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Quantum Chemistry in the Age of Quantum Computing

Yudong Cao, Jonathan Romero, Jonathan P. Olson, Matthias Degroote, Peter D. Johnson, Mária Kieferová, Ian D. Kivlichan, Tim Menke, Borja Peropadre, Nicolas P. D. Sawaya, Sukin Sim, Libor Veis, Alán Aspuru-Guzik

arXiv:1812.09976v2quant-ph

TL;DR

Classical simulation of general quantum systems becomes difficult because some required representations grow exponentially, motivating quantum-computing approaches for quantum chemistry. The paper reviews chemistry-focused quantum algorithms and their development, including Hamiltonian simulation and quantum phase estimation. It concludes that quantum simulation algorithms have achieved major scaling improvements, while existing hardware has not yet solved classically intractable instances and fault-tolerant devices remain a long-term prospect.

  • Problem

    General quantum systems can be intractable to represent classically, creating a need for quantum-computing methods relevant to quantum chemistry.

  • Method

    The paper reviews quantum algorithms for chemistry, including Hamiltonian simulation, quantum phase estimation, and mappings used in quantum simulation.

  • Results

    Quantum simulation algorithms for quantum chemistry have improved asymptotic scaling from a high-degree polynomial to sublinear in the number of terms.

  • Takeaways & Limitations

    Quantum computers provide a framework for simulating quantum systems and may offer richer state spaces for ground-state searches in certain systems.

  • Takeaways & Limitations

    Existing quantum computers have yet to solve problem instances that are intractable for classical computers.

Abstract

from arXiv · show

Practical challenges in simulating quantum systems on classical computers have been widely recognized in the quantum physics and quantum chemistry communities over the past century. Although many approximation methods have been introduced, the complexity of quantum mechanics remains hard to appease. The advent of quantum computation brings new pathways to navigate this challenging complexity landscape. By manipulating quantum states of matter and taking advantage of their unique features such as superposition and entanglement, quantum computers promise to efficiently deliver accurate results for many important problems in quantum chemistry such as the electronic structure of molecules. In the past two decades significant advances have been made in developing algorithms and physical hardware for quantum computing, heralding a revolution in simulation of quantum systems. This article is an overview of the algorithms and results that are relevant for quantum chemistry. The intended audience is both quantum chemists who seek to learn more about quantum computing, and quantum computing researchers who would like to explore applications in quantum chemistry.

C From quantum chemistry to quantum computation: example of molecular

The paper includes a section titled “A brief introduction to quantum computation.”

  • The section provides a brief introduction to quantum computation.

1 Introduction and historical overview

Quantum chemistry faces computational difficulty because general quantum systems can require exponentially growing classical resources. This review surveys quantum-computing methods that may replace or augment classical chemistry techniques, while emphasizing that practical fault-tolerant devices remain distant.

  • Scope: The authors aim to bridge quantum information theory and classical quantum chemistry through a pedagogical survey of state-of-the-art techniques.
  • Outlook: Existing quantum computers have not yet solved problem instances intractable for classical computers, although experimental groups have implemented variational algorithms.
  • Motivation: General quantum-mechanical systems can require resources growing exponentially with problem size, making some simulations inaccessible to classical computers.
  • Motivation: Quantum computers may circumvent exponential simulation costs by representing and manipulating quantum states directly.
  • Scope: The review focuses on quantum algorithms for chemistry, including molecular-energy estimation, reaction-rate computation, Hamiltonian simulation, and quantum phase estimation.
  • Near-term approaches: Variational quantum algorithms use tunable operation parameters and can partly circumvent present-day hardware shortcomings.
  • Quantum-computing models: Adiabatic quantum computing is computationally equivalent to the gate model, so algorithms in either model can be translated between them.

2 Quantum chemistry in the age of quantum comput-

Quantum chemistry faces exponential classical complexity in representing and solving molecular wave functions, motivating quantum-computing approaches. The section surveys classical approximations and quantum methods for static and dynamic problems, including NISQ and FTQC techniques.

