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Quantum computational chemistry

Sam McArdle, Suguru Endo, Alan Aspuru-Guzik, Simon Benjamin, Xiao Yuan

arXiv:1808.10402v3quant-ph

TL;DR

Quantum computational chemistry seeks ways to study chemically important systems that are difficult for classical methods, despite the challenge of building sufficiently large quantum computers. This review bridges computational chemistry and quantum computing, surveys near-term methods, and concludes that some calculations may suit early quantum hardware but substantial scaling remains necessary.

  • Problem

    Preparing quantum registers with sufficient overlap with target ground states is non-trivial because random states have exponentially vanishing success probability as systems grow.

    Source: §2 · p. 27

  • Method

    The review synthesizes computational chemistry and quantum-computing methods, illustrating mappings of chemical problems to quantum algorithms including variational and subspace approaches.

    Source: B. Variational algorithms · p. 32, §1 · p. 37, §VIII · p. 46

  • Results

    Calculations involving around 100 spin-orbitals may suit early quantum computers, with condensed-phase materials and Fermi-Hubbard simulations requiring fewer resources than individual-molecule simulations.

    Source: B. Quantum resources: medium to… · p. 49

  • Takeaways & Limitations

    Quantum computational chemistry provides a framework for assessing which chemical calculations are most promising for near-term and future quantum hardware.

    Source: §VIII · p. 46, D. Summary and outlook · p. 51

    Surpassing classical techniques still requires roughly 100,000 physical qubits, while current quantum computers have around 100 and no fully protected logical qubit has been demonstrated.

    Source: B. Quantum resources: medium to… · p. 49

Abstract

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One of the most promising suggested applications of quantum computing is solving classically intractable chemistry problems. This may help to answer unresolved questions about phenomena like: high temperature superconductivity, solid-state physics, transition metal catalysis, or certain biochemical reactions. In turn, this increased understanding may help us to refine, and perhaps even one day design, new compounds of scientific and industrial importance. However, building a sufficiently large quantum computer will be a difficult scientific challenge. As a result, developments that enable these problems to be tackled with fewer quantum resources should be considered very important. Driven by this potential utility, quantum computational chemistry is rapidly emerging as an interdisciplinary field requiring knowledge of both quantum computing and computational chemistry. This review provides a comprehensive introduction to both computational chemistry and quantum computing, bridging the current knowledge gap. We review the major developments in this area, with a particular focus on near-term quantum computation. Illustrations of key methods are provided, explicitly demonstrating how to map chemical problems onto a quantum computer, and solve them. We conclude with an outlook for this nascent field.

I. INTRODUCTION · II. QUANTUM COMPUTING AND SIMULATION · A. Quantum computing

Quantum computational chemistry addresses the exponential difficulty of exact classical quantum simulation by using quantum hardware, while remaining constrained by major hardware and algorithmic challenges. This review focuses on electronic structure, near-term and fault-tolerant methods, and practical mappings between chemistry and quantum circuits.

I. INTRODUCTION

  • Quantum computational chemistry is motivated by the exponential growth of quantum wavefunctions, which makes exact simulation of many-particle systems inefficient on classical computers.Feynman proposed quantum hardware as a simulation platform for such problems.Source: §I · p. 2
  • Quantum simulation could enable more accurate and realistic models across chemistry, biology, medicine, and materials science, potentially supporting the design of useful compounds.Reaching this goal requires substantial advances in quantum hardware, algorithms, and physical models.Source: §I · p. 2
  • Although polynomial-scaling chemistry algorithms and proof-of-principle experiments exist, transformative simulations likely require error correction with far more qubits and lower error rates than currently available.The review aims to connect quantum computing and computational chemistry to develop more realistic approaches.Source: §I · pp. 2–3
  • The review focuses primarily on the electronic structure problem because its energy eigenstates predict reaction rates, stable structures, and optical properties, while its methods generalize to related problems.Its main methodological emphasis is on variational algorithms with error mitigation and quantum phase estimation with quantum error correction.Source: §I · p. 3
  • The review provides an accessible, practical account of quantum computational chemistry, explicitly translating classical chemistry methods into quantum algorithms and discussing resource reduction and error mitigation.It complements Cao et al. (2019), which emphasizes classical electronic structure expertise and asymptotic computational complexity.Source: §I · p. 3

II. QUANTUM COMPUTING AND SIMULATION

  • Section II introduces the basic elements of quantum computing and quantum simulation, with more detailed background delegated to Georgescu et al. (2014) and Nielsen and Chuang (2002).The broader review proceeds from these foundations to chemistry methods, quantum algorithms, noise mitigation, worked examples, and resource comparisons.Source: §I · p. 3, §II · p. 4

