^{1}

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^{2}

We review the new type of Deutsch-Jozsa algorithm proposed in [K. Nagata and T. Nakamura, Int. J. Theor. Phys. 49, 162 (2010)]. We suggest that the Deutsch-Jozsa algorithm can be used for quantum key distribution. Alice sends input
N
1 partite uncorrelated state to a black box. Bob measures output state. Now, Alice and Bob have promised to use a function
f
which is one of two kinds: either the value of
f
is constant or balanced. To Eve, it is secret. Alice’s and Bob’s goal is to determine with certainty whether they have chosen a constant or a balanced function. Alice and Bob get one bit if they determine the function
f
. The speed to get one bit improves by a factor of 2
^{N}
. This may improve the speed to establish quantum key distribution by a factor of 2
^{N}
.

The quantum theory (cf. [

As for the foundations of the quantum theory, Leggett-type non-local variables theory [

The most well known and developed application of quantum cryptography is quantum key distribution (QKD), which is the process of using quantum communication to establish a shared key between two parties without a third party (Eve) learning anything about that key, even if Eve can eavesdrop on all communication between Alice and Bob. This is achieved by Alice encoding the bits of the key as quantum data and sending them to Bob; if Eve tries to learn these bits, the messages will be disturbed and Alice and Bob will notice. The key is then typically used for encrypted communication using classical techniques. For instance, the exchanged key could be used as the seed of the same random number generator both by Alice and Bob.

The security of QKD can be proven mathematically without imposing any restrictions on the abilities of an eavesdropper, something not possible with classical key distribution. This is usually described as “unconditional security”, although there are some minimal assumptions required including that the laws of quantum mechanics apply and that Alice and Bob are able to authenticate each other, i.e. Eve should not be able to impersonate Alice or Bob as otherwise a man-in-the-middle attack would be possible.

To date, the relation between quantum computer and QKD is not reported. The earliest quantum algorithm, the Deutsch-Jozsa algorithm, is representative to show that quantum computation is faster than classical counterpart with a magnitude that grows exponentially with the number of qubits.

Recently, it is discussed that von Neumann’s theory does not meet the Deutsch-Jozsa algorithm [

In this paper, we review the Deutsch-Jozsa algorithm. We suggest that the Deutsch-Jozsa algorithm can be used for improving quantum key distribution. Alice sends input

The earliest quantum algorithm, the Deutsch-Jozsa algorithm, is representative to show that quantum computa- tion is faster than classical counterpart with a magnitude that grows exponentially with the number of qubits.

Let us follow the argumentation presented in [

and replies with the result, which is either 0 or 1. Now, Bob has promised to use a function f which is of one of two kinds; either the value of

is, equal to 1 for exactly half of all the possible x, and 0 for the other half. Alice’s goal is to determine with certainty whether Bob has chosen a constant or a balanced function, corresponding with him as little as possible. How fast can she succeed?

In the classical case, Alice may only send Bob one value of x in each letter. At worst, Alice will need to query Bob at least

times, since she may receive

If Bob and Alice were able to exchange qubits, instead of just classical bits, and if Bob agreed to calculate

Alice has an N qubit register to store her query in, and a single qubit register which she will give to Bob, to store the answer in. She begins by preparing both her query and answer registers in a superposition state. Bob will evaluate

on the query register, and finishes by performing a suitable measurement to determine whether f was constant or balanced.

Let us follow the quantum states through this algorithm. The input state is

Here the query register describes the state of N qubits all prepared in the

state. After the Hadamard transformation on the query register and the Hadamard gate on the answer register we have

The query register is now a superposition of all values, and the answer register is in an evenly weighted superposition of

and

Next, the function f is evaluated (by Bob) using

giving

Here

is the bitwise XOR (exclusive OR) of y and

By checking the cases

Thus

This can be summarized more succinctly in the very useful equation

where

is the bitwise inner product of x and z, modulo 2. Using this equation and (10) we can now evaluate

Alice now observes the query register. Note that the absolute value of the amplitude for the state

is

Let’s look at the two possible cases―f constant and f balanced―to discern what happens. In the case where f is constant the absolute value of the amplitude for

is

is of unit length it follows that all the other amplitudes must be zero, and an observation will yield

times for all N qubits in the query register. Thus, global measurement outcome is

If f is balanced then the positive and negative contributions to the absolute value of the amplitude for

cancel, leaving an amplitude of zero, and a measurement must yield a result other than

that is,

on at least one qubit in the query register. Summarizing, if Alice measures all

We suggest that the Deutsch-Jozsa algorithm can be used for quantum key distribution.

• First Alice prepares the qubits in (6) and sends the

• Next, Bob picks a random function “f” that is either balanced or constant and Bob applies

• Finally, Alice applies the Hadamard transformation to each of the qubits and measures. She learns whether f was balanced or constant-Alice and Bob now share a random bit of information (the “type” of

On safety, a questionable point is left in various ways, but this is a future problem. For example, we can consider the following situation:

Alice has to send the Query (N-qubit) and Answer (1-qubit) registers to Bob. Bob will then apply

Alice will then apply

In conclusion, we have reviewed the new type of Deutsch-Jozsa algorithm. We have suggested that the Deutsch- Jozsa algorithm can be used for quantum key distribution. Alice has sent input

On safety, a questionable point has been left in various ways, but this has been a future problem.

Koji Nagata,Tadao Nakamura, (2015) The Deutsch-Jozsa Algorithm Can Be Used for Quantum Key Distribution. Open Access Library Journal,02,1-6. doi: 10.4236/oalib.1101798