/qc_mentorship_task2

Primary LanguagePython

QC Mentorship Problem 2

Problem Statement:

To find a circuit that returns |00> and |11> with equal probability where the circuit is parameterized, contains only RX, RY and CNOT gates and all parameters are determined using Gradient Descent

Approach to Solution:

The RX, RY and CNOT gates are defined as follows:

equation

equation

We consider two qubits q0 and q1 both initially in the state |0>. A circuit that returns a state of the form a|00> + b|11> can be constructed by applying an RX gate to qubit q0 and then applying a CNOT gate with q0 as control and q1 as target. The amplitudes a and b depend on the angle, theta of the RX rotation. For theta = n*pi/2 where n is any positive or negative integer, this produces a circuit that returns |00> and |11> with equal probability, i.e., 1/2. However, it adds a phase of -i to |11>. For the given problem, we use RY gate instead of RX to perform the rotation. From the gate definition given above, applying RY and then CNOT also produces a state of the form a|00> + b|11> without an additional complex phase.

For determining the parameter using gradient descent, we start with the parameter initialized to 0 and use a cost function of the form (prob00 - prob11)^2 where prob00 and prob11 are probabilities of the states |00> and |11> respectively when the circuit is run for given number of trials. The cost function encodes the condition that the probabilities of |00> and |11> must be equal which is the objective of the given problem.

Code Structure:

The code in this repository is structured as follows:

  1. equal_probability_circuit_util/equal_probability_circuit.py : Includes utilities for constructing a circuit that returns |00> and |11> with equal probabilities using the appraoch described above, running the circuit for a limited number of trials and verifying the final state produced using the parameter determined from gradient descent
  2. equal_probability_circuit_util/parameter_optimizer.py : Includes utilities for performing gradient descent to determine the parameter for equal probability circuit and a cost function that measures the performance
  3. main.py : Generates a comparison of results produced by applying gradient descent for the parameter of the equal probability circuit. The circuit is run for 1, 10, 100 and 1000 trails each time with or without noise

Bonus Question:

To ensure that our circuit returns only states of the form a|00> + b|11> and not a|00> - b|11> we use a different cost function, prob11 - 0.5 where prob11 is the probability of the state |11>. This cost function encodes the condition that the probability of the state |11> must be 1/2 for it to be equal to the probability of |00> and vice versa.