A. Quantum computing

  • The review adopts the qubit-based circuit model, where single- and two-qubit gates manipulate superpositions and entanglement before measurement extracts computational-basis outcomes.With single-qubit operations and CNOT gates, arbitrary multi-qubit gates can be approximated to any desired accuracy.Source: A. Quantum computing · pp. 4–5
  • 50% probability is obtained for each of the |00⟩ and |11⟩ outcomes when the example circuit measures its generated entangled two-qubit state.Quantum observables are estimated from repeated measurements, while Pauli strings provide a complete basis for multi-qubit Hermitian operators.Source: A. Quantum computing · pp. 5–6

B. Quantum simulation

This section focuses on digital quantum simulation for chemistry, emphasizing its need for accurately simulating large systems despite greater noise sensitivity than analog approaches. It introduces digital methods for dynamics and static electronic-structure problems, including Trotterization, variational circuits, and phase estimation.

Digital versus analog simulation

  • Digital quantum simulation maps chemical systems onto quantum gates and is likely required for accurate simulations of large systems, although it is more vulnerable to noise than analog simulation.Error correction can address imperfections but requires additional qubits and stringent gate fidelities.Source: B. Quantum simulation · pp. 6–7

Quantum dynamics

  • Hamiltonian dynamics can be simulated by mapping the system Hamiltonian and initial wavefunction to qubits, then approximating time evolution with a Lie-Trotter-Suzuki gate decomposition.For chemically relevant systems, the Hamiltonian has polynomially many terms because Coulomb interactions are two-body, and the resulting dynamics problem belongs to BQP.Source: B. Quantum simulation · p. 7
  • Variational circuits may simulate open and closed-system time evolution with fewer gates than Trotterization, but their parameters are tailored to specific initial states rather than arbitrary valid states.After evolution, measurements can extract quantities such as electronic charge densities and particle correlation functions.Source: B. Quantum simulation · p. 7

Chemical reaction dynamics

  • Chemical reaction dynamics require mappings that accurately describe electrons and nuclei throughout the reaction, with grid methods treating both on an equal footing and basis-set methods requiring additional care.Basis-set approaches may need precise classical nuclear dynamics, additional orbitals for nuclear motion, or time-dependent orbitals; Berry et al. (2019a) is identified as the only known basis-set calculation on a quantum computer.Source: B. Quantum simulation · p. 7

Static properties

  • Static properties can be obtained by mapping target wavefunctions to qubits and measuring observables, while phase estimation can determine energies and collapse the system into corresponding eigenstates.Variational algorithms provide an alternative approach to the ground-state problem.Source: B. Quantum simulation · p. 8

Electronic structure

  • Electronic structure focuses on low-lying energies of physically relevant chemical systems, which are widely believed to be efficiently solvable quantum-computationally and support calculations of reaction rates, molecular geometries, thermodynamic phases, and optical properties.The general local-Hamiltonian ground-state problem is QMA-complete, but electronic structure is treated as a central application of quantum computational chemistry.Source: B. Quantum simulation · p. 8

III. CLASSICAL COMPUTATIONAL CHEMISTRY … a. Grid based methods

The review formulates molecular electronic structure, identifies strongly correlated systems as important quantum-computing targets, and surveys classical representations and approximations. In grid-based first quantisation, antisymmetrised wavefunctions require exponentially scaling memory, limiting classical simulations to very small systems.

A. The electronic structure problem

  • The electronic structure problem seeks molecular energy eigenstates, especially ground and low-lying excited states, across nuclear configurations to map potential energy surfaces.Explicitly mapping these surfaces is exponentially costly in the molecule’s degrees of freedom.Source: A. The electronic structure problem · p. 9

1. Chemical systems of interest

  • Strongly correlated transition-metal systems such as nitrogenase’s FeMo-co and high-temperature-superconductor models remain difficult for classical methods, motivating quantum simulation.FeMo-co models exceed current classical capabilities, while the Fermi-Hubbard model remains elusive near half filling and intermediate interaction strength.Source: §1 · p. 10

B. First and Second quantisation

  • First and second quantisation encode electron antisymmetry differently: first quantisation antisymmetrises the wavefunction, whereas second quantisation assigns exchange statistics to operators.These representations affect how physical systems are simulated on quantum computers.Source: B. First and Second quantisation · p. 10
  • Grid-based methods differ from basis-set methods: Galerkin discretisation guarantees energy convergence from above, a property unavailable to grids.The distinction is separate from whether a simulation uses first or second quantisation.Source: B. First and Second quantisation · p. 10

