Variational Quantum Algorithms
Marco Cerezo,Marco Cerezo,Andrew Arrasmith,Andrew Arrasmith,Ryan Babbush,Simon C. Benjamin,Suguru Endo,Keisuke Fujii,Jarrod R. McClean,Kosuke Mitarai,Kosuke Mitarai,Xiao Yuan,Xiao Yuan,Lukasz Cincio,Lukasz Cincio,Patrick J. Coles,Patrick J. Coles +16 more
TL;DR: An overview of the field of Variational Quantum Algorithms is presented and strategies to overcome their challenges as well as the exciting prospects for using them as a means to obtain quantum advantage are discussed.
read more
Abstract: Applications such as simulating complicated quantum systems or solving large-scale linear algebra problems are very challenging for classical computers due to the extremely high computational cost. Quantum computers promise a solution, although fault-tolerant quantum computers will likely not be available in the near future. Current quantum devices have serious constraints, including limited numbers of qubits and noise processes that limit circuit depth. Variational Quantum Algorithms (VQAs), which use a classical optimizer to train a parametrized quantum circuit, have emerged as a leading strategy to address these constraints. VQAs have now been proposed for essentially all applications that researchers have envisioned for quantum computers, and they appear to the best hope for obtaining quantum advantage. Nevertheless, challenges remain including the trainability, accuracy, and efficiency of VQAs. Here we overview the field of VQAs, discuss strategies to overcome their challenges, and highlight the exciting prospects for using them to obtain quantum advantage.
read more
Chat with Paper
AI Agents for this Paper
Find similar papers on Google Scholar, PubMed and Arxiv
Write a critical review of this paper
Analyze citations of this paper to find unaddressed research gaps
Figures
FIG. 1. Schematic diagram of a Variational Quantum Algorithm (VQA). The inputs to a VQA are: a cost function C(θ), with θ a set of parameters that encodes the solution to the problem, an ansatz whose parameters are trained to minimize the cost, and (possibly) a set of training data {ρk} used during the optimization. Here, the cost can often be expressed in the form in Eq. (3), for some set of functions {fk}. Also, the ansatz is shown as a parameterized quantum circuit (on the left), which is analogous to a neural network (also shown schematically on the right). At each iteration of the loop one uses a quantum computer to efficiently estimate the cost (or its gradients). This information is fed into a classical computer that leverages the power of optimizers to navigate the cost landscape C(θ) and solve the optimization problem in Eq. (1). Once a termination condition is met, the VQA outputs an estimate of the solution to the problem. The form of the output depends on the precise task at hand. The red box indicates some of the most common types of outputs. ![FIG. 5. Quantum Approximate Optimization Algorithm (QAOA). a. Schematic representation of the Trotterized adiabatic transformation in the ansatz. The algorithm only loosely follows the evolution of the ground state of H(t) = (1−t)HM+tHP for every t ∈ [0, 1], as one is interested in making the final state close to the ground state of the problem Hamiltonian HP , with HM being a mixer Hamiltonian. The free parameters {βl}pl=1 and {γl} p l=1 are trained, with p being the number of QAOA rounds. b. Problem Hamiltonian HP and graph 〈jk〉 for a Max-Cut task. Each node in the graph (circle) represents a spin. Vertices connecting two nodes indicate an interaction σzjσzk in HP , with σ z k the Pauli z operator on spin k. The solution is encoded in the ground state of HP where some spins are pointing up (green) whereas others point down (blue).](/figures/figure5-1-2ctjdzik41gj.png)
FIG. 5. Quantum Approximate Optimization Algorithm (QAOA). a. Schematic representation of the Trotterized adiabatic transformation in the ansatz. The algorithm only loosely follows the evolution of the ground state of H(t) = (1−t)HM+tHP for every t ∈ [0, 1], as one is interested in making the final state close to the ground state of the problem Hamiltonian HP , with HM being a mixer Hamiltonian. The free parameters {βl}pl=1 and {γl} p l=1 are trained, with p being the number of QAOA rounds. b. Problem Hamiltonian HP and graph 〈jk〉 for a Max-Cut task. Each node in the graph (circle) represents a spin. Vertices connecting two nodes indicate an interaction σzjσzk in HP , with σ z k the Pauli z operator on spin k. The solution is encoded in the ground state of HP where some spins are pointing up (green) whereas others point down (blue). ![FIG. 4. Variational Quantum Eigensolver (VQE) implementation. The VQE algorithm can be used to estimate the ground state energy EG of a molecule. The interactions of the system are encoded in a HamiltonianH, usually expressed as a linear combination of simple operators hk with coefficients ck. Taking H as input, VQE outputs an estimate ẼG of the