft sum would