1. First quantisation

  • First-quantised classical simulation uses the position representation and explicitly antisymmetrises the N-electron wavefunction over spatial and spin coordinates.The wavefunction is evaluated on a discretised spatial grid with coordinates x_i = (r_i, σ_i).Source: a. Grid based methods · p. 11

a. Grid based methods

  • P^3N × 2^N complex amplitudes make grid-based wavefunction storage exponential, rendering classical simulation intractable beyond a few particles.The discretised representation uses P spatial points per axis and two spin states per electron.Source: a. Grid based methods · p. 11
  • Despite their classical cost, grid methods are useful for reaction dynamics and systems beyond Born–Oppenheimer validity because they can treat nuclear and electronic motion together.This avoids separately obtaining exponentially costly electronic potential energy surfaces.Source: a. Grid based methods · p. 11

b. Basis set methods · 2. Second quantisation

Basis-set methods reduce simulation resources by representing many-electron states with antisymmetrized Slater determinants over a finite set of spin-orbitals. Second quantisation compresses this representation into occupation-number vectors and fermionic operators, yielding an electronic Hamiltonian whose exact expansion becomes exponentially costly for large systems.

b. Basis set methods

  • Basis-set methods project the Hamiltonian onto M spin-orbital basis functions and represent the many-electron wavefunction as an antisymmetrized Slater determinant.This exploits known orbital structure to reduce resources relative to grid-based methods.Source: b. Basis set methods · p. 11
  • Slater determinants enforce fermionic exchange symmetry, while each determinant contains only the N occupied spin-orbitals among the M available orbitals.Exchanging two electrons changes the determinant sign, producing the required antisymmetry.Source: b. Basis set methods · p. 11

a. Basis set methods

  • Second quantisation encodes each Slater determinant as an occupation-number vector |f⟩ in Fock space, with f_p = 1 for occupied and f_p = 0 for empty spin-orbitals.This provides a compact shorthand for the determinant representation.Source: a. Basis set methods · p. 12
  • Fermionic creation and annihilation operators excite and de-excite electrons while obeying anticommutation relations that preserve exchange antisymmetry.The occupation operator returns f_i and therefore counts electrons in spin-orbital i.Source: a. Basis set methods · p. 12
  • Representation-independent observables require second-quantised operators to contain equal numbers of creation and annihilation operators, producing the electronic structure Hamiltonian.The Hamiltonian contains one- and two-electron terms corresponding to kinetic, nuclear-attraction, and electron-electron Coulomb interactions.Source: a. Basis set methods · p. 12
  • Because electron-electron Coulomb repulsion is a two-body interaction, the Hamiltonian contains at most two creation and two annihilation operators and up to M^4 terms.For the Fermi-Hubbard model, the corresponding Hamiltonian has only O(M) terms.Source: a. Basis set methods · pp. 12–13
  • Exact eigenstates can be expanded over Slater determinants, but full configuration interaction requires exponentially many determinants as electron number grows, making large calculations classically intractable.Second-quantised basis-set methods therefore underpin most quantum computational chemistry work and motivate approximate wavefunction and orbital-selection methods.Source: a. Basis set methods · p. 13

b. Grid based methods … 3. Configuration interaction

The section surveys grid-based quantum chemistry and classical wavefunction methods, emphasizing how Hartree–Fock, MCSCF, and CI approximate correlation with distinct computational limitations. CI truncations recover dynamic correlation but converge slowly and are neither size extensive nor size consistent.

b. Grid based methods

  • Second-quantized grid methods represent electrons at three-dimensional delta-function grid points and use finite differences for the kinetic operator.The reviewed discussion follows Babbush et al. (2018c), and the method had not yet been used in classical or quantum computational chemistry algorithms to the authors’ knowledge.Source: b. Grid based methods · p. 13

C. Classical computation methods

  • The classical methods are parametrized trial-state approaches intended to approximate ground states, considered here in a full molecular-orbital basis despite classical intractability for large systems.They are presented in second-quantized basis-set form because this translates most directly to quantum computational chemistry methods.Source: C. Classical computation methods · pp. 13–14

1. Hartree–Fock

  • Hartree–Fock iteratively diagonalizes a Fock operator to optimize orbitals in an average electron-charge potential, producing a self-consistent single-determinant reference.The resulting canonical molecular orbitals are linear combinations of atomic orbitals.Source: §1 · p. 14
  • Hartree–Fock neglects dynamic and static correlation, performing poorly when multiple determinants are comparably important, including in excited states, transition states, dissociation limits, and transition metals.Its determinant commonly serves as the reference for post-Hartree–Fock methods, although its optimized occupied orbitals are unsuitable for virtual orbitals.Source: §1 · p. 14

2. Multiconfigurational self-consistent field

  • MCSCF treats strong static correlation by variationally optimizing multiple determinant amplitudes and molecular orbitals simultaneously.CASSCF restricts determinants to an active space, while tensor-network replacements can enable larger active spaces; these methods remain weaker for dynamic correlation.Source: §2 · p. 15

