Variational Quantum Algorithms

M. Cerezo; Andrew Arrasmith; Ryan Babbush; Simon C. Benjamin; Suguru Endo; Keisuke Fujii; Jarrod R. McClean; Kosuke Mitarai; Xiao Yuan; Lukasz Cincio; Patrick J. Coles

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Variational Quantum Algorithms

M. Cerezo, Andrew Arrasmith, Ryan Babbush, Simon C. Benjamin, Suguru Endo, Keisuke Fujii, Jarrod R. McClean, Kosuke Mitarai, Xiao Yuan, Lukasz Cincio, Patrick J. Coles

arXiv:2012.09265v2Dec 16, 2020quant-phcs.LGstat.ML

TL;DR

Variational quantum algorithms target computationally expensive quantum problems on near-term hardware, but their trainability, accuracy, and efficiency remain challenging. This review surveys VQAs, examines strategies for overcoming these challenges, and discusses prospects for quantum advantage.

Problem
VQAs still face unresolved challenges in trainability, accuracy, and efficiency that affect prospects for scaling near-term architectures toward quantum speedups.
Method
The review synthesizes VQA applications, challenges, and strategies including ansatz initialization, measurement reduction, and noise-resilience approaches.
Results
The review identifies barren plateaus, measurement costs, and hardware noise as central constraints while highlighting locality, classical shadows, and variational calibration as responses.
Takeaways & Limitations
Understanding VQA limitations and developing targeted strategies are important for constructing better algorithms, proving performance guarantees, and improving quantum hardware.

Abstract

from arXiv · show

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.

I. INTRODUCTION · II. BASIC CONCEPTS AND TOOLS · A. Cost function

Variational Quantum Algorithms are presented as a leading strategy for using constrained, noisy NISQ devices toward quantum advantage, while retaining challenges in trainability, accuracy, and efficiency. Their common architecture combines a cost function, parametrized quantum ansatz, quantum cost estimation, and classical parameter optimization, with cost evaluation constrained by NISQ hardware limitations.

I. INTRODUCTION: Quantum advantage remains unrealized, while fault-tolerant quantum computers may be years or decades away and NISQ devices have limited qubits, connectivity, and circuit depth.These constraints arise from coherent and incoherent errors and motivate strategies tailored to current hardware.
I. INTRODUCTION: VQAs have emerged as the leading strategy for obtaining quantum advantage on NISQ devices by combining parametrized quantum circuits with classical optimization.Their optimization-based or learning-based structure accounts for multiple NISQ constraints within one framework.
I. INTRODUCTION: VQAs have been considered for essentially all applications envisioned for quantum computers, but trainability, accuracy, and efficiency remain important challenges.The review discusses strategies to overcome these challenges and prospects for achieving quantum advantage.
II. BASIC CONCEPTS AND TOOLS: Most VQAs share basic elements despite differing algorithmic structures and complexity, providing a general framework for solving varied problems.The framework begins with a problem description and possibly training data, followed by defining a cost function and proposing an ansatz.
II. BASIC CONCEPTS AND TOOLS: The VQA trademark is using a quantum computer to estimate the cost function C(θ) or its gradient while classical optimizers train the parameters θ.The cost encodes the problem solution, and the ansatz is a parametrized quantum operation.
A. Cost function: The cost function maps trainable parameters θ to real numbers, defining a cost landscape whose global minimum represents the optimizer’s target.Its specific mathematical form depends on functions, parametrized unitaries, input states, observables, and the task; costs or gradients are estimated statistically.
A. Cost function: A useful cost function must be faithful, efficiently estimable on a quantum computer, and operationally meaningful, while not being efficiently computable classically.Faithfulness requires the minimum to correspond to the problem solution; classical efficient computability would preclude quantum advantage.
A. Cost function: NISQ cost-evaluation circuits must keep circuit depth and ancilla requirements small because devices have gate errors, limited qubit counts, and short decoherence times.Constructing efficient cost-evaluation circuits is therefore an important aspect of VQA research.

B. Ansatzes … 4. Variational Hamiltonian ansatz

Ansatz design determines how variational parameters are trained and can be tailored to a problem or adapted to hardware. The review covers hardware-efficient, chemistry-inspired, alternating-operator, and variational Hamiltonian architectures.