  • Classical challenges: Exact molecular wave functions become intractable on classical computers because their size grows exponentially with particle number.The article identifies this growth as a central challenge for quantum chemistry.
  • Quantum-computing approaches: Quantum-computing methods are organized around static and dynamic problems and include techniques for both NISQ and fault-tolerant quantum computers.The review emphasizes quantum methods that may overcome classical limitations, including state representations and time evolution under Hamiltonians.
  • Problem formulation: The review focuses primarily on quantum chemistry within the Born-Oppenheimer approximation, while noting that some quantum formalisms also address non-BOA cases.Under this approximation, nuclei are treated as stationary point charges and the electronic problem is solved separately for each nuclear configuration.
  • Classical approaches: Classical quantum chemistry methods balance accuracy against computational cost through approximations such as DFT, SCF, coupled cluster, QMC, and FCI.DFT reduces the electronic description to density, SCF uses a self-consistent mean field, coupled cluster uses an exponential cluster parametrization, QMC estimates energies stochastically, and FCI represents all determinants.
  • Classical approaches: DFT and mean-field approaches have scope limitations: functional choice lacks a uniform rule, strong correlations can produce unpredictable results, and SCF neglects correlation effects.These limitations are especially relevant for bond breaking, solvation chemistry, and other systems beyond routine equilibrium geometries.
  • Quantum-computing approaches: Quantum computers provide access to quantum states with no known efficient classical representation and support time evolution methods used across NISQ and FTQC quantum chemistry algorithms.The reviewed approach includes propagating states by e^-iHt|ψ⟩ and leverages richer state spaces for ground-state searches.

Appendix C.

Quantum phase estimation extracts eigenvalue phases and, with Hamiltonian simulation, can estimate quantum-system spectra. Quantum chemistry applications may achieve accuracy comparable to FCI, but practical use depends on fault-tolerant hardware because phase-estimation circuits are often too deep for current NISQ devices.

  • Phase estimation: Quantum phase estimation estimates an eigenvalue phase by encoding an approximation in a finite qubit register and measuring that register.The finite register limits precision because a continuous phase is represented in a finite-dimensional system.
  • Phase estimation: For Hamiltonian eigenspectrum extraction, phase estimation uses U = e−iHt and requires an efficiently implementable unitary operator.The method targets eigenvalues of the Hamiltonian through time evolution.
  • Phase estimation: Ground-state estimation requires an initial state with sufficient overlap: |β0|2 must be at least inverse-polynomial in n for polynomial-time energy estimation.The probability of obtaining the ground-state phase is governed by |β0|2.
  • Quantum chemistry applications: Quantum chemistry has efficiently computable ansatzes for initial-state preparation and established implementations of molecular Hamiltonian time evolution.These ingredients support using phase estimation to compute spectra accurately.
  • Quantum chemistry applications: Phase estimation can compute quantum-system spectra with accuracy comparable to FCI, but its circuits are often too deep for today’s NISQ devices.The approach instead requires fault-tolerant quantum computers, whose realization still faces significant technical challenges.
  • Hybrid methods: Hybrid quantum-classical algorithms shift much of the computational burden to classical processors while using quantum devices to prepare and measure entangled states.The classical processor updates parameters iteratively; VQE is a central chemistry example for static problems.
  • Hamiltonian simulation: Quantum Hamiltonian simulation can implement e−iHt in polynomial time for broad Hamiltonian classes, whereas a classical computer takes time at least ∼2n.For molecular electronic structure, Trotter-Suzuki methods decompose evolution into local terms; newer paradigms have yielded exponential precision improvements applied to quantum chemistry.
  • Hamiltonian simulation: Recent Hamiltonian-simulation paradigms achieve exponential precision improvements over predecessors and have produced similar improvements in quantum chemistry.These methods are studied in both oracle-based and molecular electronic-structure settings.

3 Computational complexity

Computational complexity classes provide a rigorous, worst-case framework for characterizing the difficulty of quantum-physics and quantum-chemistry problems. The section introduces classical and quantum classes and relates representative problems to them.

