Optical waveguides – Having nonlinear property
Reexamination Certificate
1999-04-26
2004-01-13
Cherry, Euncha (Department: 2872)
Optical waveguides
Having nonlinear property
C359S326000
Reexamination Certificate
active
06678450
ABSTRACT:
BACKGROUND OF THE INVENTION
The invention relates to quantum computing and, more specifically, is a novel optical method for constructing a quantum computer.
Many kinds of numerical problems cannot be solved using conventional computers because of the time required to complete the computation. For example, the computer time required to factor an integer containing N digits is believed to increase exponentially with N. It has been estimated that the time required to factor a 150-digit number using the fastest supercomputers currently available would be longer than the age of the universe. Future increases in the speed of conventional computers will clearly be inadequate for problems of that kind, which are often of considerable practical importance. For example, the difficulty in factoring large numbers forms the basis for the most commonly used methods of cryptography.
It has been shown that quantum-mechanical computers could use nonclassical logic operations to provide efficient solutions to certain problems of that kind, including the factoring of large numbers. As an example of a nonclassical logic function, consider the conventional NOT operation, which simply flips a single bit from 0 to 1 or from 1 to 0. In addition to the usual NOT, a quantum computer could also implement a new type of logic operation known as the square root of NOT. When this operation is applied twice (squared), it produces the usual NOT, but if it is applied only once, it gives a logic operation with no classical interpretation.
In addition to performing nonclassical logic operations, quantum computers will be able to perform a large number of different calculations simultaneously on a single processor, which is clearly not possible for a conventional computer. This quantum parallelism is responsible for much of the increased performance of a quantum computer.
The operation of individual quantum logic gates has been demonstrated, but no operational quantum computer has been constructed. The eventual goal is to produce large numbers of quantum logic gates on a single substrate, in analogy with current semiconductor technology, which would allow the development of quantum computers for practical applications.
Quantum computers will use a binary representation of numbers, just as conventional computers do. An individual quantum bit, often called a qubit, will be physically represented by the state of a quantum system. For example, the ground state of an atom could be taken to represent the value 0, while an excited state of the same atom could represent the value 1. In the optical approach of the invention to quantum computing, a 0 is represented by a single photon in a given path. The same photon in a different path represents a 1.
Although classical bits always have a well-defined value, qubits often have some probability of being in either of the two states representing 0 and 1. It is customary to represent the general state of a quantum system by |&psgr;>, and we will let |0> and |1> represent the states corresponding to the values 0 and 1, respectively. Quantum mechanics allows superpositions of these two states, given by
|&psgr;>=&agr;|0>+&bgr;|1>
where &agr; and &bgr; are complex numbers. The probability of finding the system in the state |0> is equal to &agr;
2
the probability of the state |1> is &bgr;
2
.
Quantum-mechanical superpositions of this kind are fundamentally different from classical probabilities in that the system cannot be considered to be in only one of the states at any given time. For example, consider a single photon passing through an interferometer, as illustrated in
FIG. 1
, with phase shifts &phgr;
1
and &phgr;
2
inserted in the two paths. A beam splitter gives a 50% probability that the photon will travel in the upper or the lower path. If a measurement is made to determine where the photon is located, it will be found in only one of the two paths. But if no such measurement is made, a single photon can somehow measure both phase shifts &phgr;
1
and &phgr;
2
simultaneously, since the observed interference pattern depends on the difference of the two phases. This suggests that in some sense a photon must be located in both paths simultaneously if no measurement is made to determine its position. In a more complicated interferometer with many paths, a single photon can simultaneously measure a linear combination of the phase shifts in all of the paths even though it can be detected in only one of the paths.
The ability of a quantum computer to perform more than one calculation at the same time is analogous to the properties of the single-photon interferometer just described. A quantum computer,can provide results that depend on having performed a large number of calculations, even though a measurement to determine exactly what the computer was doing would show that it was programmed to perform only one specific calculation. To illustrate this, consider a computer programmed to perform a specific calculation based on the value of N input bits, and assume that the result can be described by N output bits, as illustrated in FIG.
2
. There are 2
N
different combinations of input bits, each of which corresponds to a specific input state denoted by |input
j
>, where j takes on all the values from 1 to 2
N
. The equal number of specific combinations of output bits is denoted by |output
k
>. Each input state can produce a superposition of possible output states,
&LeftBracketingBar;
input
j
⟩
→
∑
k
=
1
2
N
β
jk
&LeftBracketingBar;
output
k
⟩
where the complex coefficients &bgr;
jk
describe the calculation performed. In addition, the input state can be a superposition of all of the possible inputs to the computer:
&LeftBracketingBar;
input
⟩
=
∑
j
=
1
2
N
a
j
&LeftBracketingBar;
input
j
⟩
.
P
u
=
&LeftBracketingBar;
Σ
&RightBracketingBar;
2
In that case, the linearity of quantum mechanics gives an output state of the form
&LeftBracketingBar;
output
⟩
=
∑
j
=
1
2
N
a
j
∑
k
=
1
2
N
b
jk
&LeftBracketingBar;
output
k
⟩
.
The probability P
k
of getting a specific output state k is then given by the square of its coefficient in the immediately preceding equation:
P
k
=
u
∑
j
=
1
2
N
a
j
b
jk
u
2
.
It can be seen that the probability of getting a particular output depends on all of the coefficients &bgr;
jk
, which represent the results of all possible calculations on the computer. The result also depends on interference between all of the possible inputs, in the sense that P
k
will be large if all of the input states contribute in phase with each other. Conversely, P
k
will be small if the contributions from all of the initial states cancel out. The goal of quantum computing is to program the computer in such a way that the desired result occurs with high probability while all incorrect results occur with negligible probability.
To illustrate the usefulness of superposition states of this kind, suppose that we want to calculate the quantity Q,
Q
=
∑
j
=
1
2
N
e
ij
f
(
j
)
,
where ƒ(j) is a highly nonlinear function of j. The quantity Q corresponds to a weighted average of the function ƒ over all possible inputs to the computer, which is a Fourier transform of sorts. Calculations of this kind could be implemented on a quantum computer by programming the computer itself to calculate ƒ(j) and then creating a superposition of input states corresponding to the desired weighted average.
It has been shown that quantum computers could be used to efficiently factor large numbers, which is responsible for much of the current interest in quantum computing. The algorithm involved uses interference effects to ensure that, with high probability, the output of the com
Cherry Euncha
Cooch Francis A.
The Johns Hopkins University
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