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Emerging quantum computing algorithms for quantum chemistry

Mario Motta, Julia Rice

arXiv:2109.02873v2quant-ph

TL;DR

The paper addresses how digital quantum computers can support molecular electronic-structure calculations despite the difficulty of classical many-body problems and current hardware limitations. It reviews algorithms for Hamiltonian dynamics and eigenstates, including variational and qubitization-based approaches, and identifies complexity, scalability, and hardware constraints. The review concludes that these algorithms offer structured opportunities for quantum chemistry while requiring further benchmarking and hardware development.

  • Problem

    Ground-state and Hamiltonian-dynamics problems can be exponentially expensive classically, while quantum-chemistry algorithms require systematic assessment across accuracy, cost, chemical problems, and hardware capabilities.

  • Method

    The review synthesizes algorithms for Hamiltonian dynamics and eigenstates, including variational quantum eigensolvers, linear-combination and qubitization techniques, and their theoretical and implementation considerations.

  • Results

    The review identifies product formulas, quantum walks, LCU-based algorithms, variational methods, and diagonalization algorithms as key approaches, while describing qubitization as offering optimal complexity with respect to t and ε.

  • Takeaways & Limitations

    Quantum chemistry applications can use these algorithmic families to study Hamiltonian dynamics and eigenstates, with chemical applications serving to assess algorithm and hardware performance.

  • Takeaways & Limitations

    A generic n-qubit unitary may require O(4n) elementary gates, and many algorithms have been tested mainly on small active spaces of about 10 orbitals.

Abstract

from arXiv · show

Digital quantum computers provide a computational framework for solving the Schrödinger equation for a variety of many-particle systems. Quantum computing algorithms for the quantum simulation of these systems have recently witnessed remarkable growth, notwithstanding the limitations of existing quantum hardware, especially as a tool for electronic structure computations in molecules. In this review, we provide a self-contained introduction to emerging algorithms for the simulation of Hamiltonian dynamics and eigenstates, with emphasis on their applications to the electronic structure in molecular systems. Theoretical foundations and implementation details of the method are discussed, and their strengths, limitations, and recent advances are presented.

I. INTRODUCTION

The introduction frames molecular chemistry as a major application for digital quantum simulation and reviews how quantum computation can address molecular properties. It emphasizes bridging quantum chemistry and quantum information while assessing algorithmic opportunities, hardware constraints, and collaborative needs.

  • Molecular chemistry is a prominent many-body problem where accurate predictive computation remains conceptually and technologically important.
  • Digital quantum computers can serve as controllable quantum simulators for studying molecular properties, with Hamiltonian dynamics offering favorable scaling at current knowledge.
  • The work aims to bridge quantum chemistry and quantum computation by reviewing Hamiltonian-dynamics and Hamiltonian-eigenfunction algorithms and analyzing their strengths and weaknesses.
  • The review presents simulation concepts, digital quantum computers, chemical-system requirements, hardware decoherence, and error mitigation or correction techniques.
  • Meaningful near-term hardware calculations require error mitigation and correction techniques.
  • The paper calls for synergistic work between quantum information and quantum chemistry scientists to assess applications and improve algorithm and hardware performance.

1. Universality and limitations of digital quantum computers

This section introduces digital quantum-computing models and their resource requirements, then explains why universality does not guarantee efficient access to generic quantum states. It also motivates quantum algorithms for chemistry through the classical difficulty of ground-state and Hamiltonian-dynamics problems.

  • Universality: Single-qubit gates and CNOT operations form a universal gate set, while Hadamard, S, T, and CNOT provide a countable universal set with accuracy-dependent approximations.
  • Limitations: A generic n-qubit unitary requires O(4n) single-qubit and CNOT gates, so universal quantum computers do not guarantee polynomial-cost access to generic n-qubit states.
  • Limitations: Clifford circuits with computational-basis preparation and single-Pauli measurement can be efficiently simulated classically under the Gottesman-Knill theorem.
  • Quantum computational complexity: Quantum computational complexity theory studies the resource requirements of quantum problems and relates quantum complexity classes to classical counterparts.
  • Quantum computational complexity: Ground-state and Hamiltonian-dynamics problems are worst-case exponentially expensive classically, while practical quantum chemistry requires characterizing accuracy, cost, and hardware control.