B. Ansatzes: Ansatz structure determines the parameters θ and how they can be trained to minimize the cost, with designs either problem-inspired or problem-agnostic.Problem-inspired ansatze use task information, whereas generic architectures are problem-agnostic.
B. Ansatzes: The reviewed ansatze include forms expressible as Eq. (4) and more general architectures built from parametrized and unparametrized unitaries.Each unitary U_l(θ_l) acts sequentially within U(θ), and can be decomposed into parametrized and unparametrized gates.
1. Hardware efficient ansatz: Hardware-efficient ansatze reduce circuit depth by choosing gate-alphabet unitaries according to hardware connectivity and interactions, avoiding translation overhead for arbitrary unitaries.Their versatility allows them to accommodate encoding schemes.
2. Unitary coupled clustered ansatz: The Unitary Coupled Cluster ansatz is problem-inspired for quantum chemistry and prepares a candidate fermionic ground state by exciting a reference state, usually Hartree–Fock.It uses e^{T(θ)−T(θ)†}|ψ_0⟩, with T formed from excitation operators.
2. Unitary coupled clustered ansatz: UCC implementations map fermionic operators to spin operators using Jordan–Wigner or Bravyi–Kitaev transformations, while variants can reduce circuit depth through more efficient compilation.UCCSD truncates the cluster expansion to single and double excitations.
3. Quantum alternating operator ansatz: QAOA uses an alternating operator ansatz for approximate combinatorial optimization, with U(γ, β) formed from alternating problem and mixing Hamiltonian evolutions.The parameters are θ=(γ,β).
3. Quantum alternating operator ansatz: QAOA may require lengthy native-gate circuits because of many-body problem terms and limited connectivity, but restricting optimization to a feasible subspace can improve performance.For certain problems, the feasible subspace is smaller than the full Hilbert space.
4. Variational Hamiltonian ansatz: The variational Hamiltonian ansatz Trotterizes adiabatic state preparation, using each Trotter step as a variational ansatz for trial ground-state preparation.It has been implemented in quantum chemistry, optimization, and quantum simulation.

5. Variable structure ansatz … 9. Ansatz expressibility

The paper surveys ansatz designs that optimize circuit structure, incorporate device-level controls, combine quantum and classical representations, prepare mixed states, and assess expressibility and entangling capability. These approaches expand ansatz flexibility while introducing tradeoffs such as added qubit requirements and ongoing measurement challenges.

5. Variable structure ansatz: Variable-structure ansatzes optimize circuit structure as well as continuous parameters, enabling adaptive addition or removal of circuit elements through frameworks such as ADAPT-VQE.Fixed circuit structures can miss refinements from removing unnecessary elements or adding useful ones.
5. Variable structure ansatz: Machine-learning evolutionary methods grow quantum circuits from populations, develop application-specific variational ansatzes, or evolve multiple ansatz variants simultaneously.These methods explore Hilbert space by upgrading circuit individuals or coordinating an evolving cohort.
6. Sub-logical ansatz and quantum optimal control: Including device-level parameters below the logical circuit level can increase ansatz flexibility and connect variational algorithms with quantum optimal control.Logical parameters such as rotation angles may translate directly into physical device parameters.
7. Hybrid ansatzes: Hybrid ansatzes shift complexity toward classical computation by combining quantum states with classical post-processing, trainable linear combinations, tensor networks, or jointly optimized classical and quantum parameters.Examples include free-fermion simulation, classically optimized coefficients cµ, unitary tensor-network contraction, and simultaneous optimization of J and θ.
8. Ansatz for mixed states: Mixed-state ansatzes construct ρ as a reduced state of a purification, with one approach requiring up to 2n qubits.Purification methods prepare a pure state whose subsystem represents the target mixed state.
8. Ansatz for mixed states: Alternative mixed-state ansatzes train probabilities pi(φ) and states |ψi(θi)⟩ as a statistical ensemble, using product distributions, energy-based models, or autoregressive models.These methods represent ρ(φ, {θi}) through classically parameterized mixture weights and variational states.
9. Ansatz expressibility: Ansatz quality is evaluated through expressibility and entangling capability: expressibility concerns uniform exploration of quantum-state space, while entangling capability measures average sampled-state entanglement.Quantifying expressibility remains active, and some quantum architectures show higher expressibility than classical architectures under certain measures.

