Quantum-Computation-course-Basel/exercises/Exercise1.ipynb

Exercise Sheet 1: Quantum Logic Gates

To run this notebook you'll need to install qiskit, qiskit-aer and qiskit-ibm-runtime.

In [2]:

from qiskit import QuantumCircuit, transpile
from qiskit_aer import AerSimulator
from qiskit.visualization import plot_histogram
import numpy as np

Exercise 1

See 'Part 1' below, and find the circuits required for the:

• (a) XOR gate;
• (b) AND gate;
• (c) NAND gate;
• (d) OR gate.

Exercise 2

See 'Part 2' below, and find a layout for which the AND gate compiles to 6 non-local gates for ibmqx2. Note that there is some randomness in the compiling process. So you might need to try a few times.

Part 1: Classical logic gates with quantum circuits

📓 NOT gate

In the following function we use a quantum circuit to do the simplest job in all of computation: apply a NOT gate to a bit.

In [3]:

def NOT(inp):
    """An NOT gate.
    
    Parameters:
        inp (str): Input, encoded in qubit 0.
        
    Returns:
        QuantumCircuit: Output NOT circuit.
        str: Output value measured from qubit 0.
    """

    qc = QuantumCircuit(1, 1) # A quantum circuit with a single qubit and a single classical bit
    qc.reset(0)
    
    # We encode '0' as the qubit state |0⟩, and '1' as |1⟩
    # Since the qubit is initially |0⟩, we don't need to do anything for an input of '0'
    # For an input of '1', we do an x to rotate the |0⟩ to |1⟩
    if inp=='1':
        qc.x(0)
        
    # barrier between input state and gate operation 
    qc.barrier()
    
    # Now we've encoded the input, we can do a NOT on it using x
    qc.x(0)
    
    #barrier between gate operation and measurement
    qc.barrier()
    
    # Finally, we extract the |0⟩/|1⟩ output of the qubit and encode it in the bit c[0]
    qc.measure(0,0)
    
    # We'll run the program on a simulator
    backend = AerSimulator()
    # Since the output will be deterministic, we can use just a single shot to get it
    job = backend.run(qc, shots=1, memory=True)
    output = job.result().get_memory()[0]
    
    return qc, output

Now let's see it in action.

In [ ]:

for inp in ['0', '1']:
    qc, out = NOT(inp)
    print('NOT with input',inp,'gives output',out)
    display(qc.draw())
    print('\n')

This implementation of NOT gate was an example of what you will need to do for a range of other gates below.

📓 XOR gate

Takes two binary strings as input and gives one as output.

The output is '0' when the inputs are equal and '1' otherwise.

In [5]:

def XOR(inp1,inp2):
    """An XOR gate.
    
    Parameters:
        inpt1 (str): Input 1, encoded in qubit 0.
        inpt2 (str): Input 2, encoded in qubit 1.
        
    Returns:
        QuantumCircuit: Output XOR circuit.
        str: Output value measured from qubit 1.
    """
  
    qc = QuantumCircuit(2, 1) 
    qc.reset(range(2))
    
    if inp1=='1':
        qc.x(0)
    if inp2=='1':
        qc.x(1)
    
    # barrier between input state and gate operation 
    qc.barrier()
    
    # this is where your program for quantum XOR gate goes
    
    
    
    
    
    
    
    
    # barrier between input state and gate operation 
    qc.barrier()
    
    qc.measure(1,0) # output from qubit 1 is measured
  
    #We'll run the program on a simulator
    backend = AerSimulator()
    #Since the output will be deterministic, we can use just a single shot to get it
    job = backend.run(qc, shots=1, memory=True)
    output = job.result().get_memory()[0]
  
    return qc, output

In [ ]:

## Test the function
for inp1 in ['0', '1']:
    for inp2 in ['0', '1']:
        qc, output = XOR(inp1, inp2)
        print('XOR with inputs',inp1,inp2,'gives output',output)
        display(qc.draw())
        print('\n')

📓 AND gate

Takes two binary strings as input and gives one as output.

The output is '1' only when both the inputs are '1'.

In [7]:

def AND(inp1,inp2):
    """An AND gate.
    
    Parameters:
        inpt1 (str): Input 1, encoded in qubit 0.
        inpt2 (str): Input 2, encoded in qubit 1.
        
    Returns:
        QuantumCircuit: Output XOR circuit.
        str: Output value measured from qubit 2.
    """
    qc = QuantumCircuit(3, 1) 
    qc.reset(range(2))
  
    if inp1=='1':
        qc.x(0)
    if inp2=='1':
        qc.x(1)
        
    qc.barrier()

    # this is where your program for quantum AND gate goes

    
    
    
    
    

    qc.barrier()
    qc.measure(2, 0) # output from qubit 2 is measured
  
    # We'll run the program on a simulator
    backend = AerSimulator()
    #Since the output will be deterministic, we can use just a single shot to get it
    job = backend.run(qc, shots=1, memory=True)
    output = job.result().get_memory()[0]
  
    return qc, output

In [ ]:

## Test the function
for inp1 in ['0', '1']:
    for inp2 in ['0', '1']:
        qc, output = AND(inp1, inp2)
        print('AND with inputs',inp1,inp2,'gives output',output)
        display(qc.draw())
        print('\n')

📓 NAND gate

Takes two binary strings as input and gives one as output.

The output is '0' only when both the inputs are '1'.

In [9]:

def NAND(inp1,inp2):
    """An NAND gate.
    
