12.2 Variational Quantum Eigensolver (VQE)
4 min read•july 30, 2024
The (VQE) is a hybrid quantum-classical algorithm that finds the lowest eigenvalue of a Hamiltonian. It combines a to prepare trial states with classical optimization to minimize energy, making it suitable for near-term quantum devices.
VQE has applications in quantum chemistry and materials science, where it can calculate molecular ground states and material properties. Its flexibility in ansatz design and optimization techniques allows for problem-specific tailoring, making it a versatile tool for quantum-enhanced optimization.
Variational Quantum Eigensolver
Overview and Components
Top images from around the web for Overview and Components
Variational quantum state preparation via quantum data buses – Quantum View original
Is this image relevant?
Quantum annealing initialization of the quantum approximate optimization algorithm – Quantum View original
Is this image relevant?
Variational Quantum Singular Value Decomposition – Quantum View original
Is this image relevant?
Variational quantum state preparation via quantum data buses – Quantum View original
Is this image relevant?
Quantum annealing initialization of the quantum approximate optimization algorithm – Quantum View original
Is this image relevant?
1 of 3
Top images from around the web for Overview and Components
Variational quantum state preparation via quantum data buses – Quantum View original
Is this image relevant?
Quantum annealing initialization of the quantum approximate optimization algorithm – Quantum View original
Is this image relevant?
Variational Quantum Singular Value Decomposition – Quantum View original
Is this image relevant?
Variational quantum state preparation via quantum data buses – Quantum View original
Is this image relevant?
Quantum annealing initialization of the quantum approximate optimization algorithm – Quantum View original
Is this image relevant?
1 of 3
- VQE combines preparation and with classical optimization to find the lowest eigenvalue of a given Hamiltonian
- The VQE algorithm consists of three main components:
- Parameterized quantum circuit (ansatz) prepares a trial wave function approximating the ground state of the target Hamiltonian
- (typically the of the Hamiltonian) is evaluated by measuring the output of the ansatz circuit and guides the optimization process
- Classical optimization routine iteratively updates the parameters of the ansatz to minimize the cost function, effectively finding the best approximation to the ground state
- VQE leverages the strengths of both quantum and classical computation
- Quantum hardware efficiently evaluates the cost function
- Classical resources optimize the parameters
Algorithm Workflow
- The ansatz is a quantum circuit with tunable parameters that prepares a trial wave function
- Trial wave function is an approximation of the ground state of the target Hamiltonian
- The cost function is evaluated by measuring the output of the ansatz circuit
- Expectation value of the Hamiltonian is a common choice for the cost function
- The classical optimization routine iteratively updates the parameters of the ansatz
- Goal is to minimize the cost function
- Effectively finds the best approximation to the ground state
- The process continues until criteria are met or a maximum number of iterations is reached
Variational Quantum Circuits
Ansatz Design Considerations
- Variational quantum circuits, or ansatzes, are parameterized quantum circuits designed to prepare trial wave functions for VQE
- The choice of ansatz depends on the problem at hand and the available quantum hardware
- Balances expressibility, entangling capability, and circuit depth
- Hardware-efficient ansatzes maximize the use of available quantum resources while minimizing circuit depth
- Suitable for near-term quantum devices
- Examples include the Ry variational form (layers of single- rotations and entangling gates)
- Problem-specific ansatzes incorporate knowledge about the target Hamiltonian or the expected structure of the ground state
- Potentially leads to faster convergence and better approximations
- Examples include the (UCC) ansatz (quantum chemistry) and the Hamiltonian (HVA) (general Hamiltonians)
Constructing and Analyzing Ansatzes
- Ansatzes can be constructed using a modular approach
- Combines basic building blocks such as single-qubit rotations, entangling gates (CNOT, CZ), and parameter-sharing techniques
- The expressibility and entangling capability of an ansatz can be analyzed using various measures
- Meyer-Wallach entanglement measure quantifies the amount of entanglement in the ansatz
- Expressibility score assesses the ability of the ansatz to represent a wide range of quantum states
- Analyzing the properties of ansatzes helps in designing effective circuits for VQE and understanding their limitations
Classical Optimization for VQE
Optimization Methods
- Classical optimization routines minimize the cost function in VQE by iteratively updating the parameters of the variational quantum circuit
- Gradient-based optimization methods (gradient descent) can be used when the gradient of the cost function with respect to the parameters can be efficiently computed
- allows for the estimation of gradients using a finite difference approach, requiring additional quantum circuit evaluations
- Gradient-free optimization methods (, , ) can be used when gradients are difficult to compute or the cost function landscape is complex
- techniques (Gaussian process regression) model the cost function landscape and guide the search for optimal parameters
