Variational Quantum Algorithm

Variational Quantum Algorithms (VQAs) are a class of quantum algorithms that leverage both classical and quantum computing resources to find approximate solutions to problems. They are particularly useful in solving optimization and eigenvalue problems, which have applications in various fields including chemistry, finance, and logistics. VQAs are often considered hybrid algorithms, as they combine quantum subroutines with classical optimization techniques.

The core of a VQA is the Variational Quantum Eigensolver (VQE), which aims to find the ground state energy of a given Hamiltonian. The quantum part of the algorithm prepares a parameterized quantum state, known as the ansatz, and measures the expectation value of the Hamiltonian. The classical part then adjusts the parameters of the ansatz to minimize this expectation value. This iterative process continues until convergence is reached, resulting in an approximation of the ground state energy.

VQAs are gaining prominence in the era of near-term quantum computing, where error-corrected, fault-tolerant quantum computers are still under development. They are more resilient to noise and errors, making them suitable for current quantum hardware. Applications of VQAs include simulating molecular structures in quantum chemistry, portfolio optimization in finance, and solving complex logistical problems. A specific VQA, the Variational Quantum Classifier (VQC), is applicable to classification machine learning tasks. Their hybrid nature and adaptability make VQAs a versatile tool in the quantum computing toolkit.

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What is a Variational Quantum Algorithm

A VQA is an algorithm that is based on the variational method. It is essentially a series of educated guesses, in the form of parameterized quantum circuits, with each guess being refined by classical optimizers until an approximate solution is found. The first of two key takeaways is that guessing involves trial and error, and the error might include finding a non-optimal solution. The second takeaway is that VQAs do not find exact solutions, however they efficiently find approximate solutions even when the problems are hard for classical computers. 

For a wide overview of VQAs, including their challenges and prospects, be sure to read the Nature Reviews Physics paper “Variational quantum algorithms.” This paper is also downloadable from ArXiv.

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Mechanics of Variational Quantum Algorithms

VQAs use both classical and quantum computational resources. Because of this, they are also referred to as hybrid classical-quantum algorithms and hybrid quantum-classical algorithms. Although the quantum circuits are often run on classical simulators, it is important to note that they are capable of running on near-term quantum computers.

At a high level, VQAs have four steps:

1. Start with a parameterized quantum circuit, or ansatz, in which the parameters are angles of rotation in radians for some of the gates in the circuit.
2. Use a classical optimizer to iteratively update the parameters with the goal of minimizing a cost function, which indicates that the optimal solution is nearing.
3. Stop iterating once the cost function has converged at a minimum, indicating that an approximate solution has been found.
4. Use the approximate solution with classical post-processing to solve the eigenvalue or optimization problem at hand.

Choosing an ansatz and an optimizer is not a trivial task. Research is ongoing into novel ansätze and optimizers to solve an even broader range of problems.

The Role of Variational Quantum Algorithms in Quantum Computing

VQAs are a bridge between classical algorithms and fault-tolerant quantum algorithms. The current generation of quantum computers, what is called the Noisy Intermediate-Scale Quantum (NISQ) era, have too few qubits with too short lifetimes generating way too many errors to run fault-tolerant quantum algorithms with their highly coveted speedups.

Even with these constraints, however, an open question is whether or not computational advantages can nonetheless be realized with NISQ devices. A Zapata AI article titled “Variational What Now?” phrases this well: “Variational quantum algorithms provide a framework for trying to make use of these ‘along-the-way’ quantum computers.” Application-wise, VQAs have been adapted to a wide range of chemistry, finance, machine learning, and other problems in search of provable advantages.

VQA's Significance and Impact

A notable impact of VQAs could arguably have been to make NISQ computers usable, thus allowing the global quantum ecosystem to nurture, develop, and thrive. Without them, most of the quantum algorithms that would be available would be the foundational algorithms that aren’t of direct commercial interest and the fault-tolerant quantum computing (FTQC) algorithms that are too large to run. But having NISQ-capable algorithms available has given many thousands of users the incentive to create accounts and run these algorithms. These users will enjoy heightened preparedness over the next few years as logical qubits become available.

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