Datasets:
wesley7137
/
quantum

Dataset card Data Studio Files
xet
 Community
text
stringlengths
0
8.13M
| i
namely P0 = ϕ(U0+U1) †(U0+U1)ϕ /4. With proba- a classical computer for further processing. Using this
h | | i
bility1 P0,theconditionalmeasurementwillnotobtain dualitymode,onemayperformbothtasksinaquantum
−
aresult,andifthisoccurs,thestateofthewavefunction computer.
is collapsed in a normalized state N(U0 U1)ϕ 1 . In Secondly, the duality mode has provided a new av-
− | i| i
thissimulation,theconditionalmeasurementsimulatesa enue for algorithm design. Let’s study the unsorted
measurement on a part of a wave function. The result is database search problem [7]. Quantum algorithms find
clear from the basic postulate in quantum mechanics. a marked item from an unsorted database with O(√N)
The scheme can be generalized by replacing the two steps[2,8,9,10]. Howeverthesequantumalgorithmsare
Walsch-Hadamardgateswithotherunitarygatestosim- not fixed point search algorithm, the success probability
ulate more complicated slits, such as asymmetric slits, isaperiodicfunctionofthenumberofsearchingsteps. A
3
recent fixed point quantum search algorithm uses O(3n) process until a result is obtained in the conditional mea-
number of queries in a quantum computer [11]. Though surement. This algorithm works also for an arbitrary
the number of queries is more than O(2n) in a classical database where the coefficient of each item is arbitrary.
computer, it is still advantageous because it is an algo- Using the recycling mode, the repetition process is per-
rithm in a quantum computer using only n-qubit. Here formed automatically.
we give a duality mode algorithm that uses only O(2n)
steps while being still a quantum computer algorithm.
This is a modified duality algorithm in Ref. [3]. One
1000
starts from the evenly distributed state, andswitch onto
the duality mode to give 800
N 1
1 − 600
i (0 + 1 ). (12)
√2N | i | i | i
Xi=0
400
Performingontheupperslitthefollowinggateoperation
200
1 1
(0 +...+ τ +...) ( 0 ...+ τ ...),
√2N | i | i → √2N −| i− | i− 5 10 15 20
(13)
and leaves the lower-slit sub-wave unchanged. Then re-
combine them through a Hadamard gate, we obtain FIG. 2: The repetition numberversusj for N =210.
1 1
τ 0 + i 1 . (14)
√N| i| i √N | i| i
Xi=τ Thirdly, this formulationof duality computer has pro-
6
vided a way for error correction in duality computer.
Then conditionally measure on the auxiliary state being Quantum error correction has been solved successfully
in 0 , one obtains τ with probability 1/N. Conditional in quantum computer [12], this analogy easily gives the
| i
measurement is simply achievedby measuring the auxil- schemeoferrorcorrectioninadualitycomputer. Allthe
iaryqubit. Ifitis0,thenmeasuretheworkingnqubitto goodquantumerrorcodes,canbetranslatedintoduality
readoutτ. Ifitis1,repeattheprocessagain. Repeating computer.
thisprocessO(N =2n)numberoftimes,onewillgetthe Recycling quantum computing— In Eq.(14), the
desired result. probabilityofobtainingaresultinaconditionalmeasure-
The algorithm can be further speeded up by first per-
mentissmall. Howeveritis differentfromanevenly dis-
forming the quantum amplitude amplification a number tributedstate 1 i . Whenonemeasurestheevenly
of times before switching to duality mode. After j itera- √N i| i
distributed state, Pone always obtains a result, say x ,
tion, the wave function becomes | i
howeverthe resultto be τ hasonlyaprobabilityof1/N,
afterwards the state collapses to the eigenstate of that
ψ =sin((2j+1)β)τ +cos((2j+1)β)c , (15)
j
| i | i | i measured eigenvalue. However, in Eq. (14), after the
where measurement, one has two possibilities : 1) The state
collapses into state τ with probability 1/N; 2) No re-
| i
1 sult is obtained, however the state has collapsed out
c = i . (16)
| i rN 1 | i from (14), and the state becomes
− Xi=τ
6
1
Thenswitchingtothedualitymode,itgivesforthewave i 1 .
function in the upper slit √N −1 Xi=τ | i| i
6
ψ u =sin((2j+1)β)τ . (17) This happens with probability (N 1)/N. This state
| i | i −
can be reused again as input after flipping the auxiliary
Conditionally measure it one gets τ with probability
qubit and apply a recovering unitary operation to the
sin2[(2j+1)β]. RepeatingthisO(1/sin((2j+1)β)times,
original input state. Then the calculating process is re-
the marked state will be found. When j is small, the
cycled again and again until a final result is obtained.
number of repetitions is about N/(2j+1). When j ap-
Using this observation, we propose a recycling quantum
proaches π√N/4, it finds the marked state with only
computing mode as shown in Fig. 3.
single query. A schematic plot for N = 210 is given in
Namely, before the conditional measurement, suppose
Fig. 2. When one knows the number of marked state
the wave-function of the duality computer is
in an unsorted database, one can optimize the number
of repetition. If one does not know this information, one U0+U1 U0 U1
can simply switch to the duality mode, and repeat the ϕ 0 + − ϕ 1 .
2 | i| i 2 | i| i
4
input state is restored. Meanwhile flipping 1 to 0 in