  • Motivation: Complexity theory asks whether difficult open problems reflect inherent computational difficulty rather than insufficient understanding.This distinction can provide theoretical insight and guide research toward fruitful directions.
  • Basic notions: Complexity classes characterize worst-case hardness, which may differ substantially from practical difficulty on typical instances.Hartree-Fock is NP-complete, yet heuristic methods regularly solve it in practice.
  • Classical complexity classes: P contains efficiently solvable decision problems, whereas NP contains problems whose candidate solutions can be efficiently checked; whether P=NP remains unresolved.Counting problems extend this framework through #P, which concerns counting efficiently checkable solutions.
  • Classical complexity classes: NP-complete problems are both in NP and NP-hard, providing strong evidence against provably efficient classical or quantum algorithms.#P problems are at least as hard as corresponding NP problems because counting solutions can determine whether any solution exists.
  • Quantum complexity classes: BQP contains problems efficiently solvable on quantum computers, while QMA contains problems whose quantum-state solutions can be efficiently verified.QMA-complete problems are therefore unlikely to have efficient quantum solutions.
  • Applications to quantum chemistry: Quantum-chemistry examples span NP-complete, QMA-complete, and BQP classifications, including Hartree-Fock, N-representability, and time-dependent Kohn-Sham potentials.Ground-state energies of many locally interacting systems and the Bose-Hubbard model are QMA-complete, while the time-dependent effective Kohn-Sham potential is in BQP.

4 Quantum simulation algorithms for fault-tolerant quan-

Quantum simulation algorithms map quantum systems onto qubits and implement their unitary evolution with elementary operations. For fault-tolerant computers, advances in Hamiltonian simulation and phase estimation provide efficient routes to estimating eigenenergies, although practical demonstrations remain limited by noise and circuit length.

  • Quantum simulation maps a target system onto qubits, then translates its unitary evolution into a sequence of elementary operations.
  • Classical wave-function storage scales as O(exp(N)), whereas quantum-bit storage scales as O(N), but amplitudes cannot be efficiently accessed.Extracting a classical description requires repeated simulation and tomography, which can eliminate the storage savings.
  • Quantum computers can provide an exponential improvement in memory resources for quantum simulation and support expectation estimation, sampling, and larger quantum algorithms.
  • Hamiltonian simulation: Hamiltonian simulation for physically realistic systems can use a number of gates polynomial in system size, evolution time, and inverse precision.State-of-the-art algorithms achieve scaling roughly logarithmic in inverse precision and linear in time.
  • Fault tolerance: The reviewed algorithms generally assume fault-tolerant quantum computers, while current noise and decoherence restrict experiments to proof-of-principle demonstrations on simple problems.The algorithms are typically beyond the accurately executable program length of present devices.
  • Quantum phase estimation: Quantum phase estimation combines Hamiltonian simulation with controlled operations to estimate eigenenergies, with success probabilities determined by the input state's eigenstate amplitudes.Repeated runs can obtain multiple eigenstates and eigenenergies.

Hamiltonians

Quantum chemistry simulation targets molecular electronic structure, especially ground-state energies, using state preparation, Hamiltonian simulation, and measurement. The reviewed work develops these components and reports improved scaling, while resource estimates range from challenging early requirements to feasible future error-corrected simulations.

  • Electronic structure calculations provide quantum-mechanical energies and wave functions needed to understand and predict reaction rates, binding energies, and molecular pathways.The ground-state energy manifold over nuclear coordinates is sufficient for many such properties.
  • The standard workflow prepares an approximate molecular state, simulates its Hamiltonian, and extracts properties such as ground-state energy using quantum phase estimation.
  • State preparation: Ground-state energy estimation requires an efficiently preparable state with significant overlap with the ground state; Hartree–Fock may fail for strongly correlated systems.Multiconfigurational self-consistent-field states can provide better overlap by expressing needed electron correlation.
  • State preparation: O(η log η log N) gate count and O(log η log log N) circuit depth were achieved for antisymmetrization, respectively improving polynomially and exponentially over the previous algorithm.η denotes particle number and N the number of single-particle basis functions.
  • Hamiltonian simulation: O(N^2) Hamiltonian terms are required in a plane-wave basis versus O(N^4) for Gaussian orbitals, and a single Trotter step can have O(N) depth on a nearest-neighbor line.
  • Resource estimates: Early estimates required coherence times many orders beyond current capabilities, but improved Hamiltonian-simulation algorithms drastically reduced the required time resources.The review reports future error-corrected simulations of realistic chemical systems as feasible using Trotter-based approaches.
  • Hamiltonian simulation: Later LCU and qubitization algorithms scale asymptotically better than prior Trotter approaches, with qubitization retaining the scaling with better constant factors.

