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quantum

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A potentially infinite tape in the quantumcaseimplies a potentially infinite
dimensional Hilbert space. When a new tape square is added (or activated)
during the computation, the tape Hilbert space goes from nL-dimensional to
n(L+1)-dimensional.
Classicallythis situationis describedby consideringthe setofallstrings Σ
∗
overthe alphabet Σ. The set Σ thus contains all strings that could be written
∗
on the potentially infinite tape. In the quantum case, the state ψ in 6.5
L
| i
containsalllengthLstrings,providedallthe coefficientsarenonzero. Thus the
analogueofthe classicalsetΣ oughtto be ψ , which we willdenote as
∗ iL }∞L=0
{|
Σ , the set of all quantum strings. It is the set of all superpositions of states
∗Q
for all string lengths L.
L
|S i
Classically, a language is a subset of Σ , or equivalently, an element of the
∗
power set (Σ ). By analogy, we define a quantum language as a subset of
∗
P
the set of all quantum states, i.e. as an element of the power set (Σ ) =
∗Q
P
( ψ ).
iL }∞L=0
P {|
A leaner notation
The notation introduced so far, being a generalization of classical notions, is a
bit to heavy handed. Utilizing the economy of the Dirac notation, we write i
| i
instead of S , just keeping the indices. In the case of an alphabet with, say n
i
| i
symbols, i can be thought of as ranging over the numbers 1,2,...,n. Thus the
state of equation (6.3) is written i i i . A general state is then
1 2 L
| ··· i
ψ = α i i i . (6.7)
iL i1i2 ···iL| 1 2 L
| ··· i
X
In conclusion then, i spans a n-dimensional Hilbert space isomorphic to
| i
Cn. Likewise, i i i spans a nL-dimensional Hilbert space isomorphic to
1 2 L
| ··· i
(Cn) L. A general state in this space is given by 6.7.
⊗
Qubits
Aspecialcaseofthisconstructionisthequbitandthequbitstringcorresponding
tothealphabet 0,1 . Letxdenoteasinglebit. Aclassicaln-bitstringisthen
{ }
”x x ...x ”. A single qubit is denoted by x and a multi qubit state by
1 2 n
| i
x x ...x . These states are called computational basis states or the classical
1 2 n
| i
basis [26]. Measurements on the quantum computer are often thought of as
beingmade inthis basisanditcanthereforebe consideredasthe connectionto
classical I/O-streams.
Aconvenientnotationisoftenusedforthecomputationalbasisstates. Using
the binary number representationfor naturalnumbers,we candenote the state
x x ...x = x 2n 1+x 2n 2+...x 2+x . (6.8)
1 2 n 1 − 2 − n 1 n n
| i | · · − · i
123
The subscript n is needed to indicate the number of qubits, but is often
suppressed. An example of this notation is 1001 = 9 .
4
| i | i
A general state of the quantum computer is a complex linear superposition
of the basis states
2n 1
−
α i , (6.9)
i n
| i
Xi=0
and the coefficients are again subject to the restriction
2n 1
−
α 2 =1. (6.10)
i
| |
Xi=0
The Hilbert space of one qubit is isomorphic to the complex vector space
C2, and the n-qubit Hilbert space is isomorphic to (C2) n. Thus, an n-bit
⊗
quantum state carries exponentially more information than an n-bit classical
string. Classically, it is not possible to linearly combine different bit strings.1
We alsorecordthe sometimes convenientexplicitrepresentationofthe basis
states in terms of 2n-dimensional basis vectors, for example
0
0