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s s s s s s s s s s s
p p p p p p p p p p p
1 2 3 4 5 1 2 3 4 5 1
Figure1: Thelowest“root”representstheinitialstateinterpretableasprogram. Forwardcomputa-
tionrepresentsupwardsmotionthroughasequenceofstatesrepresentedbyopencircles. Different
symbols p correspond to different initial states, that is, different programs. a) One-to-one com-
i
putation. b) Many-to-one junction which is information discarding. Several computational paths,
moving upwards, merge into one. c) One-to-many computation is allowed only if no information
is created and discarded; e.g., in copy-type operations on blank memory.
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(ii) any computation (step) can be represented by a unitary transformationof the computer as a
whole;
(iii) because of the unitarity of the quantum evolution operator, any computation is reversible.
Therefore,adeterministiccomputationcanbeperformedbyaquantumcomputerifandonly
if it is reversible,i.e., if the programdoes not involve “deletion”of information or “many-to-
one” operations (cf. [60]); only one-to-one operations are allowed;
(iv) (in contradistinction to classical reversible computation) unless classical, qbits cannot be
copied; they are context-dependent (cf. below);
(v) measurements may be carried out on any qbit at any stage of the computation. But, un-
less classical, a qbit cannot be measured by a single experiment with arbitrary accuracy (cf.
(III) and (IV)). The computation process and the measurement have to be repeated in
order to obtain sufficient statistics.–Any such single measurement will yield merely a “click”
on some counter, from which information about the qbit state must be inferred. Thereby,
any single measurement is indeterminate, and coherence is destroyed. Therefore, it seems
more proper to realize that there is no such operational concept of “a single qbit.” Be-
cause of complementarity, single qbits cannot be determined precisely. What is henceforth
called “determination” or “measurement” of a qbit is, in effect, the observation of a succes-
sive number of such qbits, one after the other, from “similar” computation processes (same
preparation, same evolution). By performing these measurements on “similar” qbits, one
can “determine” this qbit within an ε-neighborhood only. The parameter ε depends on the
number of successive measurements made;
(vi) quantum parallelism: during a computation (step), a quantum computer proceeds down all
coherent paths at once;
(vii) any subroutine must not leave around any qbits beyond it’s computed answer, because the
computational paths with different residual information can no longer interfere [13].
Inordertoappreciatequantumcomputation,oneshouldmakeproperuseofthelatterfeatures–
quantum parallelism, unerasability of information, non-copying, context-dependence and impos-
sibility to directly measure the atoms of quantum information, the qbits, related to quantum
indeterminism.
Stated pointedly: the quantum computation “solution” to a decision problem may yield the
classicalbit values at random. It may depend on other qbits of information which are inferred. It
cannot be arbitrarily copied and, in this sense, is unique.
4.2.1 Copying of quantum bits
Can a non-classical qbit be copied? No! — This answer amazes the classical mind.11 Informally
speaking, the reason is that any attempt to copy a coherent superposition of states results either
inastatereduction,destroyingcoherence,or,mostimportantofall,intheadditionofnoisewhich
manifests itself as the spontaneous excitations of previously nonexisting field modes [97, 35, 67,
71, 44, 26]. Therefore, qbits can be copied if and only if they are (known to be) classical. Only
one-to-one computation processes depicted in Fig. 1a) are allowed.
Thiscanbeseenbyashortcalculation[97]whichrequiresmulti-quantumformalismdeveloped
in appendix B. A physical realization12 of the qbit state is a two-mode boson field with the
identifications
x = αt+βf , (4)
α,β
f = 0 ,1 , (5)
1 2
| i
t = 1 ,0 . (6)
1 2
| i
11CopyingofqbitswouldallowcircumventionoftheHeisenberguncertaintyrelationbymeasuringtwoincompat-
ible observables on two identical qbit copies. It would also allow faster-than-light transmission of information, as
pointed out by Herbert [51]. Herbert’s suggestion stimulated the development of “no-cloning theorems” reviewed
here.
12themostelementaryrealizationisaone-modefieldwiththesymbol0correspondingto|0i(emptymode)and
1correspondingto|1i(one-quantum filledmode).
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The classical bit states are 0 ,1 (field mode 1 unfilled, field mode 2 filled with one quantum)
1 2
| i