    Parameters:
        inpt1 (str): Input 1, encoded in qubit 0.
        inpt2 (str): Input 2, encoded in qubit 1.
        
    Returns:
        QuantumCircuit: Output NAND circuit.
        str: Output value measured from qubit 2.
    """
    qc = QuantumCircuit(3, 1) 
    qc.reset(range(3))
    
    if inp1=='1':
        qc.x(0)
    if inp2=='1':
        qc.x(1)
    
    qc.barrier()
    
    # this is where your program for quantum NAND gate goes


    
    
    
    
    
    qc.barrier()
    qc.measure(2, 0) # output from qubit 2 is measured
  
    # We'll run the program on a simulator
    backend = AerSimulator()
    #Since the output will be deterministic, we can use just a single shot to get it
    job = backend.run(qc, shots=1, memory=True)
    output = job.result().get_memory()[0]
  
    return qc, output

In [ ]:

## Test the function
for inp1 in ['0', '1']:
    for inp2 in ['0', '1']:
        qc, output = NAND(inp1, inp2)
        print('NAND with inputs',inp1,inp2,'gives output',output)
        display(qc.draw())
        print('\n')

📓 OR gate

Takes two binary strings as input and gives one as output.

The output is '1' if either input is '1'.

In [11]:

def OR(inp1,inp2):
    """An OR gate.
    
    Parameters:
        inpt1 (str): Input 1, encoded in qubit 0.
        inpt2 (str): Input 2, encoded in qubit 1.
        
    Returns:
        QuantumCircuit: Output XOR circuit.
        str: Output value measured from qubit 2.
    """

    qc = QuantumCircuit(3, 1) 
    qc.reset(range(3))
    
    if inp1=='1':
        qc.x(0)
    if inp2=='1':
        qc.x(1)
    
    qc.barrier()
   
    # this is where your program for quantum OR gate goes


    
    
    
    
    
    qc.barrier()
    qc.measure(2, 0) # output from qubit 2 is measured
  
    # We'll run the program on a simulator
    backend = AerSimulator()
    #Since the output will be deterministic, we can use just a single shot to get it
    job = backend.run(qc, shots=1, memory=True)
    output = job.result().get_memory()[0]
  
    return qc, output

In [ ]:

## Test the function
for inp1 in ['0', '1']:
    for inp2 in ['0', '1']:
        qc, output = OR(inp1, inp2)
        print('OR with inputs',inp1,inp2,'gives output',output)
        display(qc.draw())
        print('\n')

Part 2: AND gate on Quantum Computer

Real quantum computers are not able to implement arbitary gates directly. Instead, everything needs to be compiled (or 'transpiled') to the set of basic gates that the device can use. This usually consists of a set of single qubit rotations, as well as two qubit gates like cx.

There are also limits on which cx gates can be used directly: only some pairs of control and target qubits are possible. To implement other cx gates, tricks such as using swap gates to effectively move information around must be used. The possible pairs of qubits on which cx gates can be applied is known as the 'connectivity' of the device.

We'll now look at some examples. To make sure you don't end up in a queue for a busy device, we'll be using mock backends. These are designed to act exactly like real backends.

In [ ]:

from qiskit_ibm_runtime.fake_provider import FakeYorktown
backend = FakeYorktown()

The two system we are using (or at least pretending to) is ibmqx2 (also known as ibmq_yorktown). With this system, it is not possible to peform a cx gate between any arbitrary pair of qubits. Instead only the following pairings are allowed (where the qubits are labelled from 0 to 4).

In [ ]:

backend.configuration().coupling_map

The following cell has a circuit that applies an AND gate that is constructed from single and two qubit gates. However, this circuit assumes full connectivity: that cx gates can be applied between any pair of qubits.

In [ ]:

qc_and = QuantumCircuit(3)
qc_and.ccx(0,1,2)
print('AND gate')
display(qc_and.draw())
print('\n\nTranspiled AND gate for hardware with the required connectiviy')
qc_and.decompose().draw()

This ideal transpilation requires 6 cx gates.

There are often optimizations that the transpiler can perform that reduce the overall gate count, and thus total length of the input circuits. Note that the addition of swaps to match the device topology, and optimizations for reducing the length of a circuit are at odds with each other. In what follows we will make use of initial_layout that allows us to pick the qubits on a device used for the computation and optimization_level, an argument that allows selecting from internal defaults for circuit swap mapping and optimization methods to perform.

Rather than actually running the AND function, let's just look at the transpiled circuits. The following function does this for a given set of inputs.

In [16]:

# run the cell to define AND gate for real quantum system

def AND(inp1, inp2, backend, layout):
    
    qc = QuantumCircuit(3, 1) 
    qc.reset(range(3))
    
    if inp1=='1':
        qc.x(0)
    if inp2=='1':
        qc.x(1)
        
    qc.barrier()
    qc.ccx(0, 1, 2) 
    qc.barrier()
    qc.measure(2, 0) 
  
    qc_trans = transpile(qc, backend, initial_layout=layout, optimization_level=3)
    
    return qc_trans

📓 Assign your choice of layout to the list variable layout in the cell below

In [17]:

# Assign your choice of the initial_layout to the variable layout1 as a list 
# For example
#     layout = [0,2,4]
layout = 

Compile the AND gate on ibmqx2 by running the cell below.

In [ ]:

for input1 in ['0','1']:
    for input2 in ['0','1']:
        qc_trans1 = AND(input1, input2, backend, layout)
                
        print('For input '+input1+input2)
        print('# of nonlocal gates =',qc_trans1.num_nonlocal_gates())