C. Gradients · 1. Parameter-shift rule · 2. Other derivatives

Gradients provide analytically accessible information for training VQA parameters, potentially speeding optimization and supporting convergence. The parameter-shift rule evaluates derivatives through parameter shifts, while extensions cover higher-order cost derivatives and other state-based quantities.

C. Gradients: Gradient information can speed optimization and help guarantee convergence, and many VQAs allow analytical evaluation of the cost-function gradient.This concerns training parameters after defining the cost function and ansatz.
1. Parameter-shift rule: The parameter-shift rule is a hardware-friendly protocol for evaluating a cost-function partial derivative when a parameter generates a Pauli rotation.The setup considers a cost function with f_k(x)=x and a unitary e^{iθ_lσ_l}.
1. Parameter-shift rule: The parameter-shift rule evaluates a gradient component by shifting the corresponding parameter by ±α.The shifted parameter vector is θ± = θ ± αe_l, where e_l selects the l-th parameter.
1. Parameter-shift rule: 1/sin α controls sampling-related evaluation accuracy, which is maximized at α = π/4.The ±α terms are estimated by sampling O_k, and the coefficient is 1/(2 sin α).
2. Other derivatives: Higher-order cost derivatives follow by applying the parameter-shift rule repeatedly, including second derivatives and mixed or third derivatives.Explicit formulas are cited in Refs. [84].
2. Other derivatives: Because the cost function has a trigonometric-series expansion, it can be classically approximated around a reference point to offload minimization work.This permits constructing a classical cost-function model for minimization by the supervising classical system.
2. Other derivatives: Metric tensors and related derivatives of parametrized quantum states support sophisticated optimization and variational simulation, using Hadamard-test-like protocols or parameter-shift reductions.These quantities are overlaps of different states and can be reduced to the parameter-shift technique.

D. Optimizers … A. Finding ground and excited states

VQA performance depends on optimization methods that must address generally hard, non-convex cost landscapes and limited gradient information. VQAs support broad quantum-computing applications, including near-term estimation of Hamiltonian ground and excited states through VQE.

D. Optimizers: VQA optimization is generally NP-hard because cost functions can contain many local minima, creating challenges beyond those of classical optimization.The paper identifies trainability difficulties specific to VQA training in addition to standard optimization challenges.
1. Gradient descent methods: Gradient-based methods use statistically estimated gradients; Adam adapts step sizes to improve efficiency and precision over basic SGD.Quantum natural gradient instead uses an information-geometric metric encoding the state’s sensitivity to parameter variations.
2. Other methods: Gradient-free VQA optimizers include SPSA, which estimates gradients using one finite difference along a randomly chosen direction.For objectives linear in operator expectation values, trigonometric fitting also enables sequential local parameter updates.
3. Convergence analysis: Because VQA cost landscapes are generally non-convex and complicated, computational-cost guarantees are difficult, although simplified landscapes admit SGD convergence guarantees.Within convex regions around minima, parameter-shift gradients yield smaller errors than finite-difference methods.
III. APPLICATIONS: VQAs provide task-oriented programming for a wide range of tasks and have been proposed for essentially all applications envisioned for quantum computers.The framework has also been shown to allow universal quantum computing.
A. Finding ground and excited states: Estimating low-lying Hamiltonian eigenstates and eigenvalues is the best-known VQA application, motivated by earlier algorithms requiring circuit depths unavailable in the NISQ era.The Variational Quantum Eigensolver was developed as a near-term solution for finding Hamiltonian ground states.

1. Variational quantum eigensolver … 7. Accelerated VQE

The paper presents VQE-based methods for estimating ground and excited-state energies through variational cost functions, subspace constructions, and adiabatic or measurement-accelerated extensions. These methods modify state preparation, optimization, or measurement to target multiple eigenstates or reduce measurement costs.