3. Configuration interaction

  • Configuration interaction builds correlated wavefunctions from excitations above a reference, using CIS, CISD, or CISDT truncations because full configuration interaction is classically intractable.These truncations can approximate ground-state energies when low-energy excitations dominate, recovering dynamic correlation more effectively than static correlation.Source: §3 · p. 15
  • Truncated configuration interaction is not size extensive or size consistent, and its linear parametrization causes slow convergence toward full configuration interaction.Size consistency is required for correct reaction-rate behavior involving non-interacting reactants.Source: §3 · pp. 15–16

4. Coupled cluster … 3. Plane wave basis sets

The section describes coupled-cluster approximations and conventional orbital basis sets, then motivates plane-wave bases for quantum computational chemistry by reducing Hamiltonian complexity. It also outlines accuracy–cost tradeoffs and the limitations of available basis sets.

4. Coupled cluster

  • Coupled cluster uses a product-parametrized wavefunction that can recover full configuration interaction in principle, but practical truncations such as CCSD and CCSD(T) avoid exponential cost.CCSD solving scales as O((M −N)4N 2) with O(M 4) integral storage, while CCSD(T) scales approximately as O(M 7).Source: §4 · p. 16
  • Canonical coupled cluster is size extensive and rapidly convergent but nonvariational and generally unsuitable for multireference states, whereas UCC is variational and quantum-computer efficient.UCC is exponentially costly classically but efficient to implement on a quantum computer.Source: §4 · p. 16

D. Chemical basis sets

  • Conventional chemical basis functions approximate numerically defined true orbitals, balancing orbital accuracy against the computational cost of evaluating integrals and storing basis functions.STOs offer desirable orbital behavior but expensive two-electron integrals, while GTOs simplify integral evaluation despite being more localized and requiring more functions.Source: D. Chemical basis sets · p. 17
  • The number of basis functions controls classical runtime and memory and, in quantum chemistry, the required qubits and gate operations, while conventional bases can produce Hamiltonians with up to O(M 4) terms.This makes basis selection especially important for quantum computational chemistry.Source: D. Chemical basis sets · p. 17, §3 · p. 18

1. STO-nG and split-valence basis sets

  • STO-nG minimal basis sets provide qualitative descriptions with only the orbitals needed for Hartree–Fock, while split-valence and polarization functions add radial and angular flexibility.Examples include 6-31G split-valence and 6-31G* with polarization functions, which were used in early quantum-hardware demonstrations despite limited accuracy.Source: §1 · pp. 17–18

2. Correlation-consistent basis sets

  • Correlation-consistent cc-PVnZ bases add virtual polarization orbitals selected by correlation-energy contributions, with higher n improving orbital representations but not eliminating basis-set error.Even n = 5 bases can struggle to represent true Hartree–Fock orbitals, motivating extrapolation from several basis-set energies to the basis-set limit.Source: §2 · p. 18

3. Plane wave basis sets

  • Plane-wave and plane-wave dual bases reduce Hamiltonian-term scaling from O(M 4) to O(M 3) and O(M 2), respectively, lowering the asymptotic cost of quantum chemistry algorithms.They diagonalize the kinetic and potential operators, respectively, and are well suited to periodic systems.Source: §3 · p. 18
  • Plane waves require approximately 10 to 100 times more functions than GTOs for molecular systems, while gausslet and discontinuous-Galerkin bases may use fewer functions, leaving efficient quantum-chemistry bases an open research problem.The alternative bases can provide similar Hamiltonian-term reductions while requiring fewer functions for individual molecules than plane waves.Source: §3 · p. 19

E. Reduction of orbitals · IV. QUANTUM COMPUTATIONAL CHEMISTRY MAPPINGS · A. First quantised encoding methods

The paper introduces reduced active spaces for simplifying quantum-chemistry calculations and surveys mappings that encode chemical wavefunctions into qubits. First-quantised methods represent particles directly, but grid-based approaches face severe resource limitations while basis-set encodings require antisymmetrisation.

E. Reduction of orbitals

  • Reduced active-space calculations remove spin-orbitals with expected occupations near 0 or 1, retaining only ambiguously occupied orbitals.Reduced density matrices provide the occupation information needed to identify core and virtual orbitals.Source: E. Reduction of orbitals · p. 19
  • Natural orbital occupation numbers are obtained by diagonalising the 1-RDM, whose diagonal elements are orbital number-operator expectations.The RDMs may come from an approximate ground state produced by classically tractable configuration-interaction or coupled-cluster calculations.Source: E. Reduction of orbitals · p. 19
  • Virtual-orbital energy contributions can be partially reintegrated without additional qubits, at the cost of more measurements and classical post-processing.Active spaces can also be improved through orbital rotations, as in MCSCF calculations.Source: E. Reduction of orbitals · p. 19

