Variational Quantum Eigensolver (VQE) and Chemical Simulations

The Variational Quantum Eigensolver serves as a crucial algorithm for simulating molecules and chemical reactions.

In the realm of chemistry, determining the properties of atoms and molecules involves solving the Schrödinger equation. As the number of atoms increases, the complexity of the calculations escalates, making precise evaluations extremely time-consuming beyond a few atoms. While approximate methods exist, classical computers struggle to handle these computations efficiently when dealing with dozens of atoms.

VQE efficiently computes the eigenvalues of matrices, enabling us to determine the properties of larger molecules in ways classical computers cannot.

So, what exactly is an eigenvalue?

An eigenvalue of a matrix M is a scalar ? such that the equation Mv = ?v has a non-trivial solution, meaning the matrix M multiplied by the eigenvector yields a result that is not just zero.

The matrix used in this context is the one representing the Hamiltonian, which encapsulates the total energy of the atomic and molecular system, including all interactions among the atoms.

The quantum subroutine consists of two main steps:

1. Prepare the quantum state ansatz.
2. Measure the expectation value — akin to the “cost function” in machine learning.

We begin with an Ansatz, which represents an educated hypothesis about the quantum state we wish to simulate. The challenge lies in preparing this Ansatz effectively; it needs to be relatively accurate. The VQE algorithm operates in a hybrid manner, integrating a quantum step within a classical optimization loop to refine the quantum state. This process iterates until we identify the minimum eigenvalue of the Hamiltonian, signifying the lowest ground state energy.

The term "Variational" indicates the use of mathematical analysis involving variations—small changes in functions—to locate maxima and minima. This approach is reflected in our goal of minimizing the expectation values.

Due to the variational principle, the expectation value will always exceed the smallest eigenvalue of the Hamiltonian H. The quantum subroutine measures this expectation value, gradually aligning it with the smallest eigenvalue, which corresponds to the sought-after ground state energy.

This minimum energy can then be utilized in further calculations, such as modeling how the wavefunction (representing the molecule) evolves over time.

Beyond the scope of chemistry, solving large eigenvalue problems holds vast potential for practical applications in our daily lives, such as developing new materials that can withstand higher temperatures and stresses for aviation or enhancing battery efficiency.

We are entering the era of Noisy Intermediate Scale Quantum Devices (NISQ). Current quantum chips contain dozens to a few hundred physical qubits but suffer from high error rates and short coherence times, necessitating extensive error correction.

Nevertheless, algorithms like VQE can function effectively with just a handful of qubits. Additionally, it features a "shallow" circuit, meaning it involves fewer sequential gates.

This characteristic is vital because quantum information can only be retained for a limited duration, known as coherence time. All gates must be applied and the data read before the quantum bits decohere (lose their quantum states). Shallower circuits require shorter coherence times, allowing VQE to operate by calculating certain data with a quantum computer, relaying this information back to a classical system, and iterating the process. This framework is termed a hybrid classical-quantum algorithm.

A further example of variational algorithms includes Variational Quantum Factoring (VQF), which addresses RSA encryption factoring differently, along with the Quantum Approximate Optimization Algorithm (QAOA), applicable to scheduling issues.

Quantum Unconstrained Binary Optimization (QUBO)

Quantum annealers are designed to effectively translate QUBO and Ising problems onto quantum hardware. These systems can efficiently solve challenges like the Traveling Salesman Problem, scheduling dilemmas, optimal placement issues, graph coloring tasks, and even game optimization.

This capability is tremendously beneficial, as efficiency is a common challenge.

Binary objective functions that QUBO can solve efficiently can be represented as graphs, which can then be mapped onto the topology of a quantum chip. This allows us to take an optimization problem and formulate the corresponding Hamiltonian equation to minimize.

For instance, we can express it as: (f(a,b) = 5a + 7b - 3ab)

This can be visualized with a graph where nodes represent variables a and b, the duration spent in each city corresponds to the weights of the graph, and the edge between them carries the weight of 7.

The minimization of this function represents the "lowest energy state" of the system.

How does this operate on a quantum annealer? We apply a magnetic field to regulate the superposition of the qubits, affecting their probabilities of being in a state of one or zero. The weights are adjusted through qubit superposition, allowing us to manipulate the coupler, which correlates two qubits, corresponding to distances between cities in the Traveling Salesman Problem.

Finding the optimal solution can be likened to surveying a landscape to identify its lowest point. The lowest energy configuration signifies the "optimal" answer to these QUBO challenges.

Quantum Machine Learning Algorithms

Quantum machine learning encompasses various research directions:

• Quantum machine learning algorithms executed on quantum computers
• Classical algorithms inspired by quantum principles
• Classical machine learning techniques applied to quantum data

All of these approaches fit under the umbrella of "quantum machine learning."