1. Variational quantum eigensolver: VQE minimizes C(θ) = ⟨ψ(θ)|H|ψ(θ)⟩ to estimate the ground-state energy of Hamiltonian H using a parametrized trial state.For sparse Hamiltonians, the cost can usually be estimated with computational cost growing at most polynomially with system size.
2. Orthogonality constrained VQE: Orthogonality-constrained VQE adds a penalty a⟨ψ(θ)| ˜ψG⟩⟨˜ψG|ψ(θ)⟩ to target the first excited state after estimating the ground state.The constant a is chosen much larger than the energy gap between the ground state and first excited states.
3. Subspace expansion method: Subspace expansion generates states by applying low-weight Pauli operators to the estimated ground state, then solves Hα = ESα for approximate low-energy eigenstates.The candidate eigenstates are linear combinations of the generated states with optimized coefficients α.
4. Subspace VQE: Subspace VQE trains a unitary to prepare a lowest-energy subspace from mutually orthogonal inputs, using weighted or non-weighted cost functions.Weighted costs associate states with increasing energies through decreasing weights, whereas non-weighted costs require a second rotation optimization to obtain eigenstates.
5. Multistate contracted VQE: Multistate contracted VQE obtains a lowest-energy subspace and represents each eigenstate as a coefficient combination found from a generalized eigenvalue problem.It avoids optimizing an additional unitary and uses S = 1 in the generalized eigenvalue problem.
6. Adiabatically assisted VQE: Adiabatically assisted VQE uses C(θ) = ⟨ψ(θ)|H(s)|ψ(θ)⟩ with H(s) = (1 −s)H0 + sHP to connect simple and complex Hamiltonians.The variational state is prepared as |ψ(θ)⟩= U(θ)|ψ0⟩.
7. Accelerated VQE: Accelerated VQE interpolates between VQE and QPE by using α-QPE, whose tunability allows the measurement cost to interpolate between both algorithms.The interpolation modifies the measurement process rather than the variational optimization framework.

Dynamical quantum simulation … 11. Simulating open systems

VQAs enable dynamical simulation through iterative parameter evolution, low-energy subspace decompositions, and variational fast forwarding, while also extending to dissipative open-system dynamics. These approaches can avoid depth growth with simulation time or reduce quantum-resource overhead under specific conditions.

Dynamical quantum simulation: Conventional Hamiltonian simulation discretizes time, causing circuit depth to generally increase polynomially with system size and simulated time.Accumulated hardware noise therefore constrains dynamical simulation on NISQ devices.
8. Iterative approach: Iterative variational algorithms map Schrödinger-state evolution onto parameter evolution by repeatedly updating trial-state parameters.McLachlan’s principle yields a parameter-update equation whose quantities can be efficiently measured with modified Hadamard-test circuits.
8. Iterative approach: The iterative variational framework also applies to imaginary-time Schrödinger evolution and first-order equations with non-Hermitian Hamiltonians.These extensions use analogous variational procedures for updating the parameters.
9. Subspace approach: Weighted subspace VQE simulates low-energy dynamics by rotating states into the computational basis, applying eigenvalue-dependent evolution, and rotating them back.For superpositions of low-energy eigenstates, the resulting circuit depth is independent of the simulation time because evolution is directly implemented in the subspace.
10. Variational fast forwarding: Variational fast forwarding approximates time evolution with a trainable diagonal matrix and unitary, optimized using a local Hilbert-Schmidt fidelity test for a small time step.It then extends the approximation to longer times using the Trotter-Suzuki product formula.
11. Simulating open systems: For open systems governed by dρ/dt = L(ρ), McLachlan’s principle maps mixed-state evolution to variational-parameter evolution through M · ˙θ = V.The matrix and vector terms can be computed with SWAP tests on two copies of purified states.
11. Simulating open systems: Simulating an open system of n qubits with purified states requires operations on 4n+1 qubits, whereas a stochastic Schrödinger-equation approach requires n + 1 qubits.The alternative unravels density-matrix evolution into pure-state trajectories with damping and jump processes caused by noise operators.

B. Optimization · C. Mathematical applications · 1. Linear systems

The section presents QAOA as a variational approach to classical combinatorial optimization and surveys VQAs for mathematical problems, especially linear systems. These methods seek useful near-term heuristics while confronting non-convex optimization, barren plateaus, and implementation constraints.

