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computation: The circuit
model
In this chapter, the theory of quantum computation will be outlined in an ab-
stractway,without recourseto physicalconsiderations. This is in analogywith
how classical computation theory is generally developed, relying on models of
computation which abstracts away from the details of actual physical comput-
ing machinery. We will first look at the quantum Turing machine model, but
will then turn to the quantum circuit model.
The quantum Turing Machine (QTM) is a quantum generalization of the
classical Turing machine, and as such, can be considered to be a model of
a programmable quantum computer. It is not a practical model for either
algorithmconstruction or actual physical implementation. The QTM model is,
however, useful in discussing complexity theory, and has been used extensively
in that context [40].
The quantum circuit model (QCM) is easier to work with as regards algo-
rithm construction, and is closer to a physical realization.
Themodelsareclaimedtobeequivalent,butthereseemstobeafewunclear
points, mainly having to do with subtleties as regards the QTM’s. This is an
area of active research. For that reason, QTM’s will not be treated here.
The original references are [7] and [25].
6.1 Quantum alphabets, strings and languages
As astarting point,let us seehow fara quantumgeneralizationofthe concepts
ofalphabetsandlanguageswillcarryus. Analphabetisafinite,non-emptyset
of symbols. Here we will use such an alphabet to label the quantum states of a
system. This is an abstract labeling and the realization in terms of a concrete
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physicalsystemwill,assaid,notconcernushere. Supposewehaveanalphabet
Σ= S ,S ,...,S , then the corresponding set of quantum states are
1 2 n
{ }
Σ = S , S ,..., S . (6.1)
Q 1 2 n
{| i | i | i}
These states are taken to span an orthonormal basis, i.e.
S S =δ . (6.2)
i j ij
h | i
It seems reasonable to introduce the term quantum alphabet for such sets of
states, though the term is not in common use.
Next, the classicalconcept of a string of length L, ”S S S ”, is gener-
i1 i2··· iL
alized to the corresponding composite quantum state
S S S = S S S . (6.3)
i1i| i2i···| iLi i1 i2··· iLi
| |
This notation is a shorthand for the direct product notation
S S S .
i1i⊗| i2i⊗···⊗| iLi
|
The orthonormality condition (6.2) generalizes to
=δ , (6.4)
hS|Ti ST
where the notation is used for the string ”S S S ”. If the string length
i1 i2··· iL
S
needs to be recorded we could write .
L
S
It is clear that the symbols S only serve as labels, each of them ranging
i
over the set Σ. Just as in the classical case, the actual symbols used play no
role. The main difference as compared to the classical case is the possibility to
consider linear superpositions of the states . Thus the states of Σ span a
Q
|Si
nL-dimensional Hilbert space isomorphic to the complex vector space (Cn) L.
⊗
Imposingalexicographicorderinglex: N,wehavethegeneralstate
L
{S }→
ψ
L
| i
ψ = α , (6.5)
iL lex( SL) |SL
| i
X
lex( SL)
which could be denoted a quantum string of length L. Here play the role
L
|S i
of basis states.
Normalization of the states ψ , i.e. the demand that ψ ψ = 1 leads to
L
| i h | i
the usual restriction on the coefficients
α 2 =1. (6.6)
lex( SL)
| |
X
lex( SL)
Thestatesconsideredsofarhaveafixedstringlengthandarethereforefinite
in number. This is the normal situation when describing discrete quantum
systems. In order to support Turing-like quantum computation, we have to
consider arbitrary length strings. This is because fixed string length implies a
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finite size memory. Machines with a finite size memory are not really Turing
machines, they are finite state machines and as such do not support universal
computation. So, just as in the classical case, the tape must be potentially
infinite.