III. SOME IMPORTANT PROBLEMS IN COMPUTATIONAL CHEMISTRY

Computational chemistry seeks molecular electronic states and properties, but exact solutions become combinatorially costly and remain difficult for multi-reference systems. The section reviews classical formulations and highlights basis-size and hardware-related limits affecting quantum simulations.

  • Electronic structure calculations determine ground and low-lying excited states of interacting electrons by solving the Born–Oppenheimer Schrödinger equation.
  • These states provide access to energy differences, energy gradients, and electrostatic molecular properties.
  • Electronic structure simulations approximate the electronic Hamiltonian in a finite orbital basis using first- or second-quantization formalisms.
  • Accurate calculations require large one-electron basis sets, while near-term quantum simulations have mostly used about M ≃10 orbitals because hardware limits qubit number and quality.
  • Most quantum algorithms have been tested on small active spaces or minimal bases, leaving their usefulness for larger bases uncertain.
  • Exact eigenfunction calculations grow combinatorially with system size, motivating approximate wavefunction, density-functional, embedding, and diagrammatic methods.
  • Single-reference methods become less reliable for excited states, bond stretching, transition metals, and other multi-reference systems with several important electronic configurations.

B. Electronic dynamics

Electronic dynamics connects measurable molecular properties to time evolution and correlation functions. The section emphasizes applications involving excited states, linear response, nuclear motion, and the limitations of harmonic nuclear models.

  • Oscillator strengths and related spectral properties depend on excited-state energies and transition matrix elements.
  • Dipole-dipole structure factors are Fourier transforms of time-dependent correlation functions and can be obtained by simulating time evolution.
  • Time- and frequency-dependent properties, including nonlinear optical effects and circular dichroism, are natural applications of quantum algorithms because they involve excited states and time evolution.
  • Linear response theory treats weak external perturbations by truncating the evolution and observable operators to first order in the perturbation.
  • Nuclear rovibrational simulations require solving a nuclear Schrödinger equation whose potential energy surface is obtained from electronic calculations at fixed geometries.
  • The harmonic approximation gives equally spaced mode levels and cannot describe bond dissociation, motivating higher-order or anharmonic treatments.

D. Chemical reactions

Chemical reactivity requires accurate energies and potential-energy surfaces, while quantum algorithms also depend on efficient fermion-to-qubit mappings and symmetry reduction. The section connects these computational goals with challenges from solvation, conformational diversity, and limited hardware resources.

  • D. Chemical reactions: Reaction studies target enthalpy and free-energy differences, activation energies, and potential-energy surfaces governing reactivity and reaction rates.
  • D. Chemical reactions: Solvation effects require approaches ranging from implicit solvent and QM/MM models to multiscale embedding methods.
  • D. Chemical reactions: Molecular-crystal polymorphs can differ by 0.5 kcal/mol or less, making accurate first-principles energies and configuration-space exploration important.
  • 1. Fermions in second quantization: Fermionic states and operators can be mapped to qubits through transformations such as Jordan–Wigner, parity, and Bravyi–Kitaev mappings.
  • 1. Fermions in second quantization: Jordan–Wigner operators require O(M) qubit operations, whereas Bravyi–Kitaev operators act on O(log2 M) qubits, reducing measurement and circuit costs.
  • 1. Fermions in second quantization: Parity mapping can reduce the qubit count by two without improving efficiency over Jordan–Wigner mapping.
  • 1. Fermions in second quantization: Symmetry-based tapering can reduce the H2 STO-6G energy calculation from four qubits to one while preserving the same energy.

2. Bosons in second quantization

Bosonic and other d-level degrees of freedom require encodings beyond qubit-level fermionic mappings. Binary and Gray-code mappings provide alternative representations whose efficiency depends on the system and objective.

  • Quantum simulations may need to represent d-level particles, vibrational modes, spin-s particles, and electronic energy levels with d > 2.
  • A d-level system can be encoded in nq = ⌈log2 d⌉ qubits using standard binary mapping.
  • Gray coding ensures that encodings of consecutive levels differ by one binary digit, a property useful for qubit-based simulation of truncated bosonic modes.

3. Alternative approaches

Alternative simulation approaches include product formulas, quantum walks, and LCU algorithms, with implementations exploiting fermionic mappings, symmetries, and low-rank Hamiltonian representations. Product-formula costs depend on evolution time, accuracy, Hamiltonian structure, and the cost of exponentiating mapped Pauli operators.