B. Optimization: QAOA is the best-known VQA for quantum-enhanced optimization, targeting combinatorial problems such as SAT and Max-Cut.
B. Optimization: QAOA encodes a binary objective function in a problem Hamiltonian whose ground state represents the solution.
B. Optimization: Its circuit alternates p rounds of problem- and mixer-Hamiltonian evolution, with evolution intervals optimized as variational parameters.
B. Optimization: QAOA optimization is difficult because its landscape is non-convex with many local optima, motivating gradient-based, derivative-free, and reinforcement-learning optimizers.
C. Mathematical applications: VQAs for mathematical applications aim for heuristical scalings comparable to fault-tolerant algorithms while remaining compatible with NISQ requirements.
1. Linear systems: For an N × N system Ax = b, the QLSP seeks a normalized state |x⟩ satisfying A|x⟩∝|b⟩, with |b⟩=b/∥b∥.
1. Linear systems: QLSP VQAs commonly assume A is a efficiently implementable linear combination of unitaries and minimize a Hamiltonian whose ground state is the solution.
1. Linear systems: Gradients can vanish exponentially with qubit number, but local Hamiltonians or hybrid ansätze can mitigate barren plateaus; one study observed logarithmic time-to-solution scaling in N.The study used n = 10, . . . , 30 qubits for Ising-inspired systems and n = 2, . . . , 7 qubits for random sparse systems.

2. Matrix-vector multiplication … E. Error correction

The paper surveys VQA applications spanning matrix-vector multiplication, nonlinear equations, factoring, PCA, compilation, and error correction. These approaches formulate target states, equations, optimization problems, circuit synthesis, or hardware-tailored codes as variational tasks, with correctness and protection assessed through energy, fidelity, or numerical performance.

2. Matrix-vector multiplication: Matrix-vector multiplication prepares a normalized state proportional to A|b⟩ by using the ground state of a constructed Hamiltonian HM.For A = 1 − iHδt, the task becomes Hamiltonian simulation.
2. Matrix-vector multiplication: A small HM cost certifies the approximate matrix-vector solution because its fidelity obeys |⟨ψ(θ∗)|x⟩|^2 ≥ 1 − ⟨ψ(θ∗)|HM|ψ(θ∗)⟩.The Euclidean norm is ∥A|b⟩∥ = ⟨b|A†A|b⟩.
3. Non-linear equations: VQAs address nonlinear equations by optimizing costs such as total energy for the nonlinear Schrödinger equation or residual satisfaction at selected points.A basis-function approach uses nonlinear feature maps, parametrized linear combinations, expectation values, and parameter-shift derivatives.
4. Factoring: For factoring, QAOA variationally searches for the ground state of a classical Ising model, offering a near-term alternative because large-scale Shor implementations are unavailable.The proposal relies on factoring’s formulation as an optimization problem.
5. Principal Component Analysis: VQA-based PCA diagonalizes a covariance matrix encoded as a density matrix, while a majorization-based formulation reduces the requirement from 2n to n qubits.The reduced-qubit method minimizes C(θ) = Tr[ρ̃(θ)H] with a non-degenerate Hamiltonian H.
D. Compilation and unsampling: Quantum compiling variationally transforms a target unitary V into a native-gate circuit U(θ) with optimally short depth, addressing a classically exponentially difficult task.Shorter circuits also support error mitigation because errors increase with circuit depth.
E. Error correction: QVECTOR discovers device-tailored quantum error-correcting codes by jointly optimizing parametrized encoding and recovery circuits for noisy quantum memories.The circuits encode k-qubit inputs into n qubits and implement recovery with r ancillary qubits.
E. Error correction: Numerical studies found that QVECTOR can outperform existing codes, while variational compilation of conventional codes produced five- and seven-qubit encoding circuits for different noisy hardware.The latter approach bounds fidelity by F ≥ 1 − (E − EG)/a using the discovered energy E and a = min{a0, ak}.

F. Machine learning and data science … 4. Variational Quantum Generators

Variational quantum algorithms support diverse machine-learning applications, including classifiers, quantum autoencoders, generative models, and variational quantum generators. These approaches train parametrized quantum circuits to classify data, compress quantum states, learn probability distributions, or accelerate classical GANs.

