How to take two's complement of a number using a quantum circuit

Two's complement is an arithmetic operation that converts a number $n$ to $n \mod{N}$ . It is used in digital systems to compute the difference of signed binary numbers, where subtraction with a number equals addition with its two's complement.

The general algorithm for computing the two's complement of a signed binary number is as follows:

Flip all the bits of the number, that is, 0s become 1s and 1s become 0s. This is called the one's complement of a number.
Add $1$ to the one's complement of the number.

For example, the two's complement of $+6$ can be computed as follows:

from qiskit import QuantumCircuit, QuantumRegister, ClassicalRegister, Aer, execute
from qiskit.circuit.library import XGate
qc_complement = QuantumCircuit(3, 3, name = '2s_cmp')
reg = QuantumRegister(3, 'q')
qc_complement.barrier()
# Encode a number
print('The encoded number is', bin(5)[2::])
index = [0, 2]
[qc_complement.x(i) for i in index]
qc_complement.barrier()
# Take the complement
## Flip the qubits
[qc_complement.x(i) for i in range(3)]
## Add one
for i in range(2, -1, -1):
    if i > 0:
        cnx_gate = XGate().control(i)
        qc_complement.append(cnx_gate, reg[0:i] + [reg[i]])
qc_complement.x(0)
qc_complement.barrier()
# Measure the qubits
qc_complement.measure(range(3), range(3))
print(qc_complement)
# Simulate the circuit
backend = Aer.get_backend('qasm_simulator')
counts = execute(qc_complement, backend, shots = 100).result().get_counts()
print(counts)
dec_num = int(list(counts.keys())[0], 2)
print('Twos complement of the encoded number is', dec_num)

Here's an explanation of the code:

Line 4: We create a quantum circuit, qc_complement, with three qubits and three classical bits.
Line 5: We declare a quantum register with three qubits.
Lines 9–11: We encode a number to the qubits. Since the qubits are initialized to $0$ by default, applying a CNOT gateIn quantum computing, a controlled-NOT (CNOT) gate is a two-qubit gate wherein the state of the target qubit flips only when the control qubit is in state 1. to them will flip the state to 1. Here, the CNOT gate is applied to indexes $0$ and $2$ . So, the encoded number is $(101)_2=5$ .
Lines 13–22: We compute the two's complement of the encoded number.
- Line 15: The qubit states are flipped. So, the encoded number is transformed into its one's complement.
- Lines 17–21: A ripple carry is used to flip the state of a qubit if all the previous qubit states are $1$ .
Line 25: We apply the measurement gates to the qubits.
Line 26: We display the qc_complement.
Lines 29–30: We simulate qc_complement using the QASM backend provided by Qiskit and the dictionary of the simulation results is stored in counts.
Line 32: We convert the computed two's complement in the binary form to an integer.

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How to take two's complement of a number using a quantum circuit

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