Code example: Deutsch-Jozsa algorithm (duplicate)

What does it do?

In the Deutsch–Jozsa algorithm we use an oracle to determine if a binary function f(x):{0,1}n{0,1}f(x) : {0,1}^n \rightarrow {0,1} is constant or balanced.

The function is constant if f(x)=0f(x)=0 or f(x)=1f(x)=1 for all values of xx. A function is balanced if f(x)=0f(x)=0 for half of the possible input values xx and f(x)=1f(x)=1 for the other half.

What is it used for?

The Deutsch-Josza algorithm is a simple example of a quantum algorithm that can be used to speed up a search. As will be explained below, it can determine whether or not a function has a certain property (being balanced). The algorithm achieves this by requiring that the function (more precisely, a derivation of the function) need only be called once with a quantum algorithm instead of twice with a classical algorithm. When the function is very 'expensive', e.g., in terms of computational resources, it can be very beneficial if you have to compute this function only once instead of twice.

Although the speed-up of this specific algorithm is only a factor of 2, other quantum algorithms, using the same quantum mechanical effects, can achieve a polynomial or even an exponential speed-up compared to classical algorithms.

How does it work?

In this example we consider a binary function f(x):{0,1}{0,1}f(x) : {0,1} \rightarrow {0,1}. There are four possibilities for the function f(x)f(x), which we call the Oracle function. These are:

  1. f1(x)=0f_1(x)=0
  2. f2(x)=1f_2(x)=1
  3. f3(x)=xf_3(x)=x
  4. f4(x)=1xf_4(x)=1-x

The algorithm to determine whether our function f(x)f(x) is constant or balanced requires only a single query of f(x)f(x). By changing the Oracle, the 4 different functions can be tested.

In this example a very simple Oracle is implemented for the four different functions, but a much more complex Oracle could also be used.

Examination of the code

The following code shows the implementation of the algorithm, which uses two qubits. Note that we do not include a measure statement at the end of the algorithm. In this case no measurement statement is required (see optimization of simulations), but one could be added without causing any problems because this algorithm requires just one shot for its execution.

        
          version 1.0
qubits 2

# In the Deutsch–Jozsa algorithm we use an oracle to determine if a binary function f(x) is constant or balanced.
# Constant f(x)=fc1=0 OR f(x)=fc2=1
# Balanced f(x)=fb3=x OR f(x)=fb4=NOT(x)
# The algorithm requires only a single query of f(x).
# By changing the Oracle, the 4 different functions can be tested.

# Initialize qubits in |+> and |-> state
.initialize
prep_z q[0:1]
X q[1]
{H q[0]|H q[1]}
 
.oracle_fc1
# do nothing or I q[0:1]
 
#.oracle_fc2
# X q[1]
 
#.oracle_fb3
# CNOT q[0],q[1]
 
#.oracle_fb4
# CNOT q[0],q[1]
# X q[1]
 
.measurement
H q[0]
measure q[0]
        
      
initialize measurement
q[0]
 
 
 
 
 
 
 
q[1]
 
 
 
 
 
 
 

Examination of the results

Let us run the code above, using Oracle fc1f_c1. You can do this by copying the code above to the editor, executing the algorithm (1 shot is sufficient) and examining the results. You should get the following result:

1.000
00
1.0
0.8
0.6
0.4
0.2
0.0

We can now run the same code again, but this time we implement the 2nd Oracle, by uncommenting line 20. If we now examine the histogram we get exactly the same result as for the first oracle.

Likewise we can run the algorithm for Oracle 3 or Oracle 4 by commenting out line 20 and uncommenting one of the other Oracle implementations; in these cases we will get a different result:

1.000
01
1.0
0.8
0.6
0.4
0.2
0.0

As shown, with a single function call we can determine from the outcome whether a function is constant (binary register is 00) or balanced (binary register is 01).

Want to know more?

For more information see Deutsch–Jozsa algorithm on Wikipedia.