Optimization Techniques and Considerations
- The choice of optimizer depends on various factors
- Number of parameters
- Complexity of the cost function landscape
- Available computational resources
- Techniques to improve the convergence and robustness of the optimization process include:
- Parameter initialization strategies (random, heuristic)
- Learning rate scheduling (constant, adaptive)
- Early stopping criteria (based on convergence or maximum iterations)
- Careful selection and tuning of optimization techniques are crucial for the success and efficiency of VQE
VQE Applications in Science
Quantum Chemistry
- VQE has been successfully applied to solve quantum chemistry problems
- Calculating the ground state energies of molecules
- Studying chemical reactions
- The electronic structure Hamiltonian is mapped to a qubit Hamiltonian using techniques like the Jordan-Wigner or
- The UCC ansatz is commonly used for quantum chemistry problems
- Incorporates the structure of the electronic wave function
- VQE can be extended to using techniques like the (QSE) or the (FSM)
Material Science
- VQE can be used to study materials properties
- Band structure
- Density of states
- Transport properties of solid-state systems
- The Hamiltonian of the material system is discretized and mapped to a qubit Hamiltonian
- Often uses the tight-binding or
- Problem-specific ansatzes can be employed for material science applications
- Hamiltonian variational ansatz (HVA)
- (FermiNet)
- techniques (, ) can be incorporated into VQE
- Reduces the impact of hardware noise on the computed energies and properties
Key Terms to Review (35)
Accuracy: Accuracy is the measure of how close a predicted value is to the actual value in a dataset. It reflects the percentage of correct predictions made by a model compared to the total number of predictions, serving as a key performance metric in various machine learning algorithms.
Bayesian Optimization: Bayesian Optimization is a probabilistic model-based optimization technique that is particularly useful for optimizing expensive-to-evaluate functions. It builds a surrogate model, often a Gaussian process, to predict the function's behavior and then uses this model to select the most promising points to evaluate, balancing exploration and exploitation. This approach is especially relevant in contexts where evaluations are costly, like in the Variational Quantum Eigensolver, where finding optimal parameters can be computationally intensive.
Bravyi-Kitaev Transformation: The Bravyi-Kitaev transformation is a mathematical technique used to map quantum states into a different basis, particularly in the context of qubit systems and fermionic systems. This transformation is essential for efficiently representing fermionic operators as qubit operators, which simplifies computations in quantum algorithms such as the Variational Quantum Eigensolver.
Classical optimizer: A classical optimizer is an algorithm or method used in classical computing to find the best solution to a given problem, often by minimizing or maximizing a cost function. These optimizers play a crucial role in hybrid quantum-classical frameworks, where they are employed to fine-tune parameters of quantum circuits or variational methods to achieve optimal results. Classical optimizers are especially significant in the context of variational quantum algorithms, where they iteratively adjust parameters based on feedback from quantum computations.
Cobyla: Cobyla (Constrained Optimization BY Linear Approximations) is a numerical optimization algorithm used to minimize a function subject to constraints, particularly useful in problems with nonlinear objective functions and constraints. This algorithm is widely applied in variational quantum algorithms, such as the Variational Quantum Eigensolver (VQE), where it helps optimize the parameters of quantum circuits to find the ground state energy of quantum systems efficiently.
Convergence: Convergence refers to the process by which a sequence of approximations or iterative results approaches a final value or solution. In the context of quantum algorithms, particularly in variational methods, convergence is essential for ensuring that the computed results reliably reflect the target eigenvalues and eigenstates of a quantum system. The speed and reliability of convergence can significantly influence the performance and accuracy of quantum computing applications.
Cost function: A cost function is a mathematical representation used to measure the difference between the predicted output of a model and the actual output. It quantifies how well a model performs, guiding the optimization process to minimize this difference, which is crucial in various optimization techniques.
Error mitigation: Error mitigation refers to techniques used to reduce the impact of errors in quantum computing, particularly during computations. These errors can arise from various sources, such as decoherence or imperfect gate operations. Effective error mitigation is crucial for improving the reliability of quantum algorithms and achieving accurate results in processes like optimization and simulation.
Excited States: Excited states refer to the higher energy levels of a quantum system, where electrons or other particles have absorbed energy and moved from their ground state to a state of higher energy. This concept is crucial in understanding how quantum systems, such as molecules or atoms, behave and interact under various conditions, particularly in quantum algorithms designed to find the ground state energy of systems using variational methods.