F. Machine learning and data science: VQA-based quantum machine learning trains parametrized quantum circuits to learn patterns in quantum data for accurate predictions on unseen data.The review presents several QML applications that readily implement the VQA framework.
1. Classifiers: Parametrized quantum circuits implement quantum classifiers by embedding inputs, applying trainable transformations, and optimizing prediction error against measured observables.Alternative approaches include data re-uploading and quantum-kernel methods, with variational classification experimentally demonstrated.
2. Autoencoders: Quantum autoencoders train an ansatz to compress ensembles of bipartite pure states into subsystem A while preserving high-fidelity recovery.Subsystem B is discarded as ‘trash’, and the cost function uses overlap with a fixed pure state.
2. Autoencoders: A local autoencoder cost function was proposed and shown to train well for large-scale problems.This extends the quantum autoencoding framework beyond the original global cost-function construction.
3. Generative models: Quantum generative models learn a probability distribution from data using a parametrized circuit Born machine, with samples obtained by computational-basis measurement.Training minimizes negative log-likelihood because the target distribution is unavailable.
3. Generative models: Quantum circuit Born machines were demonstrated on bars-and-stripes, cat-state, and coherent-thermal-state datasets, while implicit models used Gaussian-kernel feature-space distances.These studies investigated the representation power of quantum generative models.
4. Variational Quantum Generators: Variational quantum generators adapt the GAN framework by encoding classical data, generating quantum states, and producing fake samples for discriminator-based optimization.The discriminator may be classical or quantum, and optimization minimizes discrimination probability relative to real samples.

5. Quantum Neural Network architectures … 1. Barren plateaus

VQAs span diverse architectures and applications, while barren plateaus remain a major trainability bottleneck whose severity depends on cost-function locality, circuit structure, and noise. The reviewed results motivate strategies for overcoming limitations that threaten scalable quantum speedups.

5. Quantum Neural Network architectures: Perceptron-based QNNs represent network nodes as qubits connected by parameterized unitaries, while QCNNs have been proposed for error correction, image recognition, and discrimination.The passage identifies these as examples among several proposed QNN architectures.
G. New frontiers: VQAs are being applied to quantum foundations, information theory, entanglement spectroscopy, and metrology to exploit mathematical and physical structure in quantum states.Examples include testing foundational ideas, computing information-theoretic quantities, extracting entanglement spectra, and variationally finding optimal probe states under noise.
1. Quantum foundations: NISQ devices may provide computationally tractable platforms for studying quantum-mechanical foundations, including classicality emergence and Variational Consistent Histories.The passage also cites VQAs for foundational insights such as the inability of FUMC to efficiently learn a scrambling unitary.
2. Quantum information theory: Quantum autoencoders may learn encodings and achievable quantum-channel transmission rates, while VQAs can compute von Neumann entropy and trace distance.These are identified as potential NISQ applications in quantum information theory.
3. Entanglement Spectroscopy: VQAs can extract entanglement spectra, which characterize condensed-matter entanglement and help study topological order.The passage connects entanglement spectra with principal components of reduced density matrices and variational PCA algorithms.
4. Quantum metrology: Variational-state quantum metrology searches for optimal probe states when physical noise makes analytical solutions difficult to obtain.The objective is probing a parameter, such as a magnetic field, with minimal shot noise.
1. Barren plateaus: Barren plateaus cause cost-function gradients to vanish exponentially with system size, making landscapes flat and requiring exponentially large precision to overcome sampling noise.This phenomenon is identified as a major VQA bottleneck and can occur in deep unstructured randomly initialized circuits that form 2-designs.
IV. CHALLENGES AND POTENTIAL SOLUTIONS: Global costs exhibit barren plateaus, whereas local costs have gradients vanishing polynomially in n when circuit depth is at most logarithmic in n; noise can induce plateaus regardless of ansatz.Barren plateaus also arise in scrambler learning and perceptron-based QNNs, where intrinsic randomness or entanglement drives the effect.

2. Ansatz and initialization strategies … 2. Optimized sampling

The paper addresses barren plateaus through parameter initialization and structured ansatz choices, then improves VQA measurement efficiency through commuting-group partitioning and optimized shot allocation. These strategies target trainability and the resource costs of estimating expectation values and cost functions.