5 Quantum algorithms for noisy intermediate-scale quan-

NISQ quantum-chemistry algorithms divide work between quantum state preparation and measurement and classical optimization, targeting approximate solutions despite noisy, resource-limited hardware. The section reviews VQE, related variational simulation methods, ansatz design, energy estimation, optimization, and alternative photonic approaches.

  • NISQ constraints: NISQ devices motivate algorithms that tolerate imperfect, noisy quantum computers while seeking approximate solutions to relevant problems.Such algorithms require low circuit depth because execution must fit within limited device coherence.
  • Hybrid quantum-classical algorithms: Hybrid quantum-classical algorithms use quantum circuits for state preparation and measurement, then classical feedback or optimization to improve circuit parameters.The framework allocates tasks according to the inherent advantages of quantum and classical devices.
  • Variational algorithms: VQE optimizes a parametrized quantum-circuit ansatz using the time-independent variational principle to obtain approximate solutions to the time-independent Schrödinger equation.VQE estimates an objective from measurements and iteratively updates circuit parameters.
  • Optimization and measurement: Analytical gradients can require orders-of-magnitude fewer measurements than numerical gradients, while optimizer performance depends on the problem and optimizer choice.Studies also report that COBYLA and L-BFGS-B performed better in one comparison, whereas numerical gradients required fewer function calls in another setting.
  • Ansatz design: Ansatz and Hamiltonian-design strategies can reduce resources: one representation has O(N^2) Hamiltonian terms, while LDCA has linear depth and O(N) parameters.LDCA studies reported good ground-state descriptions and superior accuracy to hardware-efficient ansatze for lithium hydride.
  • Beyond ground-state VQE: The reviewed approaches include excited-state extensions, imaginary-time variational simulation, and linear-optics simulations demonstrated experimentally for the small molecule tropolone.Some methods account for experimental imperfections and operate without error-correcting techniques.

6 Summary and outlook

Quantum computing offers new approaches to quantum chemistry, especially electronic-structure calculations, while its long-term practical advantage remains uncertain. Near-term work centers on variational methods and collaboration between chemistry, algorithms, and hardware.

  • Motivation: Approximate post-Hartree-Fock calculations can suffer inaccuracies because they approximate the electronic wave function.Quantum computers naturally handle wave functions spanning the full Hilbert space.
  • Quantum chemistry applications: Quantum computers can estimate ground- and excited-state energies and properties including polarization, magnetic dipoles, and reduced density matrices.These calculations support quantities such as reaction pathways, binding energies, and rates.
  • Fault-tolerant methods: Quantum phase estimation is promising but requires quantum error correction, making it infeasible on near-term devices.With error correction, it could become an alternative to VQE for ground-state energy estimation.
  • Near-term methods: The variational quantum eigensolver was developed for currently available hardware without dependence on quantum error correction and has become central to state-of-the-art experiments.It may be among the first commercial quantum-computing applications for predicting small-molecule electronic structures.
  • Outlook: Advances in variational algorithms, hardware, and error mitigation may enable commercial utility sooner than anticipated, although the timing of useful quantum advantage is difficult to predict.The authors emphasize that close collaboration among quantum chemists, information scientists, and device engineers will be important.
  • Cross-disciplinary innovation: Quantum algorithms for chemistry may develop methods with no classical analogs, while chemistry concepts have already informed quantum state preparation and expectation-value estimation.Examples include unitary coupled cluster ansatze, quantum subspace expansion, and N-representability-inspired methods.

A Quantum chemistry basis sets

Quantum chemistry basis sets balance physical accuracy, systematic convergence, integral-evaluation efficiency, and the ability to describe molecular properties. Their design depends on whether systems are periodic or isolated and on the trade-offs between STOs and GTOs.