IV. QUANTUM COMPUTATIONAL CHEMISTRY MAPPINGS

  • Quantum-chemistry mappings encode fermions into qubits using first- or second-quantised representations, distinguished by whether antisymmetry is enforced in the wavefunction or operators.The framework considers M spin-orbitals and N electrons and summarizes representation-specific storage requirements in Table I.Source: §IV · p. 19, Table I · p. 20

A. First quantised encoding methods

  • First-quantised simulation represents an N-particle wavefunction using either a discrete single-particle basis or a real-space grid.Grid-based wavefunctions assign each particle a position register, while basis encodings assign each electron a register identifying its orbital.Source: A. First quantised encoding methods · pp. 20–21
  • Grid-based methods are generally unsuitable for near-term quantum computers because accuracy and spatial resolution can require exponentially increasing gate counts and many qubits.Storing one spinless particle to 32-bit positional accuracy requires 96 logical qubits, while grid spacing may need to shrink exponentially with particle number in some systems.Source: A. First quantised encoding methods · p. 21
  • A first-quantised basis encoding stores each of N electrons in ⌈log2(M)⌉ qubits, requiring N⌈log2(M)⌉ qubits for M single-particle basis functions.Product states must be antisymmetrised; the original construction used O(N^2log2(M)) gates and O(Nlog2(M)) ancilla qubits, with later improvements described in the paper.Source: A. First quantised encoding methods · p. 21
  • For the illustrative system with N = 2 electrons and M = 4 spin-orbitals, the first-quantised basis mapping stores the wavefunction using 4 qubits.After mapping the Hamiltonian to Pauli operators, time evolution preserves the required antisymmetry and supports chemical ground-state algorithms.Source: A. First quantised encoding methods · pp. 21–22

B. Second quantised basis set encoding methods · 1. Jordan-Wigner encoding

Second-quantised basis-set encodings map fermionic Fock-space states and operators onto qubit Hilbert spaces, with Jordan–Wigner encoding storing spin-orbital occupations directly. JW is simple and supports compact FCI-wavefunction storage, but its parity information is non-local, producing generally O(M)-weight Pauli strings.

B. Second quantised basis set encoding methods

  • Second-quantised basis-set encodings map operators on indistinguishable fermions to operators on distinguishable qubits, representing every fermionic state in qubit Hilbert space.The review focuses here on basis-set methods rather than second-quantised grid-based methods.Source: B. Second quantised basis set encoding… · p. 22

1. Jordan-Wigner encoding

  • In Jordan–Wigner encoding, each spin-orbital’s occupation number is stored locally as an unoccupied |0⟩ or occupied |1⟩ qubit state.The mapping represents fermionic occupation-bit strings as corresponding computational-basis qubit strings.Source: §1 · p. 22
  • JW maps fermionic creation and annihilation operators to qubit operators that change target occupation while preserving the exchange phase through a string of Z operators.The target operation is represented by Q or Q†, while the Z string computes the state parity.Source: §1 · p. 22
  • Under JW, the second-quantised Hamiltonian becomes a linear combination of Pauli strings, whose Pauli weight counts non-identity single-qubit Pauli operators.All second-quantised encoding methods discussed in this section produce Hamiltonians of this form.Source: §1 · p. 22
  • Using M qubits, a quantum computer can store the full configuration-interaction wavefunction, whose determinants occupy computational-basis states in the JW basis.Classically, the number of determinants scales exponentially with electron number, roughly O(M N), whereas an M-qubit register spans 2M computational-basis states.Source: §1 · p. 22
  • JW’s primary advantage is simplicity, but its local occupation encoding requires non-local parity strings, so mapped fermionic operators generally have O(M) Pauli weight.Each Z operator in the string acts on a different qubit.Source: §1 · pp. 22–23
  • Parity encoding provides an alternative that stores parity locally and occupation non-locally, although it had not yet found particular use in quantum computational chemistry.The pth qubit stores the parity of the first p modes; transformed creation and annihilation operators are described by Seeley et al. (2012).Source: §1 · p. 23

2. Bravyi–Kitaev encoding … 2. Low rank decomposition techniques

The reviewed mappings and resource-reduction techniques target lower quantum costs while balancing locality, qubit count, circuit complexity, and symmetry constraints. Bravyi–Kitaev methods provide logarithmic Pauli weight, locality-preserving alternatives trade qubits for reduced gate requirements, and decomposition or symmetry methods reduce simulation resources with stated limitations.