  • Jordan–Wigner mappings can support local two-qubit implementations in one dimension, whereas higher dimensions introduce non-local spin couplings.
  • Symmetry-based reductions can reduce the number of qubits, including reductions based on discrete and continuous symmetries.
  • Electronic-structure Hamiltonian simulation is classified into product-formula, quantum-walk, and linear-combination-of-unitary algorithms.
  • Product formulas: Product formulas approximate time evolution by composing exponentials of Hamiltonian terms over many short time steps.
  • Product formulas: Primitive and Trotter–Suzuki formulas attain accuracy ε with step counts whose scaling depends polynomially on evolution time and ε^-1.
  • Product formulas: Mapping the electronic Hamiltonian to Pauli operators yields a per-step cost between ˜O(M^4) and O(M^5), while low-rank decompositions provide alternative cost reductions.

2. Quantum walks

Quantum-walk simulation encodes a rescaled Hamiltonian in a larger unitary operator whose spectrum can be converted into time evolution. The approach achieves linear scaling in evolution time, while LCU methods address its remaining algebraic accuracy scaling through non-unitary approximations.

  • Quantum-walk algorithms achieve linear scaling with evolution time, unlike product formulas whose cost is super-linear in time.
  • A quantum walk constructs a unitary on an extended Hilbert space whose spectrum is connected to that of the Hamiltonian time-evolution operator.
  • The walk operator preserves two-dimensional subspaces and has eigenvalues determined by the rescaled Hamiltonian eigenvalues.
  • Constructing the walk requires an oracle exposing binary representations of Hamiltonian matrix elements, and phase estimation converts its spectrum into simulated evolution.
  • LCU algorithms: LCU algorithms implement non-unitary approximations as linear combinations of unitary operations, using ancilla preparation, selection, and measurement.
  • LCU algorithms: LCU success probability can decay exponentially with repeated application, while oblivious amplitude amplification improves success when the target operator is unitary.

4. LCU based algorithms

LCU-based Taylor-series methods simulate Hamiltonian dynamics by decomposing the Hamiltonian into unitary terms and implementing preparation and selection operations. Qubitization and QSP improve the dependence on simulation time and precision, while time-dependent Hamiltonians remain a limitation.

  • Taylor-series Hamiltonian simulation: Berry et al.'s Taylor-series method achieves computational-cost scaling linear in t and logarithmic in ε^-1.The method applies to Hamiltonians represented as linear combinations of unitaries.
  • Taylor-series Hamiltonian simulation: The truncated Taylor expansion represents e^-i∆tH as an LCU and uses K(1 + log2 L) ancillae for probabilistic implementation.The preparation unitary creates coefficient-weighted states, while the selection unitary applies products of Hamiltonian unitaries.
  • Taylor-series Hamiltonian simulation: The selection unitary applies ordered products of unitary terms at cost O(L(n + log2 L)K) operations.Its input includes the term indices and Taylor order, and its output applies the corresponding product to the system state.
  • Taylor-series Hamiltonian simulation: The logarithmic dependence on ε^-1 follows from rapid Taylor-series convergence together with ancillary qubits and controlled operations.The polynomial order K controls the approximation accuracy ε/K.
  • Qubitization and QSP: Qubitization combines an LCU Hamiltonian decomposition with QSP to achieve O(t + log ε^-1) complexity, optimal in both time and accuracy.A qubiterate encodes the Hamiltonian, and QSP uses it to approximate e^-itH.
  • Qubitization and QSP: Qubitization is formulated for time-independent Hamiltonians, whereas product formulas and Taylor-series methods can simulate time-dependent Hamiltonians in the interaction picture.This distinction is especially relevant for electronic structure, where one- and two-body terms can be transformed efficiently.

6. Applications of Hamiltonian dynamics, and open problems

Hamiltonian-dynamics algorithms support chemical observables, eigenvalue estimation, and ground-state preparation, but practical costs depend on implementation resources and spectral properties. Open problems include fair comparisons across chemical regimes and integrating complementary algorithmic techniques.