  • Basis-set design: A good basis set should represent the physics accurately, converge systematically toward the basis-set limit, and facilitate molecular-integral evaluation.It should describe total energies and other properties, but existing basis sets compromise among these goals.
  • System types: Periodic systems are traditionally described with plane waves, whereas isolated molecules use atom-centered functions because their densities have different spatial behavior.Periodic densities do not vanish exponentially far from nuclei, unlike isolated-molecule densities.
  • Atomic-orbital basis sets: LCAO constructs molecular spin-orbitals as linear combinations of atomic basis functions, providing a systematic framework for building molecular basis sets.Basis-set flexibility increases through multiple contractions, polarization functions, and diffuse functions.
  • Slater-type orbitals: Slater-type orbitals capture nuclear cusps and exponential density tails, but their molecular integrals require numerical evaluation because analytical solutions are unavailable.This computational difficulty restricted STO use and motivated Gaussian-type orbitals.
  • Gaussian-type orbitals: Gaussian-type orbitals enable efficient analytical molecular-integral evaluation but do not correctly reproduce electronic-density cusps and exponential tails.Contracted GTOs provide a compromise between representational accuracy and computational ease.
  • Plane-wave bases: Plane-wave bases diagonalize kinetic and potential operators in dual representations, with the dual plane-wave basis offering a more compact representation.This makes plane-wave approaches relevant to quantum-computing implementations.

B.2 Bravyi-Kitaev mapping

The Bravyi-Kitaev mapping transforms fermionic product states into qubit product states while reducing the operator weight to logarithmic scaling under its standard power-of-two assumption. A tree variant relaxes that assumption and enables qubit-reduction techniques.

  • Mapping construction: The Bravyi-Kitaev mapping combines advantages of the Jordan-Wigner and parity mappings for representing fermionic states on qubits.The presented transformation maps Jordan-Wigner product states to Bravyi-Kitaev product states.
  • Operator weight: For N = 2^n spin-orbitals, the standard Bravyi-Kitaev encoding gives creation and annihilation operators maximum weight log2 N.Lower operator weight matters because larger-weight Hamiltonian terms require longer simulation circuits.
  • Assumption: The standard Bravyi-Kitaev transformation applies only when the spin-orbital number is a power of two.This is an explicit scope condition of the presented construction.
  • Tree variant: The Bravyi-Kitaev tree method uses Fenwick trees and produces creation and annihilation operators acting non-trivially on O(log N) qubits.Although its practical operators can have higher weight than those of the standard mapping, it supports non-power-of-two systems.
  • Tree variant: Unlike the standard mapping, the Bravyi-Kitaev tree method enables qubit-reduction techniques when the spin-orbital number is not a power of two.This extends the mapping's applicability beyond the standard construction's assumption.

C.1 Introduction

The appendix presents a concrete VQE quantum-chemistry calculation as a step-by-step workflow. It defines the chemistry problem, maps it to a quantum computer, introduces circuit-model computation, and applies VQE.

  • Workflow: The appendix gives a detailed workflow for converting a quantum-chemistry problem into results on a quantum computer.It explicitly follows the assumptions, simplifications, and calculations used in a concrete VQE calculation.
  • Workflow: The workflow defines the chemistry problem before mapping it onto the quantum computer.These are the first two stated stages of the procedure.
  • Workflow: It introduces circuit-model quantum computation and then applies the variational quantum eigensolver to treat the problem.The appendix is intended for both quantum chemists and quantum information scientists.

C.2 Defining the chemistry problem

The chemistry problem is to determine molecular electronic ground-state energies and their dependence on nuclear geometry. Accurate energy surfaces matter for understanding phenomena such as bond breaking and reaction dynamics, while practical calculations use finite-basis, second-quantized formulations.

  • The central task is determining the electronic ground-state energy of molecular systems.
  • The ground-state energy as a function of internuclear distance defines a molecular dissociation curve.
  • Accurate energy surfaces provide insight into chemical phenomena including bond breaking and reaction dynamics.
  • The electronic problem is simplified with the Born-Oppenheimer approximation, treating nuclei as stationary classical particles because electronic and nuclear dynamics occur on separated timescales.The stated electronic-to-nuclear mass ratio is roughly 1:1000.
  • Quantum-computing treatments commonly use a finite basis and second quantization, with molecular hydrogen represented here in the minimal STO-6G basis.