2. Bravyi–Kitaev encoding

  • Bravyi–Kitaev encoding balances occupation-number and parity information, producing fermionic operators with Pauli weight O(log2 M).For non-power-of-two orbital counts, the standard construction uses the next-largest power-of-two matrix and retains the first M rows.Source: §2 · p. 23
  • Across 86 molecular systems, the BK transform was at least as efficient as JW for ground-state calculations and often considerably more efficient.The BK-tree variant also has O(log2 M) operator scaling but can produce greater Pauli weight, while enabling symmetry-based qubit reduction for arbitrary orbital counts.Source: §2 · pp. 23–24

3. Locality preserving mappings

  • Locality-preserving mappings remove Jordan–Wigner Z strings and match fermionic locality, but typically require more qubits, including qubit doubling in the Verstraete–Cirac scheme.Related constructions generalize these ideas while retaining locality.Source: §3 · p. 24
  • The Bravyi–Kitaev superfast transform represents spin-orbitals as graph vertices and interactions as edges, with gate scaling O(d) and potentially fewer gates than JW for local molecular interactions.Generalizations can reduce Pauli weight to O(log d) or provide error protection, while related stabilizer codes preserve locality.Source: §3 · p. 24

C. Resource reduction

  • Resource-reduction methods focus on removing qubits through symmetries and lowering circuit costs through low-rank decompositions.The review emphasizes these approaches as general techniques for reducing quantum chemistry simulation resources.Source: C. Resource reduction · p. 24

1. Hamiltonian reduction

  • Z2 symmetry reduction removes two qubits from parity, BK, or BK-tree encodings by fixing conserved electron-number and spin-parity eigenvalues.The ground state is unchanged, but accessible excited states are restricted to the specified electron number and total sz; the standard BK case requires a power-of-two orbital count.Source: §1 · pp. 24–25

2. Low rank decomposition techniques

  • Low-rank doubly factorised Cholesky decompositions reduce molecular time-evolution costs with controllable error by truncating decomposition terms.The same technique reduces circuit repetitions for variational ground-state energy estimation, while second-quantised methods remain preferable for the smallest experimentally relevant systems.Source: §2 · p. 25

V. QUANTUM COMPUTATIONAL CHEMISTRY ALGORITHMS … a. Product formulae

The section reviews quantum algorithms for electronic structure, including phase estimation, VQE, and excited-state methods, while explaining state preparation and Hamiltonian simulation requirements. It emphasizes phase estimation’s eigenvalue resolution, state-overlap constraints, and product-formula tradeoffs for implementing time evolution.

1. Implementation

  • Phase estimation finds ground- and excited-state energies by encoding a fermionic Hamiltonian into a qubit Hamiltonian and measuring phases from controlled time evolution.The procedure uses ancillas, controlled evolutions, an inverse quantum Fourier transform, and measurement to estimate eigenvalues and collapse the register into corresponding eigenstates.Source: §1 · pp. 25–26, Figure 5 · p. 26
  • Phase-estimation energy error is bounded by ϵ_PE + ϵ_U + ϵ_CS, combining finite-precision, controlled-evolution, and circuit-synthesis errors.Optimally allocating resources among these error budgets is generally difficult.Source: §1 · pp. 26–27

2. State preparation

  • Phase estimation requires an initial state with nonzero target-eigenstate overlap and coherent implementation of a suitable efficiently invertible Hamiltonian function, often e^−iH.Random states can have exponentially vanishing ground-state overlap, making state preparation a central challenge as system size grows.Source: §1 · p. 27, §2 · p. 27
  • Adiabatic state preparation interpolates from a simple Hamiltonian, often Hartree–Fock, to the molecular Hamiltonian, with efficiency governed by the minimum spectral gap along the path.Alternative paths and driving Hamiltonians can improve efficiency, but the gap is difficult to know in advance.Source: §2 · p. 27
  • For methylene (CH2), choosing an initial state with larger ground-state overlap reduced simulated annealing times by up to four orders of magnitude (Veis and Pittner, 2014).If sufficiently large overlap is already available, phase estimation can bypass adiabatic state preparation.Source: §2 · p. 27

a. Product formulae

  • Hamiltonian simulation supplies the time-evolution operator required by phase estimation and adiabatic preparation, with product formulae approximating evolution by sequential local-term operators.First-order product formulas require S = O(t^2/ε) steps for accuracy ε; higher-order and randomized variants can improve simulation accuracy, while gate counts scale as O(poly(1/ϵ)).Source: §3 · p. 27, a. Product formulae · p. 28
  • Product formulas also simulate time-dependent Hamiltonians, whose accuracy depends on Hamiltonian derivatives, while randomization can mitigate this dependence and extrapolation can suppress finite-step errors.The ordering of terms and chosen Trotter formula additionally determine simulation error.Source: a. Product formulae · p. 28

b. Advanced Hamiltonian simulation methods … B. Variational algorithms

Advanced Hamiltonian simulation methods improve asymptotic chemistry-simulation scaling, but substantial implementation costs and unresolved tradeoffs limit near-term applicability. Variational quantum eigensolvers provide a lower-coherence hybrid alternative, while retaining important optimization, measurement, and noise challenges.

