Grover Search Marking Circuit

Design a quantum circuit with $44$ input bits $0,1,2,3$b$0,$b$1,$b$2,$b$3$ such that it "marks" the amplitudes corresponding to the states

$||0123$ in $\{||0011,||1100,||0101\}|$b$0$b$1$b$2$b$3$ in $\{|0011,|1100,|0101\}$

by inverting their amplitudes, leaving all other states unchanged. In this problem, you are asked to achieve this without using ancillary qubits or a result bit.

Let us consider the simple problem of inverting the amplitude corresponding to $||1111|1111$ leaving all the others unchanged. Show that this can be performed using a "multi controlled phase gate" with a phase of $$ radians.

Using the idea of a multi$-$controlled phase gate, now use X$-$gates $($quantum not$)$ to convert each of the pure states $$\backslash$ket$\{0011\},\backslash$ket$\{1100\},\backslash$ket$\{0101\}$$ each to $$\backslash$ket$\{1111\}$$ and use MCP gate above to mark the phase. Remember to use X$-$gates to convert the inputs back.

what I have so far :

from qiskit import QuantumCircuit, Aer, execute

# Create the quantum circuit with $4$ qubits

qc $=$ QuantumCircuit$(4)$

# Step $1$: Create uniform superposition of all states

qc$.$h$([0,1,2,3])$ # Apply Hadamard to all qubits

# Check state after superposition

qc$.$barrier$()$

# Function to mark a specific state to $|1111>$

def mark$\_$state$($state$\_$bits$)$:

# Convert bits to $|1111>$

control$\_$qubits $=[]$

for i in range$(4)$:

if state$\_$bits$[$i$]==\text{'}0\text{'}$:

qc$.$x$($i$)$ # Flip to $1$ if needed

control$\_$qubits.append$($i$)$

# Apply the multi$-$controlled phase gate with a phase of

qc$.$mcx$($control$\_$qubits$[$:$-1],$ control$\_$qubits$[-1])$ # Use the last qubit as target

# Revert the X$-$gates to get back to the original state

for i in range$(4)$:

if state$\_$bits$[$i$]==\text{'}0\text{'}$:

qc$.$x$($i$)$ # Revert X$-$gate to flip back to $0$

# Step $2$: Mark states $|0011>,|1100>,|1010>$

mark$\_$state$(\text{'}0011\text{'})$

qc$.$barrier$()$

mark$\_$state$(\text{'}1100\text{'})$

qc$.$barrier$()$

mark$\_$state$(\text{'}1010\text{'})$

# Barrier for clarity

qc$.$barrier$()$

# Step $3$: Measurement

qc$.$measure$\_$all$()$

$2$ cells that I can't manipulate :

$1$:

from qiskit import QuantumCircuit, QuantumRegister, Aer, execute

b $=$ QuantumRegister$(4,\text{'}$b$\text{'})$

qc $=$ QuantumCircuit$($b$)$

# create uniform super position

qc$.$h$($b$[0])$

qc$.$h$($b$[1])$

qc$.$h$($b$[2])$

qc$.$h$($b$[3])$

qc$.$barrier$()$

mark$\_$pure$\_$states$($qc$,$ b$[0],$ b$[1],$ b$[2],$ b$[3])$

display$($qc$.$draw$(\text{'}$mpl$\text{'},$ style$=$'iqp'$))$

# use a state vector simulator to obain the marked states

backend $=$ Aer.get$\_$backend$(\text{'}$statevector$\_$simulator'$)$

job $=$ execute$($qc$,$ backend$)$

result $=$ job.result$()$

state$\_$vector $=$ result.get$\_$statevector$()$

print$($state$\_$vector$)$

for i in range$(16)$:

if i $==3$ or i $==10$ or i $==12$: # are we marking the correct basis states

assert abs$($state$\_$vector$[$i$]+0\mathrm{.}25)<=0\mathrm{.}001$

else:

assert abs$($state$\_$vector$[$i$]-0\mathrm{.}25)<=0\mathrm{.}001$

$2$:

# Let's run Grover's algorithm

# let's now implement Grover's search to find a value

import numpy as np

from qiskit import QuantumCircuit, QuantumRegister, Aer, execute

def Uf$($qc$,$ b$)$:

mark$\_$pure$\_$states$($qc$,$ b$[0],$ b$[1],$ b$[2],$ b$[3])$

def apply$\_$reflection$\_$about$\_$uniform$\_$state$($qc$,$ input$\_$registers$)$:

# qc is a previously created quantum circuit

# input$\_$registers are the registers that measure the input

# n is the number of input qubits

#

# $1.$ Apply Hadamard on each of the input registers.

# $2.$ Invert all the qubits

for i in input$\_$registers:

qc$.$h$($i$)$

qc$.$x$($i$)$

n $=$ len$($input$\_$registers$)$

# $3.$ apply a multi controlled Z gate

qc$.$mcp$($np$.$pi$,$ input$\_$registers$[0$:n$-1],$ input$\_$registers$[$n$-1])$

for i in input$\_$registers:

qc$.$x$($i$)$ # invert back

qc$.$h$($i$)$ # apply Hadamard back

def Grover$\_$diffuse$($qc$,$ inputs$)$:

Uf$($qc$,$ inputs$)$

apply$\_$reflection$\_$about$\_$uniform$\_$state$($qc$,$ inputs$)$

qc$.$barrier$()$

def create$\_$quantum$\_$circuit$\_$for$\_$grover$($n$\_$iters$)$:

inputs $=$ QuantumRegister$(4,\text{'}$b$\text{'})$

cbit $=$ ClassicalRegister$(4,\text{'}$z$\text{'})$

qc $=$ QuantumCircuit$($inputs$,$ cbit$)$

for i in inputs:

qc$.$h$($i$)$ # apply hadamard

qc$.$barrier$()$

for i in range$($n$\_$iters$)$:

Grover$\_$diffuse$($qc$,$ inputs$)$

qc$.$measure$($inputs$,$ cbit$)$

return qc

qc$2=$ create$\_$quantum$\_$circuit$\_$for$\_$grover$(4)$

print$(\text{'}$Grover Search After Four Iterations'$)$

display$($qc$2.$draw$(\text{'}$mpl$\text{'},$ style$=$'iqp'$))$

from qiskit.tools.visualization import plot$\_$histogram

# lets test after one iteration

simulator $=$ Aer.get$\_$backend$(\text{'}$aer$\_$simulator'$)$

circ $=$ transpile$($qc$2,$ simulator$)$

# Run and get counts

result $=$ simulator.run$($qc$2).$result$()$

counts $=$ result.get$\_$counts$($circ$)$

display$($plot$\_$histogram$($counts$,$ title$=$'result counts $(1024$ simulations$)\text{'}))$

assert counts$[\text{'}0011\text{'}]+$ counts$[\text{'}1010\text{'}]+$ counts$[\text{'}1100\text{'}]>=500,$ "The marked states should account for more than $50\%$ of the counts. Are you implementing the Marking circuit correctly?"

I am getting this assertion error:

AssertionError: The marked states should account for more than $50\%$ of the counts. Are you implementing the Marking circuit correctly?

Thank you for your help!