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quantum

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nomical as it requires exponential sized vectors and matrices, most of which
entries are zero anyway. It corresponds to a unary notation for numbers.
Note,however,theveryclosecorrespondencewiththecircuitmodel. Infact,
this model is a realization of the circuit model, hence the name computational
(or better classical ) basis for the vectors (3.9 ).
Intheclassicalcase,wedon’treallyneedthisexpansionoutofthebitstrings
into exponential sized vectors. There are much more efficient ways to process
bit strings.
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Transition to quantum computations
The possibility to have linear combinations, called superpositions, of the basis
vectors,marksthe point of departure into quantum computations.3 A complex
vectorspaceis builtonthe basisvectors(whichby the way,formsanormalized
andorthogonalset). As anexample, consider the case of 3-qubitstates. Linear
combinations can now be written nicely,
7
ψ = α i =α 0 +α 1 +...+α 7 =
i 3 0 3 1 3 7 3
| i | i | i | i | i
Xi=0
α 000 +α 001 +...+α 111 .
0 1 7
| i | i | i
The general case is
2n 1
−
ψ = α i . (3.12)
i n
| i | i
Xi=0
The coefficients are normalized
2n 1
−
α 2 =1. (3.13)
i
| |
Xi=0
Havingintroducedthecomplexnumbersαintothetheory,thereisnoreason
toworkwiththeveryrestrictivesetofmatricesusedinclassicalcomputation. A
priori,anymatricC withcomplexentriescouldbecontemplatedasacandidate
for a computation. There are however restrictions even in the quantum case
that we will come to. But first, let us consider a few examples.
One-bit quantum computations
Consider first a one-qubit space. Define two new matrices Y and Z by
0 i
Y = − (3.14)
(cid:18)i 0 (cid:19)
1 0
Z = . (3.15)
(cid:18)0 1(cid:19)
−
Acting with Z on the 1-qubit basis vectors yields
1 1
Z 0 =Z = = 0 ,
| i (cid:18)0(cid:19) (cid:18)0(cid:19) | i
0 0
Z 1 =Z = = 1 .
| i (cid:18)1(cid:19) −(cid:18)1(cid:19) −| i
3Atleastinthisapproachtothetheory. Thereareotherwaystolookatit.
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Such a computationhasno meaningclassically. But quantummechanically,
the state (call it φ ) resulting from acting with Z on the state ψ of (3.1)
| i | i
Z ψ =Z α0 +β 1 =α0 β 1 = φ ,
| i | i | i | i− | i | i
(cid:0) (cid:1)
is, so to speak, no worse than ψ itself.
| i
Ifonehidestheintermediatesteps,thissimplecalculationZ ψ = φ ,shows
| i | i
that Z transforms the state ψ in one computational step.
| i
Letusalsointroduceonefurthermatrix,ofoutmostimportanceinquantum
computation, This is the so called Hadamard matrix
1 1 1
H = . (3.16)
√2(cid:18)1 1(cid:19)
−
This matrix is used build the linear superpositions of (3.3) and (3.4) out of
the basis states, or as can be checked by a simple calculation
H 0 = + (3.17)
| i | i
H 1 = . (3.18)
| i |−i
Multi qubit computations
Generalizington-qubitstatesand2n 2n dimensionalcomputationalmatrices,
×
we have C ψ = φ . Thus an exponential number of classical computational
| i | i
steps are performed in parallel.
Ifthisistobesimulatedonaclassicalcomputer,thenofcourse,anexponen-
tial number of operations have to be performed anyway, and nothing is gained
as compared to performing the classical computations
The situation is drastically changed if quantum devices can be built that
actually performs the operation C on the state ψ .
| i
Restriction on quantum computation matrices
A general quantum computation can now be written as
out =C in . (3.19)
| i | i