## Quantum ARITHMETICS

Increment and decrement operations are fundamental operations in classical computing, and they are also important in quantum computing. In quantum computing, increment and decrement operations are implemented using quantum gates that enable changes to the quantum state of qubits.

The increment and decrement operations on a quantum register are performed by manipulating the state of the qubits in the register. In a quantum register, each qubit can exist in a superposition of two states, |0> and |1>. The increment operation on a quantum register adds 1 to the value of the register, while the decrement operation subtracts 1 from the value of the register.

In quantum computing, the increment and decrement operations are implemented using a series of controlled operations, where the target qubit depends on the state of the control qubits. The controlled operations are typically implemented using the CCX gate (also known as the Toffoli gate), which flips the state of the target qubit if and only if the two control qubits are both in the state |1>.

To increment a quantum register, a series of CCX gates is used to add 1 to the value of the register. The first CCX gate increments the least significant bit of the register if the two least significant bits are both in the state |1>. This is repeated for each bit in the register until the most significant bit has been incremented.

The same series of CCX gates are used to decrement a quantum register but in reverse order. The first CCX gate decrements the least significant bit of the register if the two least significant bits are both in the state |1>. This is repeated for each bit in the register until the most significant bit has been decremented.

In quantum computing, the increment and decrement operations are important building blocks for various quantum algorithms, including arithmetic operations and quantum error correction. These operations can be implemented efficiently on quantum hardware and are an important part of the quantum computing toolbox.

The code below is written with OPENQASM and can be run on IBM Q composer.

` ````
```OPENQASM 2.0;
include "qelib1.inc";
qreg a[4];
qreg scratch[1];
x a[0];
h a[2];
rz(0.785398163397448) a[2];
// Increment
barrier a[0],a[1],a[2],a[3],scratch[0];
ccx a[0],a[1],scratch[0];
ccx scratch[0],a[2],a[3];
ccx a[0],a[1],scratch[0];
ccx a[0],a[1],a[2];
cx a[0],a[1];
x a[0];
// Decrement
barrier a[0],a[1],a[2],a[3],scratch[0];
x a[0];
cx a[0],a[1];
ccx a[0],a[1],a[2];
ccx a[0],a[1],scratch[0];
ccx scratch[0],a[2],a[3];
ccx a[0],a[1],scratch[0];

This code implements a quantum circuit that performs an increment-decrement operation on a 4-qubit register a. The qelib1.inc file is included, which provides a standard set of quantum gates.

The circuit first applies an X gate to the first qubit a[0] and a Hadamard gate followed by a rotation around the Z axis to the third qubit a[2]. Then a barrier is added to separate the increment and decrement parts of the circuit.

The increment part of the circuit is implemented using a series of controlled operations, where the target qubit depends on the state of the control qubits. The first three CCX gates increment the fourth qubit a[3] if the first two qubits a[0] and a[1] and scratch[0] are all in the state |1>. Then two CX gates and an X gate are applied to complete the increment operation.

The decrement part of the circuit is similar to the increment part, but the operations are performed in reverse order. The first operation is an X gate on a[0], followed by a CX gate between a[0] and a[1], and then two CCX gates that decrement a[3] if the first two qubits a[0] and a[1] and scratch[0] are all in the state |1>. Finally, two more CCX gates and an X gate are applied to complete the decrement operation.

Note that scratch[0] is used as an ancilla qubit for the controlled operations and is not part of the register being incremented or decremented.

You should expect the below output when running the code on the IBMQ composer.