Datasets:
wesley7137
/
quantum

Dataset card Data Studio Files
xet
 Community
text
stringlengths
0
8.13M
position and the inner state of the head/processor. Some of the arrangements are
valid protocols, others are not, but the local nature of the Turing computation
allows one to produce a Boolean polynomial b in appropriate variables such that
u
the validprotocolsarerecognized by the fact that thispolynomialtakes value1. For
detailed explanations see e.g. [GaJ], sec. 2.6. This defines the function f reducing
D to E. The construction is so direct that the polynomial time computability of f
is straightforward.
Many natural problems are known to be NP–complete, in particular 3–SAT. It
is defined as the subset of SAT consisting of those u for which card(S T ) = 3
i i
∪
for all i.
1.6.3. Remark. Most of Boolean functions are not computable in polynomial
time. Several versions of this statement can be proved by simple counting.
First of all, fix a finite basis of Boolean operations as in 1.4.1, each acting
B
upon a bits. Then sequences of these operations of length t generate O((bna)t)
≤
Boolean functions Fn Fn where b = card . On the other hand, the number of
n 2 → 2 B
all functions 2n2 grows as a double exponential of n and for large n cannot be
obtained in time t polynomially bounded in n.
Thesameconclusionholdsifweconsider notallfunctions but onlypermutations:
Stirling’s formula for cardS 2n = 2n! involves a double exponential.
Here is one more variation of this problem: define the time complexity of a
conjugacy class in S 2n as the minimal number of steps needed to calculate some
permutation in this class. This notion arises if we are interested in calculating
automorphisms of a finite universe of cardinality 2n, which is not supplied with a
specific encoding by binary words. Then it can happen that a judicious choice of
encoding will drastically simplify the calculation of a given function. However, for
most functions we still will not be able to achieve polynomial type computability,
because the asymptotical formula for the number of conjugacy classes (partitions)
exp(π 2(2n 1 )
p(2n) q3 − 24
∼ 4√3(2n 1 )
− 24
again displays the double exponential growth.
2. Quantum parallelism
In this section we will discuss the basics: how to use the superposition principle
in order to accelerate (certain) classical computations.
14
2.1. Description of the problem. Let N be a large number, F : 0,...,N
{ −
1 0,...,N 1 a function such that the computation of each particular value
} → { − }
F(x) is tractable, that is, can be done in time polynomial in logx. We want to
compute (to recognize) some property of the graph (x,F(x)), for example:
(i) Find the least period r of F, i.e. the least residue rmodN such that F(x+
rmodN) = F(x) for all x (the key step in the Factorization Problem.)
(ii) Find some x such that F(x) = 1 or establish that such x does not exist
(Search Problem.)
As we already mentioned, the direct attack on such a problem consists in com-
piling the complete list of pairs (x,F(x)) and then applying to it an algorithm
recognizing the property in question. Such a strategy requires at least exponential
time (as a function of the bit size of N) since already the length of the list is N.
Barring a theoretical breakthrough in understanding such problems, (for example
a proof that P = NP), a practical response might be in exploiting the possibility
of parallel computing, i.e. calculating simultaneously many – or even all – values
of F(x). This takes less time but uses (dis)proportionally more hardware.
A remarkable suggestion due to D. Deutsch (see [DeuJ], [Deu]) consists in using
a quantum superposition of the classical states x as the replacement of the union
| i
of N classical registers, each in one of the initial states x . To be more precise,
| i
here is a mathematical model formulated as the definition.
2.2. Quantum parallel processing: version I. Keeping the notation above,
assume moreover that N = 2n and that F is a bijective map (the set of all outputs
is a permutation of the set of all inputs).
(i) The quantum space of inputs/outputs is the 2n–dimensional complex Hilbert
space H with the orthonormal basis x , 0 x N 1. Vectors x are called
n
| i ≤ ≤ − | i
classical states.
(ii) The quantum version of F is the unique unitary operator U : H H
F n n
→
such that U x = F(x) .
F
| i | i
Quantum parallel computing of F is (a physical realization of) a system with the
state space H and the evolution operator U .
n F
Naively speaking, if we apply U to the initial state which is a superposition
F
of all classical states with, say, equal amplitudes, we will get simultaneously all
classical values of F (i.e. their superposition):
1 1
U x = F(x) . (14)
F
(cid:18)√N | i(cid:19) √N | i
X X
We will now discuss various issues related to this definition, before passing to its
more realistic modification.
15
(A) We put N = 2n above because we are imagining the respective classical
system as an n–bit register: cf. the discussion of Boolean circuits. Every number
0 x N 1 iswritteninthebinarynotationx = ǫ 2i andisidentifiedwiththe
≤ ≤ − i i
pure (classical) state ǫ n−1,...,ǫ where ǫ = 0 or 1Pis the state of the i–th register.