b. Advanced Hamiltonian simulation methods

  • Taylor-series and qubitization with quantum signal processing offer exponentially better accuracy scaling than product formulas, using LCU or sparse-Hamiltonian query models.Taylor simulation expands time evolution into unitary terms implemented through LCU oracles, while qubitization uses block encoding and quantum signal processing to recover the desired eigenphases.Source: b. Advanced Hamiltonian simulation… · p. 28

4. Phase estimation for chemistry simulation

  • Phase-estimation costs are analyzed across Gaussian-orbital, plane-wave, Fermi-Hubbard, and first-quantized grid simulations, with explicit gate and qubit counts treated separately.The review uses M for orbitals or lattice sites and N for electrons, and points to tabulated asymptotic developments for phase-estimation approaches.Source: §4 · pp. 28–29

a. Gaussian orbital basis sets

  • Gaussian-basis phase-estimation scaling improved from O(M 11) to empirically O(M 5), while Taylor-series methods achieved O(M 5) and O(N 2M 3) using Hamiltonian sparsity.The O(N 2M 3) result accesses nonzero CI-matrix elements, whose sparsity is O(N 2M 2).Source: a. Gaussian orbital basis sets · p. 29
  • Plane-wave representations enable scalings including O(M 2) for interaction-picture simulation, O(M 3) for qubitized condensed-phase problems, and O(N 4/3M 2/3 + N 8/3M 1/3) in first quantization.Plane waves reduce Hamiltonian-term counts, but require approximately 10 to 100 times as many basis functions as Gaussian orbitals for accurate molecular simulations.Source: a. Gaussian orbital basis sets · pp. 29–30

c. Fermi-Hubbard model

  • For the Fermi-Hubbard model, Trotterization costs effectively between O(1) and O(M 1/2) under extensive energy error, whereas qubitization achieves O(M) gates and outperforms it for intensive error.Lattice-specific algorithms can simulate time evolution in O(Mt) gates, with related product-formula bounds that are nearly asymptotically optimal.Source: c. Fermi-Hubbard model · p. 30

5. Outstanding problems

  • Although advanced methods are asymptotically superior, loose Trotter-error bounds and lower numerical gate counts leave the best practical method for chemical systems unresolved.Potential advantages of Trotterization arise because chemistry often concerns restricted states or observables rather than worst-case state errors.Source: §5 · p. 31
  • Large gate counts make these simulation methods generally fault-tolerant and unsuitable for near-term hardware lacking enough physical qubits for error correction.The review therefore motivates alternative approaches for near-future chemistry simulation.Source: §5 · p. 31

B. Variational algorithms

  • VQE uses quantum state preparation and measurement with classical parameter optimization to estimate ground-state energies, reducing coherence requirements through repeated measurements and processing.It has been demonstrated on superconducting, trapped-ion, and photonic platforms, but remains heuristic and faces optimization, measurement, and noise challenges.Source: B. Variational algorithms · pp. 31–32, Figure 6 · p. 31

1. Ans¨atze · c. Hamiltonian variational ansatz

VQE ansätze span hardware-efficient, chemically inspired, and intermediate Hamiltonian variational constructions, each trading implementability, chemical structure, scalability, and expressivity differently. The Hamiltonian variational ansatz offers problem-structured evolution with favorable resource scalings for several molecular and lattice Hamiltonians, but its efficiency depends on Hamiltonian structure and implementation choices.

1. Ans¨atze

  • Hardware-efficient ansätze use repeated, easily implementable parametrized-gate blocks suited to current hardware but generally lack chemical structure and may not scale.They have found small-molecule ground-state energies, while random initialization can make energy gradients essentially zero and classical optimization exponentially difficult as qubit number and depth increase.Source: §1 · p. 33
  • Chemically inspired ansätze adapt classical chemistry methods to quantum computers, most notably unitary coupled cluster (UCC), which builds trial states from excitations above a reference state.UCC is classically intractable but efficiently implementable on a quantum computer, and retains coupled-cluster advantages while remaining variational and compatible with multireference initial states.Source: §1 · p. 33
  • Canonical UCCSD requires O(M^3N^2) Jordan–Wigner gates, improved to O(M^3N) gates with swap networks or O(M^3)−O(M^4) with low-rank tensor decomposition.These bounds assume one Trotter step; operator ordering can nevertheless change optimized energies beyond chemical accuracy for strongly correlated systems, motivating further ordering research.Source: §1 · p. 33
  • Symmetries, reduced active spaces, and classical parameter initialization can substantially reduce the practical excitation count and ease VQE optimization.For LiH in STO-3G, symmetry reduction and an active space reduce approximately 200 candidate excitation operators to approximately 12 while retaining accurate results.Source: §1 · pp. 33–34
  • Alternative UCC-related ansätze include Bogoliubov-UCC, low-depth circuits, orbital-optimized UCCD, k-UpCCGSD, k-uCJ, adaptive-VQE, and qubit coupled-cluster.These variants target broader Hamiltonians, lower depth or gate counts, repeated selected operators, adaptive correlation recovery, or classically treated single excitations.Source: §1 · p. 34