2. Ansatz and initialization strategies: Barren-plateau mitigation focuses on two main approaches: parameter initialization and ansatz choice.Both strategies aim to break assumptions that lead to barren plateaus.
2. Ansatz and initialization strategies: Parameter initialization matters because random seeds can start optimization far from solutions, near local minima, or in barren-plateau regions.QAOA optimal parameters have also been observed to exhibit persistent patterns.
2. Ansatz and initialization strategies: Correlated parameters reduce hyperparameter dimension and can produce large gradients, while layer-by-layer training progressively expands initially shallow circuits.These methods restrict ansatz randomization or circuit growth to mitigate barren plateaus.
2. Ansatz and initialization strategies: Problem-inspired structured ansatzes restrict the explored optimization space and are usually trainable under random initialization, including UCC and quantum alternating operator ansatzes.Other proposals include learning a mixture of states determined by the problem.
B. Efficiency: Expectation-value estimation is essential for quantum advantage because barren plateaus can exponentially increase precision requirements and early chemistry estimates required astronomical measurement counts.More reasonable resource estimates are available for restricted problems such as the Hubbard model.
B. Efficiency: For large molecules, chemical Hamiltonians contain n4 distinct Pauli strings, motivating measurement-frugal cost-function estimation.The number of Pauli strings scales as n4 with the number of orbitals and qubits n.
1. Commuting sets of operators: Partitioning Pauli strings into commuting subsets reduces measurements, but qubit-wise commuting methods do not change quantum-chemistry asymptotic scaling.Partitioning can be formulated as graph coloring, minimum clique cover, or maximal-flow problems.
2. Optimized sampling: For fermionic VQE, tensor factorizations reduce measurement scaling to quadratic or, in simpler cases, linear in n; shot allocation further improves efficiency.Optimal shots scale with |ci|√Var(σi), while randomized allocation with the same probabilities gives unbiased estimates with as little as one shot.

3. Classical shadows … 3. Error mitigation

The section surveys efficient partial-tomography methods, hardware-noise effects and resilience in VQAs, and quantum error-mitigation strategies that improve observable estimates through classical processing. It emphasizes both noise-related limitations and methods for preserving or recovering accurate results.

3. Classical shadows: Classical shadows construct an approximate classical representation of a quantum state by combining measurements in randomly chosen bases.This provides a form of partial tomography for efficient measurement.
4. Neural network tomography: RBM tomography estimates operator expectation values from an approximate neural-network state, substantially reducing sampling variance while introducing a small positive bias.The RBM is fitted using measurements of the Pauli operators required for the target expectation value, so it does not inherently reduce the number of operators measured.
C. Accuracy: VQAs can minimize circuit depth and combine with error mitigation to improve accuracy on noisy intermediate-scale quantum devices, but hardware noise can still alter training and optimal costs.Noise may slow training, bias the optimization landscape, or change the final optimal cost.
1. Impact of hardware noise: Noise typically slows VQA training because it flattens the cost landscape and reduces gradient magnitudes.Noise-free training has been heuristically observed to reach lower noise-free costs than noisy training.
1. Impact of hardware noise: Under local Pauli noise, the cost landscape concentrates exponentially with ansatz depth around the maximally mixed-state cost, corrupting final cost values.The effect has also been observed for QAOA, although it is not important for VQAs whose costs can be evaluated classically after optimization.
2. Noise resilience: VQAs exhibit non-trivial resilience to coherent and incoherent noise, including optimizer calibration of coherent parameter shifts and movement toward noise-resilient subspaces.Quantum compiling can additionally exhibit Optimal Parameter Resilience, where noisy and noise-free global minima coincide.
3. Error mitigation: Quantum error mitigation suppresses physical errors in observable expectation values through classical post-processing, with zero-noise extrapolation increasing noise levels to estimate the zero-noise result.Extrapolation cannot fully mitigate physical errors, whereas probabilistic error cancellation can theoretically produce unbiased expectation values given complete noise characterization.
3. Error mitigation: Additional QEM methods use near-Clifford classical simulability, symmetry verification, subspace expansion, and specialized procedures for correlated measurement errors or photon loss.Symmetry verification can recover the quantum state itself, while subspace expansion supports Hamiltonian eigenstates and excited states and works better for coherent than stochastic noise.

