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τ Z, can be described by a quantum state C(τ,p ). Let C(p) = s stand for a computer C with
i
∈
programpwhichoutputssinarbitrarylongtime. Inwhatfollowsweshallassumethattheprogram
p is coded classically. That is, we choose a finite code alphabet A and denote by A the set of all
i ∗
strings overA. Any programp is coded as a classicalsequence pp q=s s s A , s A.
i i 1i 2i ni ∗ ji
Whenever possible, pp q will be abbreviated by p . We assume prefix coding· [· 4· 9, 28∈ , 86, 24];∈ i.e.,
i i
the domain of C is prefix-free such that no admissible programis the prefix of another admissible
program. Furthermore,withoutlossofgenerality,weconsideronlyempty inputstrings. p stands
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for the length of p.
4.6 Diagonalization
This is neither the place for a comprehensive review of the diagonalization method (cf. [80, 73]),
nor suffices the author’s competence for such an endeavor. Therefore, only a few hallmarks are
stated. As already G¨odel pointed out in his classical paper on the incompleteness of arithmetic
[45],theundecidabilitytheoremsofformallogic[29](andthetheoryofrecursivefunctions[80,73])
arebasedonsemanticalparadoxessuchasthe liar[16]orRichard’sparadox. Apropertranslation
of the semantic paradoxes results in the diagonalization method. Diagonalization has apparently
firstbeenappliedbyCantortodemonstratetheundenumerabilityofrealnumbers[25]. Ithasalso
been used by Turing for a proof of the recursive undecidability of the halting problem [92].
A brief review of the classical algorithmic argument will be given first. Consider a universal
computerC. Forthe sakeofcontradiction,consideranarbitraryalgorithmB(X)whoseinputis a
string of symbols X. Assume that there exists a “halting algorithm” HALT which is able to decide
whether B terminates on X or not. The domain of HALT is the set of legal programs. The range
of HALT are cbits (classical case) and qbits (quantum mechanical case).
Using HALT(B(X)) we shall construct another deterministic computing agent A, which has as
input any effective program B and which proceeds as follows: Upon reading the program B as
input, A makes a copy of it. This can be readily achieved, since the program B is presented to A
in some encoded form pBq, i.e., as a string of symbols. In the next step, the agent uses the code
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pBq as input string for B itself; i.e., A forms B(pBq), henceforth denoted by B(B). The agent
now hands B(B) overto its subroutine HALT. Then, A proceeds as follows: if HALT(B(B)) decides
that B(B) halts, then the agent A does not halt; this can for instance be realized by an infinite
DO-loop; if HALT(B(B)) decides that B(B) does not halt, then A halts.
The agent A will now be confronted with the following paradoxicaltask: take the own code as
input and proceed.
4.6.1 Classical case
Assume thatAis restrictedtoclassicalbits ofinformation. Tobe morespecific, assumethatHALT
outputs the codeofacbitasfollows( and standsfordivergenceandconvergence,respectively):
↑ ↓
0 if B(X)
HALT(B(X))= ↑ . (10)
1 if B(X)
(cid:26) ↓
Then, whenever A(A) halts, HALT(A(A)) outputs 1 and forces A(A) not to halt. Conversely,
whenever A(A) does not halt, then HALT(A(A)) outputs 0 and steers A(A) into the halting mode.
In both cases one arrives at a complete contradiction. Classically, this contradiction can only be
consistently avoidedby assuming the nonexistence ofA and, since the only nontrivialfeature of A
is the use of the peculiar halting algorithm HALT, the impossibility of any such halting algorithm.
4.6.2 Quantum mechanical case
Recall that a quantum computer C evolves accordingto a unitary operator U such that (τ stands
for the discrete time parameter) C(τ,p )=UC(τ 1,p )=UtC(0,p ).
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As has been pointed out before, in quantum information theory a qbit may be in a coherent
superpositionofthetwoclassicalstatestandf. Duetothispossibilityofacoherentsuperposition
of classical bit states, the usual reductio ad absurdum argument breaks down. Instead, diagonal-
ization procedures in quantum information theory yield qbit solutions which are fixed points of
the associated unitary operators.
In what follows it will be demonstrated how the task of the agent A can be performed con-
sistently if A is allowed to process quantum information. To be more specific, assume that the
output of the hypothetical “halting algorithm” is a halting qbit
HALT(B(X))=h . (11)
α,β
OnemaythinkofHALT(B(X))asauniversal“watchdog”computerC simulatingC andcontaining
′
a dedicated halting bit, which it outputs at every (discrete) time cycle [31]. Alternatively, it can
be assumedthat the computer C contains its own halting bit indicating whether it has completed
its task or not. Note that the halting qbit h can be represented by a normalized13 vector in
α,β
twodimensional complex Hilbert space spanned by the the orthonormal vectors “t” and “f.” Let
the halting state h =t (up to factors modulus 1) be the physical realization that the computer
1,0
has “halted;” likewise let h = f (up to factors modulus 1) be the physical realization that
0,1
the computer has not “halted.” Note that, since quantum computations are governed by unitary
evolution laws which are reversible, the halting state does not imply that the computer does not
changeastimeevolves. Itjustmeansthatithassetasignal—thehaltingbit—toindicatedthatit
hasfinisheditstask. αandβ arecomplexnumberswhichareaquantummechanicalmeasureofthe
probability amplitude that the computer is in the halting and the non-halting states, respectively.
The corresponding halting and non-halting probabilities are a2 and a2, respectively.
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Initially, i.e., at t=0, the halting bit is prepared to be a 50:50 mixture of the classicalhalting
and non-halting states t and f; i.e., h . If later C finds that C converges (diverges) on
1/√2,1/√2 ′
B(X), then the halting bit of C is set to the classical value t (f).
′
The emergence of fixed points can be demonstrated by a simple example. Agent A’s diagonal-
ization task can be formalized as follows. Consider for the moment the action of diagonalization
onthe cbit states. (Since the qbit states aremerelya coherentsuperpositionthereof,the actionof
diagonalizationonqbitsisstraightforward.) Diagonalizationeffectivelytransformsthe cbitvaluet
into f andvice versa. Recallthatin equation(10), the state t has beenidentified with the halting
state and the state f with the non-halting state. Since the halting state and the non-halting state
13 (h α,β,h α,β)=1.
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excludeeachother,f,tcanbeidentifiedwithorthonormalbasisvectorsinatwodimensionalvector