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whole circuit requires O(L+m+n) gates if we allow the application of them to not
necessarily neighboring bits. Otherwise we must insert gates for local permutations
which will replace this estimate by O((L+m+n)2).
3.3. Fast Fourier transform. Finding theleast periodofafunction ofonereal
variablecanbedonebycalculatingitsFouriertransformsandlookingatitsmaxima.
The same strategy is applied by Shor in his solution of the factorization problem.
19
We will show now that the discrete Fourier transform Φ is computationally easy
n
(quantum polynomial time). We define Φ : H H by
n n n
→
N−1
1
Φ ( x ) = c exp(2πicx/N) (17)
n
| i √N | i
Xc=0
In fact, it is slightly easier to implement directly the operator
N−1
1
Φt( x ) = ct exp(2πicx/N). (18)
n | i √N | i
Xc=0
where ct is c read from the right to the left. The effects of the bit reversal can be
then compensated at a later stage without difficulty.
(kj)
Let U : H H , k < j, be the quantum gate which acts on the pair of the
2 n → n
k–th and j–th qubits in the following way: it multiplies 11 by exp(iπ/2j−k) and
| i
leaves the remaining classical states 00 , 01 , 10 intact.
| i | i | i
3.3.1. Lemma. We have
n−1 n−1
Φt = U(k) U(kj) . (19)
n 1 2
Y Y
k=0 j=k+1
 
By our rules of the game, (19) has polynomial length in the sense that it involves
only O(n2) gates. However, implementation of U(kj) requires controlling variable
2
phase factors which tend to 1 as k j grows. Moreover, arbitrary pairs of qubits
−
must allow quantum mechanical coupling so that for large n the interaction be-
tween qubits must be non–local. The contribution of these complications to the
notion of complexity cannot be estimated without going into the details of physical
arrangement. Therefore I will add a few words to this effect.
Theimplementationofquantumregistersuggested in[CZ]consistsofacollection
of ions (charged atoms) in a linear harmonic trap (optical cavity). Two of the elec-
tronic states of each ion are denoted 0 and 1 and represent a qubit. Laser pulses
| i | i
transmitted to the cavity through the optical fibers and controlled by the classical
computer are used to implement gates and read out. The Coulomb repulsion keeps
ions apart (spatial selectivity) which allows the preparation of each ion separately
in any superposition of 0 and 1 by timing the laser pulse properly and preparing
| i | i
its phase carefully. The same Coulomb repulsion allows for collective excitations
of the whole cluster whose quanta are called phonons. Such excitations are pro-
duced by laser pulses as well under appropriate resonance conditions. The resulting
resonance selectivity combined with the spatial selectivity implements a controlled
20
entanglement of the ions that can be used in order to simulate two and three bit
gates. For a detailed and lucid mathematical explanation, see [Ts], Lecture 8.
Another recent suggestion ([GeC]) is to use a single molecule as a quantum regis-
ter, representing qubits by nuclear spins of individual atoms, and using interactions
through chemical bonds in order to perform multiple bit logic. The classical tech-
nique of nuclear magnetic resonance developed since the 1940’s, which allows one
to work with many molecules simultaneously, provides the start up technology for
this project.
3.4. Quantum search. Allthesubroutines described up tonow boileddown to
some identities in the unitary groups involving products of not too many operators
acting on subspaces of small dimension. They did not involve output subroutines
and therefore did not “compute” anything in the traditional sense of the word. We
will now describe the beautiful quantum search algorithm due to L. Grover which
produces a new identity of this type, but also demonstrates the effect of observation
and the way one can use quantum entanglement in order to exploit the potential
of quantum parallelism.
We will treat only the simplest version. Let F : Fn 0,1 be a function
2 → { }
taking the value 1 at exactly one point x . We want to compute x . We assume
0 0
that F is computable in polynomial time, or else that its values are given by an
oracle. Classical search for x requires on the average about N/2 evaluations of F
0
where N = 2n.
In the quantum version, we will assume that we have a quantum Boolean circuit
(or quantum oracle) calculating the unitary operator H H
n n
→
I : x eπiF(x) x .
F
| i 7→ | i
In other words, I is the reflection inverting the sign of x and leaving the re-
F 0
| i
maining classical states intact.
Moreover, we put J = I , where δ : Fn 0,1 takes the value 1 only at 0,
− δ 2 → { }