Ground State

The ground state of a quantum system refers to its lowest energy state. It is the state of the system when it is in a stable equilibrium, with no excess energy. In this state, the system's wave function is described by a specific mathematical form, and the corresponding energy is known as the ground state energy.

To help build an intuition for why there is such a thing as a ground state energy level, user Quirinus on StackExchange Physics offers up a brief proof. Simply put, an assumption that energy E must be greater than negative infinity, or E>−∞, means that there must be a minimum to the amount of energy a particle can have. And to help build an intuition as to why Quirinus uses negative infinity instead of zero, answerer Eric Toombs notes on Quora that the ground state energies of atoms and molecules are usually negative, describing the amount of energy required to ionize the system, which is also known as the ionization energy. The long answer on Quora also discusses zero potential energy and positive potential energy.

In quantum mechanics, the energy levels of a system are quantized, meaning they can only take specific discrete values. The ground state is the state corresponding to the lowest of these energy levels. It is often denoted by the symbol |0> and is the state with the highest probability of being observed when the system is at absolute zero temperature.

User Adam on StackExchange Physics, notes that energies can be shifted. In doing so, however, the difference in energy between the ground state versus excited state principally remains unchanged. The difference is quantized and must be a discrete value.

In quantum computing, the ground state plays a vital role in various algorithms and computational models. For example, in adiabatic quantum computation and quantum annealing, the goal is often to find the ground state of a specific Hamiltonian, which corresponds to the solution of a computational problem. The ground state energy of a ground state atom or system is one of its ground state quantum numbers, which describe the fundamental properties of an atom’s or system’s electrons.

Different quantum systems have distinct ground states, depending on their Hamiltonians and physical properties. For example, the ground state of a simple harmonic oscillator is a Gaussian wave function, while the ground state of a hydrogen atom is described by a specific spherical harmonic function.

Above the ground state, a quantum system may have various excited states, each corresponding to a higher energy level. Transitions between the ground state and excited states can occur through interactions with external fields or particles, leading to phenomena such as absorption and emission of photons.

In quantum chemistry, the ground state of a molecule or atom is of particular interest, as it provides essential information about the system's stability, reactivity, and other chemical properties. Computational methods to accurately determine the ground state are central to modern quantum chemistry.

Finding the exact ground state of complex quantum systems can be a highly non-trivial problem, especially for systems with strong correlations and interactions. Techniques such as variational methods, Monte Carlo simulations, and tensor network algorithms are used to approximate the ground state in various contexts.

The concept of the ground state extends beyond quantum physics, influencing areas such as materials science, nanotechnology, and even biology. Understanding and manipulating the ground state is key to developing new materials, technologies, and insights into the fundamental behavior of matter. 

On Quora, answerer Keith Allpress offers up a couple of analogies that are extended here:  

• Except for the fact you can’t be completely broke, quantization is like having coins and notes in that a particle can only add and lose energy in discrete amounts; particles also live paycheck-to-paycheck, always decaying to this minimum amount, the ground state

• Gaseous molecules are “little springy dumbells” that never stop vibrating; the minimum vibration is the ground state, and heating causes the molecules to become excited

What is Ground State in Quantum Computing

In quantum computing, the term “ground state energy” can have two different meanings. The first meaning is as described above, which is finding the ground state quantum numbers of a ground state atom with its ground state electron. While this has notable applications in chemistry, the relevant algorithms can also be used to solve completely unrelated problems. If a problem, such as an optimization problem, can be formulated as a Hamiltonian, which is the total energy of a system in chemistry, then the optimization problem can be solved, as well. The process of calculating ground state energies, therefore, has potential applicability across a broad number of industries. 

The algorithms that would be used for this are:

• Quantum Phase Estimation (QPE) and its variations, which cannot be run on today’s noisy hardware, but will solve chemistry and other problems once fault tolerance is achieved
• Variational Quantum Eigensolver (VQE) and its variations, which are usually found calculating the ground state energies of small molecules for chemistry applications 
• Quantum Approximate Optimization Algorithm (QAOA) and its variations, which are used to solve, as the name suggests, a wide range of optimization problems 
• Quantum Annealing, which uses different hardware than the other three algorithms, uses quantum superposition and quantum tunneling to solve various optimization problems

It’s worth noting that these applications are all computationally intensive problems. As these problems scale in size and complexity, the classical computing resources required to find solutions grows rapidly, perhaps even exponentially. Exact solutions then become classically intractable, and approximate solutions have to be settled for. One of the promises of quantum computing is to efficiently and precisely solve these types of problems.

The second meaning of “ground state energy” is the 0 when the measurement of a qubit returns a 0. Computationally, the difference between ground state versus excited state is the 0 and 1 we hear about when we hear such phrases as “0 and 1 at the same time.” A measurement collapses the wave function of the qubit, and it is found in one of these two states. As previously noted, the ground state is usually denoted as |0>, but this is simplified to 0 for classical programming purposes.

The two possible measurement outcomes of each qubit, therefore, are:

• 0, which means the qubit was found in its ground state |0>
• 1, which means the qubit was found in its excited state |1>

Some quantum computers allow the exploration of higher energy states, in which case measurements may include 2, 3, 4, or higher. The vast majority of current hardware and algorithms only support 0 and 1, hence the term “qubit,” with a “b” referring to binary. Adding a third energy level, the 2, would make the term “qubit” a misnomer, and so the term “qutrit” is used instead. Adding even higher energy levels requires use of the term “qudit.”

Quantum computers have what is called an initial state. The states evolve through laser or microwave pulses, and then select qubits are measured. In most cases, the qubits start off in their ground states, at |0>, and then they are evolved into initial states for computation. The initial states may be as simple as a uniform superposition, but they may also encode classical information within amplitudes or angles, or they might encode quantum information from Quantum Random Access Memory (QRAM). 

In rare cases, from a user’s perspective, anyway, qubits don’t have to start in their ground states, and can be initialized in other basis states. For example, when qubits can be initialized in their |+> states, and uniform superpositions are required, the algorithms proper can commence immediately. The quantum phase estimation (QPE) algorithm starts qubits in a mix of |0>, |1>, and |+> states, and some hardware may support initializing qubits in this mix of states. But, again, the vast majority of hardware and algorithms start their qubits at |0>, which represents the ground states of those qubits.

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