What is Hamiltonian in quantum mechanics?

A fundamental idea guiding the total energy of a quantum system in quantum computing is the Hamiltonian. It is a Hermitian operator that controls system quantum state evolution over time. Developing novel quantum algorithms and modeling intricate quantum systems across many spheres of research and technology depend on an understanding and ability to control Hamiltonians.

Aspects of the Hamiltonian

Energy Operator: Mathematical operator corresponding to total energy of the quantum system; commonly abbreviated as H

Hermitian Nature: The Hamiltonian is a Hermitian operator, hence the eigenvalues—that is, the energies—are real.

Time Evolution: By use of the Schrödinger equation, the Hamiltonian controls the temporal development of a quantum state. The time-dependent Schrödinger equation is specifically

iℐ d|ψ(t)〉/dt = H(t)|ψ(t)〉,

where ℐ is the reduced Planck constant, H(t) is the time-dependent Hamiltonian, and ψ(t) is the quantum state at time t. Should the Hamiltonian be time-independent, the unitary time-evolution operator e^iHt provides the solution to the Schrödinger equation:

|ψ(t)〉 = e^iHt|ψ(0),

where ψ(0) is the starting state.

Components: Usually consisting of many terms, the Hamiltonian relates to various energy sources within the system. These could comprise particle kinetic, potential, and interaction energy. Many physical systems have the Hamiltonian as the sum of local terms, each operating on a tiny number of qubits.

Expectation Value: Given by 〈ψ|H|ψ〉, the Hamiltonian’s expectation value provides the system’s energy while it is in the state |ψ〉.

Quantum Simulation: Since quantum computers directly represent quantum states, they can effectively simulate quantum systems while simulating the behavior of quantum systems on classical computers is challenging since the number of parameters needed to describe a quantum state grows exponentially with the number of particles. In quantum computing, a major objective is Hamiltonian simulation—that is, the modeling of the temporal evolution controlled by the Hamiltonian of the system.

Importance of Hamiltonian in Quantum Computing

• Understanding Quantum Systems: Prediction of the behavior of quantum systems depends on the Hamiltonian. The interactions and energy of intricate quantum systems are modeled by their characteristics and framework.
• Quantum Algorithm Design: Especially in the framework of quantum simulation, quantum walks, and adiabatic quantum computing, the Hamiltonian is fundamental for designing numerous quantum algorithms. Quantum algorithms are designed and implemented with an awareness of and ability to control the Hamiltonian to produce desired computational results.
• Often including directly either directly implementing or estimating the time evolution produced by the system’s Hamiltonian, quantum simulation is the use of a quantum computer to replicate the behavior of another quantum system.
• In variational quantum algorithms (VQAs), when the aim is to identify the ground state of a Hamiltonian, for example, by reducing the energy of a parametrized state, the Hamiltonian is absolutely essential. Optimization, materials science, and quantum chemistry all depend on these methods in great part.
• Adiabatic quantum computing depends on progressively transforming the ground state of an initial Hamiltonian into the ground state of a problem Hamiltonian. This method has a time-dependent Hamiltonian, hence its design is absolutely important for the solution of optimization issues.
• Continuous-time quantum walks are described by a Hamiltonian grounded on the adjacency matrix of a graph. Several quantum algorithms depend on the simulation of the temporal evolution based on this kind of Hamiltonian.

How Hamiltonians Are Used in Simulations

• Using the Lie-Suzuki-Trotter approach—which divides the Hamiltonian into smaller, more doable bits—helps one replicate the time evolution under a Hamiltonian. Simulating the time evolution operator e-iHt requires a simple method.
• The Hamiltonian can be expressed as a linear combination of unitary operators in linear combination of Unitaries (LCU). This method enables the implementation of the time evolution utilizing controlled unitaries and ancilla qubits, which, in some situations can offer greater performance than Trotter techniques.
• Often with block-encodings of the Hamiltonian, quantum signal processing techniques may be applied effectively to perform Hamiltonian simulation.

Examples of Hamiltonians in Quantum Computing

1. Electronic Structure Hamiltonians: Usually in quantum chemistry, the Hamiltonian is of an electronic system nature. By modeling electron energy in molecules, this Hamiltonian allows molecular characteristics and chemical reaction simulations.
2. Local Hamiltonians, in condensed matter physics, are crucial for simulating many-body systems and phase transitions as interactions in condensed matter physics are restricted to close particles. Furthermore crucial for the study of Hamiltonian complexity are these Hamiltonians.
3. Problem Hamiltonians: Designed in the framework of optimization, Hamiltonians may be made to match the ground state of a computational problem’s solution.

What is Hamiltonian Formula?

The Hamiltonian formula is central to quantum mechanics, where it represents the observable corresponding to the total energy of a system.

Schrödinger Equation: The time evolution of a quantum state ∣ψ(t)⟩ is determined by the Hamiltonian H through the Schrödinger equation:

iℏd∣ψ(t)⟩/dt=H∣ψ(t)⟩

where H(t) is the Hamiltonian and |ψ(t) is the state of the system at time t. The constant ħ is Planck’s constant

When the Hamiltonian is not time-dependent, the Schrödinger equation can be written as

H|ψ(r)⟩ = E|ψ(r)⟩,

where H is the Hamiltonian, |ψ(r)> is the wave function, and E is the energy eigenvalue. In this case, the solution to the Schrödinger equation is of the form

∣ψ(t)⟩=e^iHt∣ψ(0)⟩

represents the unitary time evolution operator.