C.3 Mapping the problem

The second-quantized chemistry Hamiltonian must be encoded into qubit operators before quantum algorithms can simulate it. For molecular hydrogen, Bravyi-Kitaev mapping and symmetry reduction produce a compact two-qubit Hamiltonian for ground-state calculations.

  • Encoding maps fermionic creation and annihilation operators to qubit operators while preserving their canonical commutation relations.This mapping is a prerequisite for Hamiltonian simulation, quantum phase estimation, and VQE.
  • The molecular-hydrogen minimal-basis Hamiltonian is mapped with Bravyi-Kitaev transformation to a four-qubit Hamiltonian.
  • The four-qubit Hamiltonian is reduced by exploiting symmetries to construct a two-qubit Hamiltonian containing Pauli-operator terms.
  • The resulting two-qubit Hamiltonian is used to determine the molecular ground-state energy as a function of nuclear separation.

C.4 A brief introduction to quantum computation

The circuit model represents quantum computations as sequences of unitary gates applied to qubits and followed by measurement. VQE combines quantum-circuit executions with classical routines to estimate molecular ground-state energies.

  • Circuit-model quantum computation: Quantum gates are unitary transformations that manipulate qubits individually or through two-qubit couplings.
  • Circuit-model quantum computation: A circuit-model computation applies quantum gates to an initialized qubit state and measures each qubit to obtain classical outcomes.
  • Hybrid quantum-classical computation: The circuit model can be combined with classical routines to perform more sophisticated computational tasks.
  • Hybrid quantum-classical computation: VQE is a hybrid quantum-classical algorithm used to estimate molecular ground-state energies.

C.5 Variational quantum eigensolver for quantum chemistry

VQE prepares a parametrized trial state on a quantum computer, estimates its energy through Pauli-term measurements, and uses classical optimization to update the parameters. In the molecular-hydrogen simulation, the method reproduced FCI energies, subject to ansatz, optimizer, and sampling requirements.

  • Algorithm: VQE allocates trial-state preparation and energy estimation to the quantum computer while a classical processor minimizes the energy expectation.
  • Algorithm: The algorithm repeatedly measures the energy, updates circuit parameters with a classical optimizer, and stops when convergence criteria are met.
  • Practical considerations: Ansatz design strongly influences VQE performance, and high-quality energy estimates require an expressive ansatz and a robust optimizer.
  • Molecular-hydrogen implementation: For molecular hydrogen, the ansatz uses a Hartree-Fock reference state followed by a unitary coupled-cluster-inspired variational circuit.
  • Energy estimation: The energy expectation is estimated by averaging measured expectation values of Pauli terms weighted by their Hamiltonian coefficients.
  • Limitations: Finite measurement counts introduce energy-estimation error, so high-precision ground-state energies require many measurements despite VQE's low coherence-time requirements.
  • Results: At every sampled H2 bond length, simulated VQE ground-state energies were numerically equal to corresponding FCI values.

C.6 Appendix Glossary

The glossary defines quantum-chemistry symbols, operators, Hamiltonians, variational circuits, optimization methods, orbital functions, and quantum-computing architectures used throughout the paper.

  • l and m denote orbital angular momentum and magnetic quantum numbers, while Y_l,m is the corresponding spherical harmonic.
  • Helec is the electronic Hamiltonian; r_i and R_i specify electron and fixed nuclear positions, respectively.
  • i denotes a fermionic annihilation or creation operator on orbital φ_i, and Z_i denotes the i-th nuclear charge.
  • The glossary identifies X_j, Y_j, and Z_j as Pauli operators, O_i as Pauli products, and h_i as their corresponding weights.
  • VQE uses a variational quantum circuit U(t), with μ_i and ν_i representing coefficients of the original and symmetry-reduced electronic Hamiltonians.
  • The glossary expands UCC, SPSA, PSO, and GTO as unitary coupled cluster, simultaneous perturbation stochastic approximation, particle swarm optimization, and Gaussian type orbitals.
  • Table 5 presents representative VQE demonstrations across quantum-computer architectures and identifies SPSA and PSO as optimization methods.
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