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8.13M
| i
N−1
1
a,0 .
√N | i
Xa=0
(ii) Reversible calculation of F processes this state into
N−1
1
a,tamodM .
√N | i
Xa=0
(iii) Partial Fourier transform then furnishes
N−1N−1
1
exp(2πiac/N) c,tamodM .
N | i
Xa=0 Xc=0
(iv) The last step is the observation of this state with respect to the system of
classical states c,mmodM . This step produces some concrete output
| i
c,tkmodM (20)
| i
23
with probability
2
1
exp(2πiac/N) . (21)
(cid:12) (cid:12)
N
(cid:12) (cid:12) a:ta≡X tkmodM (cid:12) (cid:12)
(cid:12) (cid:12)
The remaining part of th(cid:12)e run is assigned to the classica(cid:12)l computer and consists of
the following steps.
c
(A) Find the best approximation (in lowest terms) to with denominator r′ <
N
M < √N:
c d′ 1
< . (22)
(cid:12)N − r′(cid:12) 2N
(cid:12) (cid:12)
(cid:12) (cid:12)
As we will see below, we may(cid:12) hope th(cid:12) at r′ will coincide with r in at least one
run among at most polynomially many. Hence we try r′ in the role of r right away:
(B) If r′ 0mod2, calculate gcd(tr′ /2 1,M).
≡ ±
If r′ is odd, or if r′ is even, but we did not get a proper divisor of M, repeat the
run O(loglogM) times with the same t. In case of failure, change t and start a new
run.
4.3.1. Justification. We will now show that, given t, from the observed val-
ues of c,tkmodM in O(loglogM) runs we can find the correct value of r with
| i
probability close to 1.
Let us call the observed value of c good, if
r r
l , , rc lmodN.
∃ ∈ −2 2 ≡
h i
In this case there exists such d that
r r
rc dN = l
−2 ≤ − ≤ 2
so that
c d 1
< .
(cid:12)N − r(cid:12) 2N
(cid:12) (cid:12)
(cid:12) (cid:12)
Hence if c is good, then r′ found(cid:12)from (22(cid:12)) in fact divides r.
Now call c very good if r′ = r.
Estimating the exponential sum (21), we can easily check that the probability of
1
observing a goodcis .On theotherhand, therearerϕ(r)states c,tkmodM
≥ 3r2 | i
with very good c. Thus to find a very good c with high probability, O(r2logr) runs
will suffice.
24
5. Kolmogorov complexity and growth of recursive functions
Consider general functions f : N N. Computability theory uses several
→
growth scales for such functions, of which two are most useful: f may be majorized
by some recursive function (e.g. when it is itself recursive), or by a polynomial
(e.g. when it is computable in polynomial time). Linear growth does not seem
particularly relevant in this context. However, this impression is quite misleading,
at least if one allows re–ordering N. In fact, we have:
5.1. Claim. There exists a permutation K : N N such that for any partially
→
recursive function f : N N there exists a constant c with the property
→
K f K−1(n) cn for all n K(D(f)). (23)
◦ ◦ ≤ ∈
Moreover, K is bounded by a linear function, but K−1 is not bounded by any recur-
sive function.
Proof. We will use the Kolmogorov complexity measure. For a recursive func-
tion u : N N, x N, put C (x) := min k f(k) = x , or if such k does not
u
→ ∈ { | } ∞
exist. Call such a function u optimal if, for any other recursive function v, there
exists a constant c such that C (x) c C (x) for all x. Optimal functions do