ground-state energy. The lower part of the figure shows the results of a VQE implementation for the electronic structure problem of an H2 molecule, whose exact energy is shown as a dashed line. The experimental results were obtained using two of the five qubits in one of IBM’s superconducting quantum processors (the inset illustrates qubit connectivity with Q0 . . . Q4 denoting the qubits ). Due to the presence of hardware noise the estimated energy ẼG has a gap with the true energy. In fact, amplifying the noise strength (that is increasing the quantity s), deteriorates the solution quality. However, as discussed below, one can use error mitigation techniques to improve the solution quality. Figure adapted from Ref. [106], Springer Nature Limited.](/figures/figure4-1-6p1qlk7ybq2h.png)
FIG. 4. Variational Quantum Eigensolver (VQE) implementation. The VQE algorithm can be used to estimate the ground state energy EG of a molecule. The interactions of the system are encoded in a HamiltonianH, usually expressed as a linear combination of simple operators hk with coefficients ck. Taking H as input, VQE outputs an estimate ẼG of the ground-state energy. The lower part of the figure shows the results of a VQE implementation for the electronic structure problem of an H2 molecule, whose exact energy is shown as a dashed line. The experimental results were obtained using two of the five qubits in one of IBM’s superconducting quantum processors (the inset illustrates qubit connectivity with Q0 . . . Q4 denoting the qubits ). Due to the presence of hardware noise the estimated energy ẼG has a gap with the true energy. In fact, amplifying the noise strength (that is increasing the quantity s), deteriorates the solution quality. However, as discussed below, one can use error mitigation techniques to improve the solution quality. Figure adapted from Ref. [106], Springer Nature Limited. ![FIG. 7. Qubit trajectories on the Bloch sphere with the Zero-Noise Extrapolation (ZNE) technique. The accuracy of a noisy quantum computer can be improved with the ZNE error mitigation method. a. Here, one repeats a given calculation with different levels of noise. The green curve corresponds to a rotation on the Bloch sphere with a higher noise level than that leading to the red curve. b. Taking data from the red and green curves, ZNE can be used to estimate what the trajectory (blue) would be like in the absence of noise. Adapted from Ref. [106], Springer Nature Limited.](/figures/figure7-1-6q3cb3w7sqof.png)
FIG. 7. Qubit trajectories on the Bloch sphere with the Zero-Noise Extrapolation (ZNE) technique. The accuracy of a noisy quantum computer can be improved with the ZNE error mitigation method. a. Here, one repeats a given calculation with different levels of noise. The green curve corresponds to a rotation on the Bloch sphere with a higher noise level than that leading to the red curve. b. Taking data from the red and green curves, ZNE can be used to estimate what the trajectory (blue) would be like in the absence of noise. Adapted from Ref. [106], Springer Nature Limited. ![FIG. 6. Barren plateau (BP) phenomenon. a. Variance of the cost function partial derivative, Var(∂θ1,1E), for a particular parameter θ1,1 in the ansatz versus number of qubits (n). Results were obtained from a Variational Quantum Eigensolver implementation with a deep unstructured ansatz. The y-axis is on a log scale. As the number of qubits increases the variance vanish exponentially with the system size. b. Visualization of the landscape of a global cost function which exhibits a BP for the quantum compilation implementation, . The orange (blue) landscape was obtained for n = 24 (n = 4) qubits. As the number of qubits increases, the landscape becomes flatter. Moreover, this cost also exhibits the narrow gorge phenomenon [166], where the volume of parameters leading to small cost values shrinks exponentially with n. Panel a is adapted from Ref. [194], CC BY 4.0; Panel b is adapted from Ref. [166], CC BY 4.0.](/figures/figure6-1-35tu42xcg88v.png)
FIG. 6. Barren plateau (BP) phenomenon. a. Variance of the cost function partial derivative, Var(∂θ1,1E), for a particular parameter θ1,1 in the ansatz versus number of qubits (n). Results were obtained from a Variational Quantum Eigensolver implementation with a deep unstructured ansatz. The y-axis is on a log scale. As the number of qubits increases the variance vanish exponentially with the system size. b. Visualization of the landscape of a global cost function which exhibits a BP for the quantum compilation implementation, . The orange (blue) landscape was obtained for n = 24 (n = 4) qubits. As the number of qubits increases, the landscape becomes flatter. Moreover, this cost also exhibits the narrow gorge phenomenon [166], where the volume of parameters leading to small cost values shrinks exponentially with n. Panel a is adapted from Ref. [194], CC BY 4.0; Panel b is adapted from Ref. [166], CC BY 4.0.
