Introduction to Circuit Descriptors in Quantum Computing

"There must always be some quantitative measures for knowing the efficiency of something."

We are slowly approaching the NISQ(Noisy Intermediate Scale Quantum) device era. NISQ devices have a few hundred qubits and are not advanced enough to achieve fault tolerance or quantum supremacy. We do not have such quantum computers which can solve a problem by itself. Hence we also need the help of classical devices.

Such hardware is used in HQC (Hybrid Quantum Classical) Algorithms. These algorithms strategically divide the computational task into classical and quantum resources. A good example of HQC is VQE(Variational Quantum Eigensolver) which computes the ground state of a molecular system.

Hardware Workflow of VQE:

1. A parameterized trial wavefunction of the system of interest is prepared on a quantum computer by tuning a quantum circuit based on the chosen parameterization.

2. The expectation value of the energy is measured concerning the parameters of the ansatz.

3. These parameters are updated using optimization routines and are fed back to the quantum computer to prepare a "better learned" trial state.

This cycle of updating the trial state and its parameters continues until the convergence criteria are satisfied.

Other examples of HQC are:

• Quantum Approximate Optimization Algorithm (QAOA): The QAOA is a hybrid quantum-classical variational algorithm designed to tackle combinatorial optimization problems.

• Quantum AutoEncoder (QAE): Classical AutoEncoders are neural networks that can learn efficient low-dimensional representations of data in higher dimensional space. Given an input x, the task of an autoencoder is to map x to a lower dimensional point y such that x can likely be recovered from y. Quantum Autoencoder performs a similar task on quantum data. It is trained to compress a particular dataset of quantum states, where a classical compression algorithm cannot be employed.

• Quantum Variational Error Corrector (QVECTOR): It uses a quantum variational circuit to learn encoding and error recovery gate sequences. It is used for Quantum Error Correction.

• Classification via Quantum Neural Network (QNN): It combines the concept of Quantum Computing and Artificial Neural Networks.

• Generative Modeling: Generative modeling algorithm models probability distribution underlying a given data set. This probability distribution helps to characterize the generation process of a given dataset. Quantum generative modeling can employ quantum resources such as parameterized quantum channels, quantum operators etc.

Though the objective function to be optimized may differ for each problem, they share a common quantum subroutine(a function) for producing parameterized trial states or ansatz such that the parameters can be tuned to correspond to the optimal or near-optimal objective function value. Different objective functions can be:

• Energy

• Mean Squared Error

  

• Average Fidelity: Fidelity means the accuracy with which the output includes the characteristics of the input.

• KL divergence: A statistical measurement to quantify the difference between one probability distribution from a reference probability distribution. Mathematically it is given as:

  If Q is the reference probability distribution then KL divergence of the probability distribution P is given as:

  

  

  Here '||' is a symbol of divergence.

Now if there are 2 variational circuits A and B, then which circuit is more suitable for a given application and why? Are there any descriptors based on which we can answer the above question? The answer is yes.

A few of the descriptors are

• Expressibility

• Entangling Capability

• Number of Circuit Parameters

• Number of Two Qubit Operations

These descriptors help in characterizing and evaluating parameterized quantum circuits using classical simulations. Hence, whenever we have an HQC algorithm, we are equipped with tools for making decisions about and designing parameterized quantum circuits.

Similar metrics have been proposed for learning algorithms in statistical learning:

• Empirical Risk: It results from averaging the loss function over the data. We don't know how well an algorithm will perform in practice(true risk), so we will measure its performance on a known set of training data giving us the empirical risk.

• Generalization Ability: The model's ability to adapt new, previously unseen data drawn from the same distribution as the one used to create the model.

• Sample Complexity: The number of data samples needed to achieve a certain level of performance or the number of training samples needed to successfully learn a target function.

If an algorithm performs well based on one metric then it can perform poorly based on another metric. Hence any algorithm must be evaluated on multiple metrics and the performance of that algorithm must balance all the chosen metrics. A supervised learning task should minimize Empirical Risk and Maximize Generalization Ability to avoid Overfitting. The algorithm must learn the training data well, but it must also perform well on unobserved data. Efforts are made to develop NISQ analogues to descriptors in statistical learning theory and it urges us to investigate more about Parameterized Quantum Circuits and to address analogous challenges.

Credit for the above blog:

I have gained the information of the above blog from the research paper "Expressiblity and Entangling Capability of Parameterized Quantum Circuits for Hybrid Quantum-Classical Algorithms. "

References:

1) Sukin Sim, Peter D. Johnson and Alan Aspuru-Guzik, Expressibility and entangling capability of parameterized quantum circuits for hybrid quantum-classical algorithms, Advanced Quantum Technology 2 (2019) 1900070, doi:10.1002/qute.201900070, arXiv:1905.10876.