Expectation Value: The expectation value is a fundamental concept in quantum mechanics that represents the average outcome of a measurement for a quantum state. It provides a way to calculate the expected result of an observable, defined mathematically as the weighted sum of all possible measurement outcomes, each multiplied by their corresponding probability. This concept is crucial for understanding quantum measurement and plays a significant role in various algorithms used in quantum computing.
Fermionic neural network: A fermionic neural network is a type of quantum neural network specifically designed to represent and manipulate fermionic states, which obey the Pauli exclusion principle. These networks utilize quantum mechanics to efficiently model complex quantum systems, making them particularly useful in tasks such as quantum chemistry and materials science. By leveraging the unique properties of fermions, these networks can capture correlations and entanglements that are fundamental in quantum systems.
Folded spectrum method: The folded spectrum method is a technique used in quantum computing, particularly in the context of the Variational Quantum Eigensolver (VQE), that involves manipulating and analyzing the energy spectrum of a quantum system. This method helps to enhance the efficiency of energy calculations by folding the spectrum, allowing for better convergence properties and reduced computational complexity during optimization. By adjusting the parameter space effectively, it can lead to more accurate estimations of the ground state energy and other relevant eigenvalues.
Ground state energy: Ground state energy is the lowest possible energy that a quantum mechanical system can have. It serves as a fundamental concept in quantum mechanics and plays a crucial role in determining the stability and behavior of quantum systems, particularly in the context of finding solutions for complex Hamiltonians using algorithms like the Variational Quantum Eigensolver (VQE). Understanding ground state energy is essential for accurately approximating the energy levels of molecules and materials in quantum chemistry.
Hamiltonian Simulation: Hamiltonian simulation refers to the process of simulating the time evolution of quantum systems governed by a Hamiltonian operator. This technique is crucial for understanding the dynamics of quantum systems and plays a significant role in variational quantum algorithms, particularly in efficiently estimating the ground state energy of a quantum system.
Hubbard Model: The Hubbard Model is a theoretical framework used to study interacting particles in a lattice, primarily in the context of condensed matter physics. It focuses on the balance between the kinetic energy of particle hopping and the potential energy due to on-site interactions, which is crucial for understanding phenomena like magnetism and superconductivity in materials.
John Preskill: John Preskill is a prominent theoretical physicist known for his significant contributions to quantum computing and quantum information theory. He is particularly recognized for coining the term 'quantum supremacy,' which refers to the point at which a quantum computer can perform tasks that classical computers cannot achieve in a reasonable time frame. His work has implications across various areas, including algorithms, machine learning, and the development of quantum technologies.
Jordan-Wigner Transformation: The Jordan-Wigner transformation is a mathematical technique used to map spin systems onto fermionic systems, allowing the representation of spin operators in terms of fermionic creation and annihilation operators. This transformation is essential for studying quantum many-body systems, particularly in the context of quantum computing, as it enables the use of fermionic states to describe quantum states effectively.
Measurement: Measurement in quantum mechanics refers to the process of obtaining information about a quantum system, often leading to the collapse of the system's wave function into a definite state. This process is essential in determining the values of observable quantities and plays a crucial role in quantum computing, where the outcomes of measurements can influence subsequent computations and algorithms.
Nelder-Mead Simplex: The Nelder-Mead Simplex is a popular optimization algorithm used for minimizing a function in a multi-dimensional space without requiring the gradient of the function. This method works by iteratively adjusting a simplex, which is a geometric figure consisting of a set of points, to find the minimum point. In the context of variational quantum eigensolvers, it plays a crucial role in optimizing parameters to minimize the energy of quantum states efficiently.
Parameter-shift rule: The parameter-shift rule is a technique used in quantum computing to compute the gradient of a quantum circuit's output with respect to its parameters by measuring the output at shifted parameter values. This method enables efficient optimization of parameters in quantum algorithms, particularly within variational methods, where fine-tuning is essential for finding optimal solutions.
Powell's Method: Powell's Method is an optimization algorithm used to find the minimum of a function without requiring the calculation of derivatives. It combines techniques from conjugate direction methods to efficiently navigate the parameter space, making it particularly useful in variational approaches like the Variational Quantum Eigensolver (VQE), which seeks to minimize energy in quantum systems.
Quantum circuit: A quantum circuit is a model for quantum computation, where a sequence of quantum gates is applied to qubits to perform specific operations on quantum information. These circuits harness the principles of superposition and entanglement, allowing for complex computations that classical circuits cannot achieve efficiently. The design and representation of quantum circuits are fundamental in various quantum algorithms and applications, making them central to the study of quantum machine learning and its integration with classical systems.