![FIG. 5. Quantum Approximate Optimization Algorithm (QAOA). a. Schematic representation of the Trotterized adiabatic transformation in the ansatz. The algorithm only loosely follows the evolution of the ground state of H(t) = (1−t)HM+tHP for every t ∈ [0, 1], as one is interested in making the final state close to the ground state of the problem Hamiltonian HP , with HM being a mixer Hamiltonian. The free parameters {βl}pl=1 and {γl} p l=1 are trained, with p being the number of QAOA rounds. b. Problem Hamiltonian HP and graph 〈jk〉 for a Max-Cut task. Each node in the graph (circle) represents a spin. Vertices connecting two nodes indicate an interaction σzjσzk in HP , with σ z k the Pauli z operator on spin k. The solution is encoded in the ground state of HP where some spins are pointing up (green) whereas others point down (blue).](/figures/figure5-1-2ctjdzik41gj.webp)
![FIG. 4. Variational Quantum Eigensolver (VQE) implementation. The VQE algorithm can be used to estimate the ground state energy EG of a molecule. The interactions of the system are encoded in a HamiltonianH, usually expressed as a linear combination of simple operators hk with coefficients ck. Taking H as input, VQE outputs an estimate ẼG of the ground-state energy. The lower part of the figure shows the results of a VQE implementation for the electronic structure problem of an H2 molecule, whose exact energy is shown as a dashed line. The experimental results were obtained using two of the five qubits in one of IBM’s superconducting quantum processors (the inset illustrates qubit connectivity with Q0 . . . Q4 denoting the qubits ). Due to the presence of hardware noise the estimated energy ẼG has a gap with the true energy. In fact, amplifying the noise strength (that is increasing the quantity s), deteriorates the solution quality. However, as discussed below, one can use error mitigation techniques to improve the solution quality. Figure adapted from Ref. [106], Springer Nature Limited.](/figures/figure4-1-6p1qlk7ybq2h.webp)
![FIG. 7. Qubit trajectories on the Bloch sphere with the Zero-Noise Extrapolation (ZNE) technique. The accuracy of a noisy quantum computer can be improved with the ZNE error mitigation method. a. Here, one repeats a given calculation with different levels of noise. The green curve corresponds to a rotation on the Bloch sphere with a higher noise level than that leading to the red curve. b. Taking data from the red and green curves, ZNE can be used to estimate what the trajectory (blue) would be like in the absence of noise. Adapted from Ref. [106], Springer Nature Limited.](/figures/figure7-1-6q3cb3w7sqof.webp)
![FIG. 6. Barren plateau (BP) phenomenon. a. Variance of the cost function partial derivative, Var(∂θ1,1E), for a particular parameter θ1,1 in the ansatz versus number of qubits (n). Results were obtained from a Variational Quantum Eigensolver implementation with a deep unstructured ansatz. The y-axis is on a log scale. As the number of qubits increases the variance vanish exponentially with the system size. b. Visualization of the landscape of a global cost function which exhibits a BP for the quantum compilation implementation, . The orange (blue) landscape was obtained for n = 24 (n = 4) qubits. As the number of qubits increases, the landscape becomes flatter. Moreover, this cost also exhibits the narrow gorge phenomenon [166], where the volume of parameters leading to small cost values shrinks exponentially with n. Panel a is adapted from Ref. [194], CC BY 4.0; Panel b is adapted from Ref. [166], CC BY 4.0.](/figures/figure6-1-35tu42xcg88v.webp)
Citations
Practical quantum advantage in quantum simulation
Andrew J. Daley,Immanuel Bloch,Christian Kokail,Stuart Flannigan,Natalie Pearson,Matthias Troyer,Peter Zoller +6 more
TL;DR: In this article , the authors overview the state of the art and future perspectives for quantum simulation, arguing that a first practical quantum advantage already exists in the case of specialized applications of analogue devices, and that fully digital devices open a full range of applications but require further development of fault-tolerant hardware.