V. OPPORTUNITIES FOR NEAR-TERM QUANTUM ADVANTAGE … C. Optimization and machine learning

VQAs are regarded as leading candidates for practical quantum advantage across quantum-science and classical applications, supported by advances in ansätze, optimization, new algorithms, and error mitigation. The surveyed opportunities span chemistry and materials, nuclear and particle physics, and optimization and machine learning, while important challenges remain.

V. OPPORTUNITIES FOR NEAR-TERM QUANTUM ADVANTAGE: VQAs are widely regarded as the best candidate for practical quantum advantage, motivating advances in ansatz design, quantum-aware optimization, new algorithms, and error mitigation.The paper notes that many challenges remain to be addressed.
A. Chemistry and material sciences: Molecular and correlated-electronic simulations support applications including protein-folding dynamics, drug-receptor analysis, drug discovery, high-temperature superconductivity, and transition-metal materials near a Mott transition.
1. Molecular structure: Classical molecular-structure treatments include Hartree-Fock, density functional theory, and density matrix renormalization group methods using matrix-product-state ansätze.
2. Molecular dynamics: Time-dependent variational principles approximate chemical and quantum-system dynamics across electronic and longer timescales, potentially enabling proton-coupled electron transfer and organic-photovoltaic design.
3. Materials science: Strongly correlated materials exhibit effects beyond classical density-functional methods, while long-term material-simulation algorithms requiring phase estimation exceed near-term device capabilities.
1. Nuclear physics: VQE has been demonstrated for nuclear ground-state calculations, including the deuteron binding energy and extensions to the triton, 3He, and alpha particle.
2. Particle physics: VQAs may provide NISQ-era advantages for lattice gauge theories, with applications to simulation, mass gaps, Green’s functions, running coupling constants, and interpolation operators.
C. Optimization and machine learning: Quantum algorithms may address classical optimization and machine-learning tasks by encoding large problems or datasets in Hilbert spaces, leveraging coherence or entanglement to accelerate computation.

1. Optimization … KEY POINTS:

VQAs are positioned as a leading route to quantum advantage on near-term quantum computers, with applications spanning optimization and machine learning. Their future depends on improving trainability, accuracy, efficiency, and implementation as hardware and methods advance.

1. Optimization: QAOA is widely considered a leading candidate for achieving quantum advantage on NISQ devices in classical optimization problems.Examples include MaxCut and Max-Sat, which model applications such as circuitry layout, statistical physics, and automotive configuration.
2. Machine Learning: Quantum algorithms are being actively developed for machine learning applications despite ongoing challenges in loading classical data onto quantum computers.Quantum neural networks have been shown to achieve significantly higher effective-dimension capacity than comp.
VI. OUTLOOK: Future VQA research will develop analytical and heuristic scaling analyses covering gradient scaling, local-minimum density, and cost-landscape shape.These results are intended to guide the search for quantum advantage.
VI. OUTLOOK: Quantum-aware optimizers and commercial software are expected to mitigate small gradients, avoid local minima, and accelerate VQA parameter optimization.The optimizers will exploit knowledge of the cost landscape, while software packages will streamline VQA testing.
VI. OUTLOOK: Application-specific and adaptive ansatzes are expected to improve trainability by enhancing gradient magnitudes and may reduce noise impacts.Hybrid quantum-classical models that parameterize both classical and quantum ansatzes could facilitate near-term applications.
VI. OUTLOOK: New error-mitigation strategies are anticipated to improve VQA accuracy by orders of magnitude and become integrated into cloud-based quantum platforms.This integration is intended to let users obtain accurate results with ease.
VI. OUTLOOK: Near-term VQA work will shift toward larger, more realistic implementations combining strategies for accuracy, trainability, and efficiency.These implementations will test VQAs’ capabilities and push the boundaries of NISQ devices.
KEY POINTS:: VQAs are the leading proposal for near-term quantum advantage, use a quantum-evaluated parameterized cost function with classical optimization, and face trainability, accuracy, and efficiency challenges.Their adaptive structure suits near-term hardware constraints, while applications include molecular ground states, quantum dynamics, and linear systems.