Quantum gate decomposition: Quantum gate decomposition is the process of breaking down complex quantum gates into a series of simpler, more manageable gates that can be implemented on a quantum computer. This technique is crucial because many quantum algorithms, including those used in variational quantum eigensolvers, require specific gate implementations to achieve the desired quantum operations efficiently.
Quantum State: A quantum state is a mathematical representation of a quantum system, encapsulating all the information about the system’s properties and behavior. Quantum states can exist in multiple configurations simultaneously, which allows for unique phenomena such as interference and entanglement, essential for the workings of quantum computing.
Quantum subspace expansion: Quantum subspace expansion is a method used in quantum algorithms to explore and represent the solutions of a quantum system by expanding its state space into a lower-dimensional subspace. This technique is particularly relevant for optimizing variational parameters in quantum circuits, allowing for efficient approximations of the ground state energy and other properties of quantum systems.
Quantum superposition: Quantum superposition is a fundamental principle of quantum mechanics that allows quantum systems to exist in multiple states simultaneously until measured or observed. This concept underpins many unique properties of quantum systems, leading to phenomena like interference and enabling the potential for exponentially faster computations in quantum computing.
Qubit: A qubit, or quantum bit, is the fundamental unit of quantum information, analogous to a classical bit but capable of existing in multiple states simultaneously due to the principles of quantum mechanics. Unlike classical bits, which can be either 0 or 1, qubits can be in superpositions of these states, allowing for vastly more complex computations and interactions in quantum computing.
Richard Feynman: Richard Feynman was a renowned American theoretical physicist known for his fundamental contributions to quantum mechanics and quantum electrodynamics. He is celebrated for his ability to communicate complex scientific ideas in an accessible manner, bridging the gap between theoretical physics and practical applications, especially in the context of quantum computing and algorithms.
Richardson Extrapolation: Richardson extrapolation is a mathematical technique used to improve the accuracy of numerical approximations by combining results from calculations at different step sizes. It allows for the reduction of truncation errors in numerical methods, which is crucial in variational algorithms that aim to determine ground state energies. By leveraging the results from coarser and finer discretizations, this technique enhances precision and convergence in simulations, making it particularly relevant in quantum computing applications.
Tight-Binding Model: The tight-binding model is a theoretical framework used to describe the electronic properties of solids by considering the behavior of electrons in a lattice structure. It simplifies the complex interactions in solid-state systems by focusing on how electrons hop between neighboring atomic sites, making it a valuable tool for studying materials at the quantum level, especially when connected to quantum algorithms like the Variational Quantum Eigensolver (VQE). This model is particularly useful in understanding band structures and energy levels in condensed matter physics.
Unitary coupled cluster: Unitary coupled cluster is a quantum computational technique used to approximate the ground state energy of quantum systems by employing a parametrized unitary operator. This method enhances the Variational Quantum Eigensolver (VQE) by incorporating the effects of electron correlation through a series of unitary transformations, allowing for more accurate energy estimations in quantum chemistry problems.
Variational ansatz: A variational ansatz is a trial wave function used in quantum mechanics to approximate the ground state energy of a quantum system. By selecting a parameterized form for the wave function, one can optimize these parameters to minimize the energy expectation value, allowing for an efficient estimation of the ground state energy. This approach is essential in methods like the Variational Quantum Eigensolver (VQE), where quantum computers are utilized to find approximate solutions to complex quantum problems.
Variational Quantum Eigensolver: The Variational Quantum Eigensolver (VQE) is a hybrid quantum-classical algorithm designed to find the lowest eigenvalue of a Hamiltonian, which represents the energy of a quantum system. By leveraging the principles of superposition and entanglement, VQE optimizes a parameterized quantum circuit to minimize the energy expectation value, combining the strengths of quantum computing and classical optimization techniques.
Wavefunction: A wavefunction is a mathematical function that describes the quantum state of a system, encapsulating all the information about the probabilities of finding a particle in various positions and states. It serves as a crucial element in quantum mechanics, allowing for the representation of superposition and entanglement by enabling particles to exist in multiple states simultaneously. This function is central to various quantum algorithms and approaches, including techniques for finding the ground state energy of quantum systems.
Zero-noise extrapolation: Zero-noise extrapolation is a technique used in quantum computing to reduce the impact of noise on quantum computations by extrapolating results obtained at different noise levels to estimate the ideal result at zero noise. This method is particularly relevant in variational algorithms, where the presence of noise can significantly affect the accuracy of computed results. By analyzing the effects of noise and systematically reducing it through extrapolation, this approach enhances the reliability of quantum computations, particularly in methods like the Variational Quantum Eigensolver.