•Posted Content
Noise-Induced Barren Plateaus in Variational Quantum Algorithms
Samson Wang,Enrico Fontana,Marco Cerezo,Kunal Sharma,Akira Sone,Lukasz Cincio,Patrick J. Coles +6 more
TL;DR: This work rigorously proves a serious limitation for noisy VQAs, in that the noise causes the training landscape to have a barren plateau, and proves that the gradient vanishes exponentially in the number of qubits n if the depth of the ansatz grows linearly with n.
•Posted Content
Connecting ansatz expressibility to gradient magnitudes and barren plateaus.
TL;DR: In this article, the authors derive a fundamental relationship between expressibility and trainability of a quantum circuit and derive a variance in the cost gradient in terms of the expressibility of the ansatz as measured by its distance from being a 2-design.
384
Ising machines as hardware solvers of combinatorial optimization problems
TL;DR: Ising machines as discussed by the authors are special-purpose hardware solvers that aim to find the absolute or approximate ground states of the Ising model, which is of fundamental computational interest because any problem in the complexity class NP can be formulated as an Ising problem with only polynomial overhead and thus a scalable Ising machine that outperforms existing standard digital computers could have a huge impact for practical applications.
Challenges and opportunities in quantum machine learning
TL;DR: In this paper , the authors highlight differences between quantum and classical machine learning, with a focus on quantum neural networks and quantum deep learning, and discuss opportunities for quantum advantage with quantum machine learning.
References
•Proceedings Article
Adam: A Method for Stochastic Optimization
Diederik P. Kingma,Jimmy Ba +1 more
- 01 Jan 2015
TL;DR: This work introduces Adam, an algorithm for first-order gradient-based optimization of stochastic objective functions, based on adaptive estimates of lower-order moments, and provides a regret bound on the convergence rate that is comparable to the best known results under the online convex optimization framework.
138.5K
Self-Consistent Equations Including Exchange and Correlation Effects
Walter Kohn,L. J. Sham +1 more
TL;DR: In this paper, the Hartree and Hartree-Fock equations are applied to a uniform electron gas, where the exchange and correlation portions of the chemical potential of the gas are used as additional effective potentials.
Quantum Computation and Quantum Information
Michael A. Nielsen,Isaac L. Chuang +1 more
- 01 Dec 2010
TL;DR: This chapter discusses quantum information theory, public-key cryptography and the RSA cryptosystem, and the proof of Lieb's theorem.
19.6K
Algorithms for quantum computation: discrete logarithms and factoring
Peter W. Shor
- 20 Nov 1994
TL;DR: Las Vegas algorithms for finding discrete logarithms and factoring integers on a quantum computer that take a number of steps which is polynomial in the input size, e.g., the number of digits of the integer to be factored are given.
9.1K
Simulating physics with computers
TL;DR: In this paper, the authors describe the possibility of simulating physics in the classical approximation, a thing which is usually described by local differential equations, and the possibility that there is to be an exact simulation, that the computer will do exactly the same as nature.