The Deutsch problem in quantum computation

In summary, the Deutsch problem in QC involves finding the answer to the question of whether a given function f(x) is constant or balanced. This can be done with a quantum computer using Deutsch's algorithm, which takes advantage of superposition to evaluate the function only once. The Hadamard operator and the unitary operator corresponding to f are important components in this algorithm.
  • #1
ptabor
15
0
I'm doing a research paper on the deutsch problem in QC. My google inquiries have turned up numerous papers on solutions to the problem, but I am having difficulty finding a clear formulation of the problem itself.

Can someone point me in the direction of this information?
 
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  • #2
http://www.cs.xu.edu/~kinne/quantum/deutche.html

In short,
You have a function f: {0,1} --> {0,1} (there are four possible functions like this). You want to know the answer to the question - is f(0)=f(1)?. A classical circuit (and hence, any classical algorithm) must evaluate f twice: you must evaluate f(0), and also f(1); and then you compare them. With Deutsch's algorithm, if you have a quantum computer you can exploit superposition to answer the question with only one evaluation of f.
 
  • #3
To clarify what is on that page:

The Hadamard operator H operates on a single qubit, and its matrix representation is

[tex]\frac{1}{\sqrt{2}}\left( \begin{array}1 & 1 \\ 1 & -1
\end{array} \right)[/tex]

The unitary operator corresponding to f (operates on two qubits) is:

f(x,y)=(x,y+f(x)) for eigenstates x,y, and extended to arbitrary x,y by linearity.

(i.e., f(a0+b1, c0+d1) = acf(0,0)+adf(0,1)+...)
 
  • #4
The function can take input over x = 0,1,2,..2^n-1 for any integer n.

The key is that you are promised that the function is either constant, or it is balanced. "Constant" means that either f(x)=1 or f(x)=0 for every input x, "balanced" means that f(x)=1 for exactly half of the possible inputs and f(x)=0 for the other half of the inputs.
 

FAQ: The Deutsch problem in quantum computation

What is the Deutsch problem in quantum computation?

The Deutsch problem is a problem in quantum computation that aims to demonstrate the power of quantum computers over classical computers. It involves finding the parity of a function that takes in a binary input and outputs a binary value. In classical computers, this problem would require two evaluations of the function, while in quantum computers it can be solved in a single evaluation.

Why is the Deutsch problem important in quantum computation?

The Deutsch problem is important because it was one of the first problems to demonstrate the potential speed-up of quantum computers over classical computers. It also highlights the differences in how classical and quantum computers process information, with quantum computers being able to solve certain problems exponentially faster than classical computers.

What is the significance of the Deutsch problem in the field of quantum computing?

The Deutsch problem has significant implications in the field of quantum computing as it was one of the first problems to be solved using a quantum algorithm, showing the potential of quantum computers to outperform classical computers. It also paved the way for further research and development in quantum algorithms and their applications in various fields.

How is the Deutsch problem solved using quantum computation?

The Deutsch problem can be solved using a quantum algorithm known as the Deutsch-Jozsa algorithm. This algorithm uses two quantum gates, the Hadamard gate and the phase oracle, to evaluate the function in a single step. This is possible due to the phenomenon of quantum superposition and entanglement.

What are the limitations of the Deutsch problem in quantum computation?

One limitation of the Deutsch problem is that it only demonstrates a speed-up in terms of the number of function evaluations, but not necessarily in terms of overall runtime. Additionally, the Deutsch problem is a contrived problem and may not have practical applications. It also assumes an ideal quantum computer, which may not be achievable in reality due to noise and error correction challenges.

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