A significant focus within quantum machine learning research involves developing quantum versions of established classical machine learning algorithms. Another avenue is creating entirely new quantum algorithms that consist of purely quantum steps.

Additionally, quantum subroutines can enhance machine learning. One concept is for a quantum subroutine to organize data in a manner that classical computers can process more efficiently, potentially alleviating bottlenecks in training. Ongoing research aims to uncover quantum equivalents for methods like k-nearest neighbors and support vector machines.

An intriguing avenue is "quantum learning," which explores how quantum technology can process input-output relations, optimize parameters, and devise effective learning strategies. How can we effectively represent data within a quantum framework?

There are proposals for quantum neural networks, primarily based on Hopfield networks, which excel in associative memory tasks derived from neuroscience rather than traditional machine learning.

Other methods explore quantum fuzzy feedforward networks or pattern recognition via adiabatic computing.

The implications of these developments are akin to those witnessed with conventional machine learning algorithms, which have already transformed various sectors. For instance, Google PageRank was one of the earliest machine learning implementations, and today, it significantly influences our daily lives.

Now, anyone can easily train a neural network with just a few lines of code. However, the volume of data continues to expand, and there is hope that quantum technology can facilitate more efficient processing and algorithm development.

Grover’s Algorithm for Efficient Search

The complexity of searching unsorted databases is typically O(n), according to Big O notation.

Grover’s algorithm, which operates in (O(sqrt{N})) time, represents the fastest quantum algorithm available for searching unsorted databases.

While discussions around Grover’s algorithm often suggest it merely "parallelizes" the search process, this is a misunderstanding. A more accurate interpretation is that Grover's algorithm inverts a function.

Grover's algorithm is applicable when a function yields a True result for one specific input while returning False for all others. The algorithm's task is to identify the input that results in a True output. It essentially scans through the function's inputs until it locates that singular input that causes the function to return true.

The concept of function inversion is relevant to database searching, as we can construct a function that produces a specific value (like "true") when it matches a desired entry in a database, and a different value ("false") for others.

Although Grover's algorithm can enhance search efficiency, it’s essential to determine whether your application can be effectively mapped onto quantum systems. Molecular structures can be more readily aligned with quantum states, but challenges arise when attempting to map, for instance, words onto qubit states.

While Grover's algorithm has potential for improved searching, it does not guarantee acceleration for every search task. Its true power is evident in scenarios where classical search methods struggle to scale effectively.

Problems like the Traveling Salesman and graph coloring (e.g., coloring a map with three colors while ensuring adjacent areas don’t share the same color) present complexities that become exponentially more challenging classically. Research is ongoing into leveraging Grover’s algorithm for solving NP-complete problems.

Shor’s Algorithm for Fast Number Factoring

Two prevalent cryptographic systems are Rivest-Shamir-Adleman (RSA) and elliptic curve cryptography (ECC). Most online communications rely on these encryption techniques, both of which are susceptible to quantum attacks.

RSA, for instance, depends on the complexity of factoring large numbers. While multiplying two prime numbers is straightforward, decomposing a large composite number into its prime factors is a daunting task, often requiring longer than the age of the universe to accomplish with classical systems.

Shor's algorithm can rapidly identify the prime factors of a number, effectively addressing the factoring challenge far more efficiently than classical computers. Developed in 1994, Shor’s algorithm works by identifying the "period" of a function.

The period of the function represents the oscillations of a result, akin to a frequency wave.

If we have an efficient means to determine the period of a known periodic function, we can quickly derive the factors.

The function can be expressed as: (f(x) = m^x mod N)

Shor's algorithm consists of five steps, with only one being quantum in nature.

A well-known classical function relates to frequency and time, referred to as the Fourier transform. Its quantum equivalent is the Quantum Fourier Transform, which transitions from the time domain to the frequency domain—precisely what we seek.

The ramifications of breaking encryption are significant. We must devise new encryption methods resilient against quantum computing threats.

Approximately 4000 logical qubits would be necessary to compromise an RSA key. However, these are idealized, "logical" qubits. Due to the need for error correction, the actual requirement is for many more physical qubits. John Preskill noted in his lecture on quantum information that a quantum computer would need around 10 million physical qubits to achieve 10,000 logical qubits. We are still far from achieving this benchmark, but it is a crucial consideration.

This potential for quantum computing suggests a future impact that extends beyond mere cybersecurity and telecommunications applications. Although the number of quantum algorithms may be limited, their implications could be vast since optimization and search issues permeate various sectors.

For additional insights, you can explore quantumalgorithmzoo.org for a comprehensive list of quantum algorithms!

Even the ability to simulate quantum states opens numerous possibilities in chemistry, materials science, and energy. As we continue to investigate more algorithms, they will assist in optimization and scheduling challenges, and eventually, even in search capabilities. The essential question remains—how do we map problems to a Hamiltonian?

See my YouTube video discussing these quantum algorithms here:

Originally published at https://www.amarchenkova.com.