Datasets:
wesley7137
/
quantum

Dataset card Data Studio Files
xet
 Community
text
stringlengths
0
8.13M
fromtrueϕ,estimatorslikesampleaverageofϕ˜ ,...,
1
1
ϕ˜ may not estimate ϕ well. We adopt a robuststa-
b 1 log ζ+ , n
∼ − 2 4ǫlog2 tisticalmethodtoestimateϕbyα-trimmedmeanϕ¯,
which grows at a very fast rate. For example, an which is defined as follows. Ordering ϕ˜ 1,...,ϕ˜ n and
increase in success probability from 90% to 99% re- then removing [nα] largest and [nα] smallest ones,
quires eighteen times of qubit increase compared to wetaketheaverageoftheremainingϕ˜ j asα-trimmed
the change from 80% to 90%. mean, where α is chosen to be greater than ǫ/2.
Quantum algorithms are of random nature in the One example is the sample median of ϕ˜ 1,...,ϕ˜ n.
sense that they often produce correct answers only The probability that ϕ¯ is within ζ from ϕ can be
with certain probabilities. The success probabilities calculated from the binomial probability as follows:
depend upon the schemes of the algorithms as well
P(ϕ¯ ϕ ζ) P(more than n(1 2α) number
as the context of applications. Given a quantum al- | − |≤ ≥ −
gorithm forsolving aproblem,acommon practice is of ϕ˜ j are within ζ from ϕ)
to repeatedly runthe quantum algorithm to achieve n
n
high probability of successfully obtaining a correct = (1 ǫ)kǫn k.
−
(cid:18)k(cid:19) −
answer. Consider that the phase estimation proce- X
k=[n(1 2α)] 1
− −
dure is repeatedly run n times to obtain results
As n , nǫ approaches to infinity. The binomial
ϕ˜ ,...,ϕ˜ . Then ϕ˜ ,...,ϕ˜ may be treated as i.i.d. →∞
1 n 1 n probabilitycanbeapproximatedbyresortingtoanor-
random variables with each ϕ˜ satisfying
j mal approximation, yielding
P(ϕ˜ j ϕ >ζ) ǫ. n
| − | ≤ n
(1 ǫ)kǫn k
We may statistically model ϕ˜ by the gross error −
j (cid:18)k(cid:19) −
X
model (Huber and Ronchetti (2009)) as follows. As- k=[n(1 2α)] 1
− −
sumethat ϕ˜ areindependentlyand identically gen-
j √n(ǫ 2α) √n(2α ǫ)
erated from (1 ǫ)F(x)+ǫH(x), where F(x) is the 1 Φ − =Φ − ,
− ∼ − (cid:18) ǫ(1 ǫ) (cid:19) (cid:18) ǫ(1 ǫ) (cid:19)
distribution of the correct answers that are within ζ − −
p p
from true ϕ, and H(x) is the distribution of wrong whereΦ()isthestandardnormaldistributionfunc-
·
answersthatareatleastζ awayfromtrueϕ.Thenϕ˜ tion. Since 2α ǫ>0, as n increases, P(ϕ¯ ϕ ζ)
j
− | − |≤
14
Y.WANG
approaches to 1 exponentially fast. Combining the operationsofordern2lognloglogn,thecurrentbest
two cases together, we arrive at the following theo- known classical algorithm requires operations of or-
rem. der exp(n1/3log2/3n) to factor an n-bit composite
number(CrandallandPomerance(2001)).Notethat
Theorem 1. Suppose that the outcome ϕ˜ of
thenumberof operations requiredinthebestclassi-
a quantum algorithm obeys the gross error model
cal algorithm grows exponentially in the size of the
that with probability 1 ǫ it produces a correct an-
− number being factored. Because of the exponential
swer and probability ǫ it gives a wrong answer. Then
complexity, the factoring problem is generally re-
by repeatedly running the quantum algorithm we will
garded as an intractable problem on classical com-
obtain a correct answer with probability approaching
puters.
1 exponentially fast in the number of repetitive runs.
The factoring problem plays an important role in
For a quantum algorithm that produces a correct cryptography. Cryptography is to enable two par-
answer with probability 70% and α=0.2, in order ties,Alice andBob, tocommunicate privately, while
toobtain acorrect answerwith 0.999 probability we it is very difficult for the third parties to “eaves-
need to run the algorithm five times and 20 times, drop” on the contents of the communications. Ex-
respectively, for the cases that the outcome results amplesincludeATMcards,computerpasswords,in-
are verifiable and not verifiable. ternet commences, clandestine meetings and mili-
tary communications. Two cryptographic protocols
4.4 Factoring and Order-Finding Algorithms
used in the communications are private key cryp-
The factoring problem is to find all prime fac- tosystem and public key cryptosystem. A private
tors of a given positive composite number such that key cryptosystem requires the two communicating
the product of these prime numbers is equal to the parties to share a private key. Alice uses the key
composite number. Factoring is known to be a very to encrypt the information, sends the encrypted in-
hard problem for classical computers. Shor (1994, formation to Bob who uses the key to decrypt the
1997) developed a quantum algorithm for the fac- received information. The severe drawback of the
toring problem that is exponentially faster than the private key cryptosystem is that the parties have
most efficient known classical factoring algorithm. to safeguard the key transmission from being eaves-
Shor’squantumalgorithmsworkasfollows.Math- dropped.A public key cryptosystem invented in the
ematically the factoring problem is equivalent to 1970s requiresnosharingsecretkey inadvance.Bob
the order-findingproblem that for two positive inte- publishes a “public key” available to the general
gers x and N, x<N, with no common factors, find public, and Alice uses the public key to encrypt in-
the smallest integer r such that dividing xr by N formation and sends the encrypted information to
we obtain a reminder 1 (Shor (1997); Nielsen and Bob.Theencryptiontransformationisspeciallycre-
Chuang (2000)). The quantum algorithm for factor- ated such that with only the public key, it is ex-
ing is reduced to a quantum algorithm for order- tremely difficult, though not impossible, to invert
finding. The quantum algorithm for order-finding is theencryptiontransformation.Whenpublishingthe
to apply the phase estimation algorithm to the uni- public key Bob keeps a matched secret key for easy
tary operator inversion of the encryption transformation and de-
cryption of the received information. One of the
Uy = xy(modN) .