This is an example implementation of the Deutsch-Jozsa algorithm for the special case of \(2\) qubits. The algorithm allows to distinguish between a constant or balanced function \(f\) with a single application of \(f\), relying on what is called quantum parallelism.

We first prepare a state \(|\psi_0\rangle = |x, y\rangle = |0, 1\rangle\) with only 2 qubits as follows

`## ( 1 ) * |0,1>`

Note that we count the qubits from one to number of qubits, and the **least significant bit** (the right most one) is counted **first**.

Using the Hadamard gate on both qubits results in a superposition in both qubits

```
## ( 0.5 ) * |0,0>
## + ( -0.5 ) * |0,1>
## + ( 0.5 ) * |1,0>
## + ( -0.5 ) * |1,1>
```

The next step is to apply the uniform transformation \(U_f\) to the state \(|x\rangle(|0\rangle - |1\rangle)\). The action of \(U_f\) was defined as \(|x,y\rangle \to |x, y\oplus f(x)\rangle\), where \(\oplus\) is addition modulo \(2\). The function \(f\) is a function \(\{0,1\}\to\{0,1\}\).

We first consider a so-called balanced function \(f(x)\), i.e. it is equal to \(1\) for exactly half of the possible \(x\). In our case with a single qubit \(x\) this could be \(f(0)=0\) and \(f(1) = 1\).

\(U_f\) is realised in this case by CNOT\((2,1)\), where we consider the second qubit as the control qubit. For \(|x, y\oplus f(x)\rangle\), there are four different possibilities

- \(x=0, y=0\), \(U_f(|0,0\rangle) = |0, 0\oplus f(0)\rangle = |0, 0\rangle\)
- \(x=1, y=0\), \(U_f(|1,0\rangle) = |1, 0\oplus f(1)\rangle = |1, 1\rangle\)
- \(x=0, y=1\), \(U_f(|0,1\rangle) = |0, 1\oplus f(0)\rangle = |0, 1\rangle\)
- \(x=1, y=1\), \(U_f(|1,1\rangle) = |1, 1\oplus f(1)\rangle = |1, 0\rangle\)

Now,

- CNOT\((2,1)|0,0\rangle = |0,0\rangle\)
- CNOT\((2,1)|1,0\rangle = |1,1\rangle\)
- CNOT\((2,1)|0,1\rangle = |0,1\rangle\)
- CNOT\((2,1)|1,1\rangle = |1,0\rangle\)

which is what we wanted to archive. Thus, we apply it:

```
## ( 0.5 ) * |0,0>
## + ( -0.5 ) * |0,1>
## + ( -0.5 ) * |1,0>
## + ( 0.5 ) * |1,1>
```

Now apply the Hadamard gate again on \(x\) (the query register), i.e. the second qubit

```
## ( 0.7071068 ) * |1,0>
## + ( -0.7071068 ) * |1,1>
```

Now qubit \(2\) equals \(1\), thus, if we measure,

we obtain \(1\). We can also plot the corresponding circuit

On the other hand, a constant function \(f(x) = 1\) leads to

- \(x=0, y=0\), \(U_f(|0,0\rangle) = |0, 0\oplus f(0)\rangle = |0, 1\rangle\)
- \(x=1, y=0\), \(U_f(|1,0\rangle) = |1, 0\oplus f(1)\rangle = |1, 1\rangle\)
- \(x=0, y=1\), \(U_f(|0,1\rangle) = |0, 1\oplus f(0)\rangle = |0, 0\rangle\)
- \(x=1, y=1\), \(U_f(|1,1\rangle) = |1, 1\oplus f(1)\rangle = |1, 0\rangle\)

which can be realised with a NOT operation on the first qubit

- X\((1)|0,0\rangle = |0,1\rangle\)
- X\((1)|1,0\rangle = |1,1\rangle\)
- X\((1)|0,1\rangle = |0,0\rangle\)
- X\((1)|1,1\rangle = |1,0\rangle\)

So, the same algorithm again, now with the constant \(f\)

```
x <- X(1) * qstate(nbits=2, basis=genComputationalBasis(2, collapse=","))
y <- H(2) * (H(1) * x)
z <- X(1) * y
z
```

```
## ( -0.5 ) * |0,0>
## + ( 0.5 ) * |0,1>
## + ( -0.5 ) * |1,0>
## + ( 0.5 ) * |1,1>
```

```
## ( -0.7071068 ) * |0,0>
## + ( 0.7071068 ) * |0,1>
```

```
## $bit
## [1] 2
##
## $repetitions
## [1] 1
##
## $prob
## [1] 0.5 0.5 0.0 0.0
##
## $value
## [1] 0
##
## $psi
## ( -0.7071068 ) * |0,0>
## + ( 0.7071068 ) * |0,1>
##
## attr(,"class")
## [1] "measurement" "list"
```

and we obtain \(0\) for the second qubit.

In principle the code in Qiskit could look somehow like this:

```
filename <- paste0(tempdir(), "/circuit.py")
export2qiskit(u, filename=filename)
cat(readLines(filename), sep = '\n')
```

```
# automatically generated by qsimulatR
qc = QuantumCircuit(2)
qc.x(0)
qc.h(0)
qc.h(1)
qc.x(0)
qc.h(1)
```

But this does not include the measurement and the measurement cannot simply be added as an additional command. If a measurement is to be performed, one has to add corresponding classical bits. To do so export the state including the measurement:

```
# automatically generated by qsimulatR
import numpy as np
from qiskit import(QuantumCircuit, execute, Aer)
from qiskit.visualization import plot_histogram
simulator = Aer.get_backend('qasm_simulator')
qc = QuantumCircuit(2,1)
qc.x(0)
qc.h(0)
qc.h(1)
qc.x(0)
qc.h(1)
qc.measure(1, 0)
```

Then the results from the Quiskit simulation can be obtained with

```
qc.draw()
res = execute(qc, simulator, shots=1)
res.result().get_counts(qc)
```