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quantum

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| i | i most widely used public key cryptosystems is the
The eigenvectors of U are RSA cryptosystem, which is named after its cre-
r 1 ator Rivest, Shamir and Adleman (Menezes, van
1 − 2πisk
u = exp − xkmodN , Oorschot and Vanstone (1996); Rivest, Shamir and
s
| i √r (cid:18) r (cid:19)| i
Xk=0 Adleman (1978)). RSAis builton the mathematical
asymmetry of factoring: it is easy to multiply large
s=0,1,...,r 1,i=√ 1,
− − prime numbers and obtain their product as a com-
with corresponding eigenvalues exp(2πis/r). Using posite number but hard to find the prime factors
the phase estimation algorithm we can obtain the of a given large composite number. RSA encryption
eigenvaluesexp(2πis/r)withhighaccuracyandthus keeps the large primes as a secret key and uses their
find the order r with certain probability. producttomakea“publickey.”Becauseofitsexpo-
While the quantum factoring algorithm can ac- nentialcomplexity,tremendouseffortstriedtobreak
complish the task of factoring an n-bit integer with the RSA system so far have resulted in vain, and
15
QUANTUMCOMPUTATIONAND QUANTUMINFORMATION
there is a widespread belief that the RSA system is and then applying a so-called Grover iteration (or
secureagainst any classical computer based attacks. operator) repeatedly. Set
As the factoring problem can be efficiently solved
1 1
by Shor’s quantum factoring algorithm, a quantum φ = x ′ , ϕ = x ′′ ,
| i √N M | i | i √M | i
computer can break the RSA system easily. Fortu- − Xx′ Xx′′
nately, while quantum mechanics takes away with
where the summations over x and x denote sums
′ ′′
one hand, it gives back with the other. A quan-
over all non-solutions and solutions, respectively.
tum procedure known as quantum cryptography or
Then we can express ψ as follows:
quantum key distributioncan dokey distribution so | i
that the communication security cannot becompro- N M M
ψ = − φ + ϕ .
mised. The idea is based on the quantum principle | i r N | i rN | i
that observing a quantum system will disturb the
The Grover operator is to perform two reflections,
system being observed. If there is an eavesdropper
one aboutthevector φ and another aboutthe vec-
duringthetransmissionofthequantumkeybetween | i
tor ψ . The two reflections together are a rotation
Alice and Bob, eavesdropping willdisturbthequan- | i
with angle θ in the two-dimensional space spanned
tum communication channel that is used to estab-
by φ and ϕ , where
lish the key, and the disturbance will make eaves- | i | i
droppingvisible. Alice and Bob will throw away the N M
compromised key and keep only the secured key for cos(θ/2)= r − N .
their communication.
Aftertherotation,theinitialstate ψ =cos(θ/2)φ +
4.5 Quantum Search Algorithm sin(θ/2)ϕ becomes state | i | i
| i
Suppose that you would like to find the name cos(3θ/2)φ +sin(3θ/2)ϕ .
corresponding to a given phone number in a tele- | i | i
phone directory; or suppose that there are some lo- ThuseachapplicationoftheGroveroperatorisaro-
cations in a given city you would like to visit and tation with angle θ. The initial state ψ has angle
| i
wish to find the shortest route passing through all π/2 θ/2 with ϕ ; after the first rotation, the re-
− | i
the locations. If there are N names in the telephone sulted state has angle π/2 3θ/2 with ϕ ; and in
− | i
directory or N possible routes to pass through all general after the rth rotation, the resulted state has
the locations, search algorithms by classical com- angle π/2 (2r+1)θ/2 with ϕ . Repeatedly apply-
− | i
puters usually require operations of order N. One ing the Grover operator, we rotate the state vector
such simple classical algorithm is to check exhaus- near ϕ . With the initial state ψ =cos(θ/2)φ +
| i | i | i
tively all names to find a name matching with the sin(θ/2)ϕ ,weneedtorotatethrougharccos M/N
| i
given phone number or to search all possible routes radians to transform the state vector to ϕp. After
| i
and then find the shortest route among all routes. R=arccos( M/N)/θ=O( N/M) times of appli-
However, Grover (1996, 1997) developed a quantum cations of thpe Grover operatpor, we rotate the state
searchalgorithm thatneedsonlyoperationsoforder vector ψ to within an angle θ/2 of ϕ . Performing
| i | i
√N to find a solution to the search problem. measurements of the state yields a solution to the
The quantum search algorithm works as follows. searchproblemwithprobabilityatleast cos2(θ/2)
≥
Suppose that the search space has N elements and 1 M/N.
−
thesearchproblemhasexactlyM solutions.Assume The number of iterations R depends on M, the
M N/2. (For the silly case of M >N/2, we ei- number of solutions. Since R π/(2θ) and θ/2
≤ ≤ ≥
ther search for the solution by doing random selec- sin(θ/2) = M/N, R (π/4) N/M. Typically,
≤
tion from the search space or double the number of M N,θ psinθ 2 M/N,thpusR (π/4) N/M.
≪ ≈ ≈ ≈ ·
the elements in the search space by adding N extra We estimate the numpber of solutions by quapntum
non-solution elements to the search space.) The al- counting, which is to combine the Grover operator
gorithm works by creating superposition state with with the phase estimation method. Under the basis
Hadamard gate, φ and ϕ the Grover operator has eigenvalues eiθ
| i | i
and ei(2π θ).Applyingthephaseestimation method
N 1 −
1 − we can estimate the eigenvalues and thus θ with
ψ = x ,
| i N1/2 | i prescribed precision and probability, which in turn
Xx=0