Energy and Eigenvalues: The expectation value of the Hamiltonian H for a state ∣ψ⟩ gives its energy:

⟨ψ∣H∣ψ⟩

The eigenvalues of H represent the possible energy levels of the system.

Lie-Suzuki-Trotter Decomposition: In Hamiltonian simulation, when H=∑_jH_j , the unitary operator e^iHt can be approximated by:

e^iHt≈(e^iH₁t/m e^iH₂t/m …..)^m

This method is useful for efficiently simulating quantum systems on quantum computers.

Application in k-Local Hamiltonians: A Hamiltonian can describe specific computational problems, such as the k-Local Hamiltonian problem, where the goal is to determine the minimal eigenvalue of H constructed as a sum of terms, each acting on a few qubits.

The above principles are important to the Hamiltonian in governing quantum system dynamics, energy characterization, and simulation.

Hamiltonian Simulation

Hamiltonian simulation is a cornerstone of quantum computing that enables the study of complex quantum systems. By simulating the time evolution dictated by the Schrödinger equation, quantum computers can offer insights into fundamental physics and enable advancements in various scientific and technological fields. The methods for Hamiltonian simulation, such as Trotter formulas, linear combinations of unitaries, and quantum walks, continue to be refined and improved as the field of quantum computation evolves.

Methods for Hamiltonian Simulation

Various techniques are used to implement the unitary evolution described by e^iHt as a quantum circuit. These methods include:

Lie-Suzuki-Trotter Methods:

• This method approximates the time evolution operator by breaking it into small time steps.
• If the Hamiltonian H is a sum of terms H = H1 + H2 + … + Hm, the unitary operator e^iHt can be approximated as e^iH₁t/re^iH₂t/r…e^iH_mt/r repeated r times.
• First-order Trotter: A basic Lie-Suzuki-Trotter approach involves approximating e^iHt by (e^iH₁t/re^iH₂t/r)r.
• Error: The approximation introduces an error, and the number of time steps, r, must be chosen such that the error is below a desired threshold.
• The number of gates in the resulting quantum circuit scales polynomially with the simulation time (t) and the inverse of the error (ε).
• Higher-order Trotter Formulas: Higher-order Trotter formulas achieve better accuracy and efficiency, reducing the dependence on simulation time to nearly linear. For example, (e^iH₁t/2me^iH₂t/me^iH₁t/2m)^m gives a better approximation.
• The Lie-Suzuki-Trotter approach is widely used due to its simplicity and general applicability.

Linear Combination of Unitaries (LCU):

• This technique involves expressing the Hamiltonian as a linear combination of unitary operators: H = Σ_j α_jV_j, where Vj are unitary operators.
• For example, if H consists of 2-local terms, each term can be written as a sum of Pauli matrices, which are unitary.
• The method then uses a combination of controlled unitaries and ancilla qubits to simulate the time evolution.
• The complexity of this method is dependent on the time t and the error ε, with the dependence on t being linear and on ε being logarithmic.
• LCU techniques are based on block encodings of the Hamiltonian.
• A block encoding of a matrix A is a unitary U such that A is in the top left block of U.
• Hamiltonian simulation is done via singular value transformation of these block-encoded matrices.
• This method can provide better performance than Trotter methods with respect to the desired precision.

Quantum Walks:

• Quantum walks, which are the quantum analogue of classical random walks, can be used to simulate Hamiltonian dynamics.
• The continuous-time quantum walk can be defined using a Hamiltonian that is based on the adjacency matrix of a graph.
• The Hamiltonian H can be simulated by performing phase estimation on the unitary evolution operator of a related discrete-time quantum walk.
• The complexity of simulating H for time t with this approach is O(t), achieving the optimal query complexity with respect to time.
• A discrete-time quantum walk can be implemented using two applications of an isometry T.
• The method relies on performing a phase estimation on a quantum walk.
• By using quantum signal processing techniques, this method can achieve optimal query complexity in both t and the error ε.

Sparse Hamiltonian Simulation:

• Sparse Hamiltonians are those where each row has a limited number of nonzero elements.
• These Hamiltonians are common in physical systems with local interactions.
• Efficient simulation of sparse Hamiltonians is achieved using techniques based on block-encodings, linear combinations of unitaries and quantum walks.
• The complexity of the Hamiltonian simulation depends on the sparsity (s) of the matrix and is approximately O(st + log(1/ε)).
• Quantum signal processing can be used with block-encoded matrices to achieve optimal query complexity for sparse Hamiltonian simulation.

Adiabatic Quantum Computing:

• Adiabatic quantum computing uses a time-dependent Hamiltonian H(t) whose initial ground state evolves into the ground state of the problem Hamiltonian if the evolution is slow enough.
• The system starts in the ground state of a simple Hamiltonian, which then is smoothly transformed into the problem Hamiltonian.
• The quantum adiabatic theorem guarantees that if the change is slow, the system remains in the ground state throughout the evolution.
• Hamiltonian simulation is needed to implement the time evolution under the adiabatic Hamiltonian.