  • Applications and open problems: Hamiltonian-dynamics simulation is a compelling quantum-computing application with potential to yield relevant quantum simulations of chemical systems.The review places this application within BQP and emphasizes its continuing implementation and algorithmic development.
  • Applications and open problems: LCU-based algorithms require additional ancillae and controlled operations, which are challenging for near-term quantum-device implementations.Actual runtime also depends on prefactors and classical–quantum data movement, not only asymptotic complexity.
  • Applications and open problems: The review calls for comparative studies across diverse chemical problems to identify regimes where quantum-walk and LCU methods outperform product formulas.It also highlights hybrid constructions that combine elements from different algorithm families.
  • Time-dependent observables and correlation functions: Time-dependent correlation functions reduce to measurements of Pauli operators after mapping molecular Hamiltonians and dipole operators to Pauli combinations.The associated circuits also provide settings for benchmarking ancillae and controlled operations.
  • Quantum phase estimation: QPE estimates Hamiltonian eigenvalues by applying controlled powers of a unitary approximating e^-iλH, including ground- and excited-state energies.The algorithm has inverse-QFT and shallower multi-measurement variants.
  • Quantum phase estimation: With exact phase representation, QPE returns the binary phase estimate with probability 1; otherwise the nearest estimate has probability at least 4/π^2 ≃ 0.4.Increasing the ancilla count to t = O(log ε^-1) raises this probability to 1 − ε.
  • Adiabatic state preparation: Adiabatic state preparation approaches the interacting ground state by evolving from an easily solved Hamiltonian H0 to the target Hamiltonian H.The adiabatic theorem guarantees convergence in the large-T limit under appropriate conditions.
  • Adiabatic state preparation: ASP is polynomially expensive when the spectral gap remains constant or decreases as 1/poly(M), but can become exponentially expensive otherwise.Its cost also depends on derivatives of the Hamiltonian along the adiabatic path.

C. Simulation of Hamiltonian eigenstates

Hamiltonian eigenstate algorithms use heuristic approximations because exact quantum speedups are not generally expected for eigenpair problems. Variational methods trade Ansatz expressivity and accuracy against hardware feasibility and circuit cost.

  • Motivation: Hamiltonian eigenpair computation is important in chemistry, but heuristic algorithms are used because general quantum speedups over classical methods are not expected.For structured problems, these methods can approximate ground and selected excited states at polynomial cost.
  • Variational framework: Variational quantum algorithms prepare parametrized Ansätze, measure the resulting state, and update parameters with a classical optimizer.The Ansatz is generated by applying parameterized unitaries to an initial wavefunction.
  • Variational quantum eigensolver: VQE minimizes the measured energy expectation, which is an upper bound to the Hamiltonian ground-state energy.Quantum hardware evaluates the energy and, in some implementations, its derivatives; measurements yield statistical estimates.
  • Quantum unitary coupled-cluster: q-UCCSD uses single and double particle-hole excitations, but a basic Jordan–Wigner implementation scales as O(M^5), limiting present hardware implementations.The excitation circuits themselves require O(M) CNOT gates and depth O(M).
  • Hardware-efficient Ansätze: Hardware-efficient Ansätze match chip connectivity and native operations, but they are not guaranteed to contain accurate approximations to the target state.Ansatz design therefore balances hardware feasibility, state accuracy, and optimization behavior.
  • Adaptive Ansätze: ADAPT-VQE appends operators selected by energy-gradient values and was reported to improve over q-UCCSD in accuracy at a given circuit depth.The approach targets compact Ansätze combining hardware efficiency with chemical insight.

3. Variational quantum simulation

Variational simulation methods approximate dynamical trajectories or eigenstates by projecting quantum evolution into a parameterized subspace. They combine quantum measurements with classical updates, while their accuracy and resource requirements remain problem dependent.

