Datasets:
wesley7137
/
quantum

Dataset card Data Studio Files
xet
 Community
text
stringlengths
0
8.13M
P
diverge.
Nevertheless, qbits are allowed as output. Since they are objects defined in Hilbert space H,
the basic definitions of algorithmic information theory have to be slightly adapted.
The canonical program associated with an object s H representable as vector in a Hilbert
∈
space H is denoted by s and defined by
∗
s = min p . (20)
∗
C(p)=s
I.e., s is the first element in the ordered set of all strings that is a program for C to calculate
∗
s. The string s is thus the code of the smallest-size program which, implemented on a quantum
∗
computer, outputs s. (If several binary programs of equal length exist, the one is chosen which
comes first in an enumeration using the usual lexicographic order relation “0<1.”)
Let again“x” ofanobject encodedas (binary)string standfor the length ofthat string. The
| |
quantum algorithmic information H(s) of an object s H representable as vector in a Hilbert
∈
space H is defined as the length of the shortest program p which runs on a quantum computer C
and generates the output s:
H(s)= s = min p . (21)
∗
| | C(p)=s| |
If no program makes computer C output s, then H(s)= .
∞
Thejoint quantumalgorithmic informationH(s,t)oftwoobjectss Handt Hrepresentable
∈ ∈
as vectorsina Hilbert spaceH is the length ofthe smallest-sizebinary programto calculate s and
t simultaneously.
The relative or conditional quantum algorithmic information H(st) ofs H givent N is the
| ∈ ∈
length of the smallest-size binary program to calculate s from a smallest-size program for t:
H(st)= min p . (22)
| C(p,t )=s| |
∗
13
Most features and results of algorithmic information theory hold for quantum algorithmic in-
formation as well. In particular, we restrict our attention to universal quantum computers whose
quantum algorithmic information content is machine-independent, such that the quantum algo-
rithmic informationcontentofanarbitraryobjectdoes not exceeda constantindependent of that
object. That is, for all objects s H and two computers C and C of this class,
′
∈
H H =O(1) . (23)
C C
| − ′|
Furthermore, let s and t be two objects representable as vectors in Hilbert space. Then (recall
that t N),
∈
H(s,t) = H(t,s)+O(1) ; (24)
H(ss) = O(1) ; (25)
|
H(H(s)s) = O(1) ; (26)
|
H(s) H(s,t)+O(1) ; (27)
≤
H(st) H(s)+O(1) ; (28)
| ≤
H(s,t) = H(s)+H(ts )+O(1) (if s is classical) ; (29)
∗ ∗
|
H(s,t) H(s)+H(t)+O(1) (subadditivity) ; (30)
≤
H(s,s) = H(s)+O(1) ; (31)
H(s,H(s)) = H(s)+O(1) . (32)
Notice that there exist sets of objects S = s ,...,s , n< whose algorithmic information
1 n
{ } ∞
contentH(S)isarbitrarysmallcomparedtothealgorithmicinformationcontentofsomeunpecified
single elements s S; i.e.,
i
∈
H(S)<maxH(s ) . (33)
i
si∈S
6 Quantum omega
Chaitin’s Ω [28, 85, 24] is a magic number. It is a measure for arbitrary programs to take a finite
numberofexecutionstepsandthenhalt. Itcontainsthesolutionofallhaltingproblems,andhence
of questions codable into halting problems, such as Fermat’s theorem. It contains the solution of
the question of whether or not a particular exponential Diophantine equation has infinitely many
or a finite number of solutions. And, since Ω is provable “algorithmically incompressible,” it is
Martin-Lo¨f/Chaitin/Solovay random. Therefore, Ω is both: a mathematicians “fair coin,” and a
formalist’s nightmare.
Here, Ω is generalized to quantum computations.14
In the orthonormal halting basis t,f , the computer C with classical input p can be repre-
i
{ }
sented by C(τ,p )=t(t,C(τ,p ))+f(f,C(τ,p )).
i i i
Recall that initially, i.e., at time τ = 0, the halting bit is in a coherent 50:50-superposition;
i.e., in terms of the halting basis, C(0,p ) = (t +f)/√2 for all p A . This corresponds to
i i ∗
∈
the fact that initially it is unknown whether or not the computer halts on p . When during the
i
time evolution the computer has completed its task, the halting bit value is switched to t by some
internal operation. If the computer never halts, the halting bit value is switched to f by some