Auto Bell-State Generator

Contents

• Problem Statement
• Instructions to Run Program
• Circuit Design
• Ideal quantum circuit
• Design Partitions
• Cirquit and Parameter Initialization
• Cost Function
• Mean Squared Error
• Optimizers
  • Gradient Descent
  • Nesterov Accelerated Gradient
• Results and Comparison
• Bonus Question
• Conclusion and Future Scope

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Problem Statement

Implement a circuit that returns |01> and |10> with equal probability.
Requirements for the task

• The circuit should consist only of CNOTs, RXs and RYs.
• Start from all parameters in parametric gates being equal to 0 or randomly chosen.
• You should find the right set of parameters using gradient descent (you can use more advanced optimization methods if you like).
• Simulations must be done with sampling - i.e. a limited number of measurements per iteration and noise.
• Compare the results for different numbers of measurements: 1, 10, 100, 1000.

Requirements met

• Ideal circuit involves just one RY and CNOT gate each. :D
• Code has parameters, 'shots' and 'angle', initially being equal to 0 or randomly chosen.
• Ideal values of parameter 'angle' obtained using both Gradient descent and Nesterov Accelerated Gradient, followed by a performance comparison.
• Results of the optimizer have been compared for four different amounts of measurements: 1, 10, 100, 1000.

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Instructions to Run Program

1. Install qiskit
2. Clone repository and execute python file QOSF_Task2.ipynb

Circuit Design

• Ideal state

  

• The state can be obtained by first applying Hadamard and X gates to q0 and q1

  

  

• And then applying CNOT (0 -> 1) gate to the two-qubit system.

  

  

Ideal quantum circuit

• The ideal circuit can then be consolidated as follows

  

Design Partitions

• The problem statement can be approached with a Hybrid Classical-Quantum optimization model, having three major partitions.

  1. The Bell State Generator instantiates a quantum circuit required for the state generation, but with one optimizable parameter angle. It is a Quantum circuit.

  2. Cost Function calculates the cost or error for the parameter value of current iteration.It is a classical calculation.

  3. Optimizer, is a classical machine learning optimizer, updates the values of parameter angle for better performance of the circuit. Two classical optimizers namely - 'gradient descent' and 'Nesterov Accelerated Gradient' have been used and compared.

     

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Cirquit and Parameter Initialization

1. Qubit 0 is initialized to state [1,0], and Qubit 1 to [0,1] to be able to reach the aforementioned Bell State.
2. Parameter angle which is the angle by which the paraterized gate will rotate.
3. The angle is converted to radians from degreesand given a random initial value.

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Cost Function

Mean Squared Error

• The Cost function is a simple MSE function that can be used with the first order derivative optimizers like Gradient Descent very easily.

• The cost function is the squared difference of the Probability averages of both states |01> and |10>, that are obtained after the circuit is executed for a given number of shots.

  

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Optimizers

Gradient Descent

• The Gradient Descent Optimizer is used as it simplifies the process of convergence when our loss function is quadratic in nature.

• The local and global minimas are same things with same depths and the loss landscape can be easily analysed with much less compute power, and good accuracy.

• The parameter angle is being learned, using Gradient Descent and alpha is the learning_rate.

  

Nesterov Accelerated Gradient

• If the momentum is too high the algorithm may miss the local minima and may continue to rise up. So, to resolve this issue the NAG algorithm is used.

• We know we’ll be using γV(t−1) for modifying the weights so, θ−γV(t−1) approximately tells us the future location.

  

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Results and Comparison

• The results are taken for measurements 1, 10, 100 and 1000 respectively.
• The optimized value of angle over a range of 1000 iterations of given number of shots have been calculated.
• Expected output: 90 degrees or a multiple of 90 degrees

1. Gradient Descent

• The error of measurement reaches a max of 1.05% among all cases as seen.

  

1. Nesterov Accelerated Gradient

• The error of measurement reaches a max of 1.34% among all cases.

  

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Bonus Question

How to make sure you produce state |01> + |10> and not |01> - |10> ?

• To prevent the occurrence of phase inverse of the required state i.e. |01> - |10>, we will have to alter our cost function to be unsymmetrical. To alter the Mean Sqared error, we can remove the squares and keep the function linear.
• Normalising the cost function to a scale of -1 to 1 with optimum at 0 along with a linear cost function.

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Conclusion and Future Scope

• Use of the concept of momentum with NAG(Nesterov Accelerated Gradient) did not help up achieve a better accuracy and using gradient descent is equally efficient for this problem statement.
• Hessian based analysis of the Linear Loss Landscapes may give us a better intuition for getting better accuracies and preventing optimization of the |01> - |10> state.