  • Variational quantum simulation: Variational quantum simulation approximates a dynamical curve in Hilbert space using time-dependent parametrized wavefunctions.The parameter flow is represented by a differential equation rather than by minimizing a single static cost function.
  • Variational quantum simulation: McLachlan’s variational principle determines parameter evolution by minimizing the norm of the residual between the approximate and target dynamics.The resulting differential equation defines dθ/dt.
  • Quantum–classical workflow: Quantum measurements provide the quantities needed to compute dθ/dt, after which a classical computer updates the variational parameters.This establishes a quantum–classical feedback loop.
  • Quantum diagonalization: Quantum diagonalization constructs a subspace, measures overlap and Hamiltonian matrices, and solves a generalized eigenvalue problem for approximate eigenpairs.Candidate vectors can be generated by applying suitable operators to an initial state.
  • Quantum diagonalization: Hadamard tests or Pauli decompositions can evaluate the matrix elements required by quantum diagonalization algorithms.Polynomial-cost approximations to ground and selected excited states are possible depending on the generated vectors and problem structure.
  • Eigenstate algorithms: Quantum filter diagonalization projects the Hamiltonian onto a subspace generated by approximate time evolution, while q-EOM computes excitation energies through commutator-based generalized eigenvalue problems.These methods target excited-state information through different projected formulations.
  • Quantum Lanczos and imaginary time: qLANCZOS uses imaginary-time evolution to construct a subspace, but implementing imaginary-time evolution on digital quantum computers requires specialized approaches.QITE avoids ancillae and controlled operations, while its performance across problems remains insufficiently characterized.

5. Applications of variational algorithms, and open problems

Applications of variational algorithms extend across molecular properties and electronic-structure workflows, while measurement, hardware, and error-control constraints remain central open problems. Near-term devices are limited by scale, connectivity, native gates, and decoherence.

  • Open algorithmic problems: Research directions include chemically informed Ansätze, broader accessible properties, and lower-cost calculations.These directions aim to improve the practical scope of heuristic Hamiltonian algorithms.
  • Applications: VQE has been extended to excited states, orbital optimization, properties beyond ground-state energy, solid-state chemistry, transcorrelated Hamiltonians, and quantum embedding.These extensions modify cost functions or evaluate additional operators on the VQE wavefunction.
  • Measurement optimization: Measurement costs are being reduced through simultaneous measurement of commuting Pauli subsets, amplitude amplification, and machine-learning analysis of measurement data.These methods seek more information from the measurements required by variational algorithms.
  • Alternative formulations: Two-electron reduced-density-matrix approaches replace the full N-electron wavefunction in expressing ground-state energy, but their full potential requires additional assessment.Recent results identify this as a promising direction for efficient molecular quantum simulations.
  • Hardware constraints: Contemporary devices have fewer than 100 qubits, limiting the number of electrons and orbitals that can be simulated.Hardware also imposes connectivity and native-gate constraints that affect circuit compilation.
  • Hardware constraints: Decoherence and imperfect implementation accumulate with circuit depth and entangling-gate count, increasing biases and statistical uncertainties.The effect is especially pronounced when execution time approaches qubit decoherence times.
  • Error mitigation: Readout mitigation models measured probabilities as a linear transformation of ideal probabilities and reconstructs the ideal distribution through classical post-processing.Numerical studies reported approximate validity on several quantum-chip prototypes, although the cost can scale exponentially with qubit number.
  • Error mitigation: Gate-error mitigation can enhance superconducting-processor capabilities without additional quantum resources, but it only corrects expectation values and cannot indefinitely extend computation time.Richardson extrapolation also requires detailed control of the circuit gates.

VI. CONCLUSION AND OUTLOOK

The review distinguishes structured problems that may benefit from polynomial-resource quantum algorithms from eigenpair problems addressed through heuristic approximations. It calls for systematic chemical benchmarks and collaboration between quantum chemistry and quantum information researchers.

  • Conclusion: The review frames Hamiltonian dynamics as a structured problem potentially benefiting from polynomial-resource quantum algorithms, while Hamiltonian eigenpairs rely on heuristic approximations.This distinction summarizes the paper’s central scope for quantum chemistry algorithms.
  • Conclusion: It reviews product formulas, quantum walks, LCU-based methods, variational algorithms, and diagonalization algorithms for Hamiltonian dynamics and eigenstates.The review highlights applications and open problems across these algorithm classes.
  • Open problems: Systematic numerical studies across diverse chemical problems are needed to characterize accuracy and computational cost for both heuristic and non-heuristic algorithms.For heuristic methods, such studies can help establish and refine the underlying approximations.
  • Open problems: Chemists can design benchmark systems ranging from small molecules such as H2 to realistic systems such as enzymes to study scalability and accuracy.The proposed benchmarks could also demonstrate algorithms on today’s devices.

Appendix A: Glossary

Table III presents a glossary of acronyms used throughout the work.

  • Table III lists acronyms used throughout the present work.
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