c. Hamiltonian variational ansatz

  • The Hamiltonian variational ansatz Trotterizes an adiabatic path with variationally optimized evolution times, using a parameter count linear in the number of Trotter steps S.It lies between hardware-efficient and chemically inspired ansätze and was inspired by adiabatic state preparation and the quantum approximate optimization algorithm (Farhi et al., 2014).Source: c. Hamiltonian variational ansatz · p. 34
  • For Gaussian molecular orbitals, Hamiltonian variational implementation scales approximately as O(M^4), while low-rank decomposition reduces Trotter-step costs to O(M^2log^2(M)) or O(M^3).For plane-wave dual-basis Hamiltonians, Trotter steps can use O(M^2) two-qubit gates at depth O(M).Source: c. Hamiltonian variational ansatz · p. 34
  • The ansatz is especially suitable for Fermi–Hubbard Hamiltonians, whose locality enables O(M) depth and O(M^1.5) gates on linear nearest-neighbor arrays.Two-dimensional arrays reduce Trotter steps to O(M) gates, while locality-preserving mappings can achieve constant depth at the cost of extra qubits and potentially large gate overhead.Source: c. Hamiltonian variational ansatz · p. 34

2. Measurement … 1. Quantum subspace expansion

The review surveys measurement costs, classical optimisation strategies, and excited-state methods for near-term VQE chemistry. Quantum subspace expansion is especially versatile, but its measurement burden and limited excitation space remain important constraints.

2. Measurement

  • O(M^6/ϵ^2) measurements are required in a Gaussian orbital basis and O(M^4/ϵ^2) in a plane-wave dual basis for energy estimation.For a 112 spin-orbital Fe2S2 molecule, about 10^13 samples may be required per energy evaluation.Source: §2 · p. 34
  • Measurement costs can be reduced by commuting-term grouping, unitary transforms, fermionic-level grouping, and covariance-aware grouping, with some schemes using O(M^2) circuits.For energy-only estimation, low-rank decompositions and orbital rotations produce O(M) groups and reduced small-molecule measurement requirements by several orders of magnitude (Huggins et al., 2019b).Source: §2 · p. 35

3. Classical optimisation

  • VQE optimisation must be fast, accurate, and noise-robust; direct-search methods are generally more noise-tolerant, whereas gradient methods may require fewer evaluations.This tradeoff motivates comparing both optimisation classes for near-term devices.Source: §3 · p. 35

a. Previous optimisation studies

  • Small-system experiments found good noisy convergence for several direct-search methods and SPSA, while canonical gradient descent struggled under current device noise.These experiments remain limited because available qubit counts restrict the optimised parameter spaces.Source: a. Previous optimisation studies · p. 35
  • Out-of-the-box Nelder-Mead was outperformed by other direct-search and gradient-based methods, but all methods could become trapped in local minima; good initialisation partly mitigated this.Realistic noise has rarely been included in most numerical studies, leaving practical effectiveness uncertain.Source: a. Previous optimisation studies · pp. 35–36

b. Related methods of optimisation

C. Evaluation of excited states

  • Excited-state methods target spectral properties and specific spin or charge states, but no clear best method has emerged; QSE/EoM comparisons found EoM more noise-robust experimentally and more accurate for excitation energies numerically.QSE targets low-lying states through excitation-based subspaces, but generally requires O(M^8) 4-RDM elements and may fail when higher-order excitations are needed.Source: C. Evaluation of excited states · p. 36, §1 · p. 37

1. Quantum subspace expansion

  • QSE supports excited-state estimation, improved ground-state energies, and hardware-error mitigation, making it one of the most useful near-term quantum-chemistry techniques.It was experimentally demonstrated for H2 on a two-qubit superconducting system.Source: §1 · p. 37

2. Overlap-based methods … B. Probabilistic error cancellation

The reviewed methods target excited-state estimation and near-term chemistry under noise by exploiting eigenstate orthogonality or mitigating errors without full error correction. These approaches reduce resource demands but incur measurement, circuit, noise-model, or scalability limitations.

2. Overlap-based methods

  • Overlap-based methods find excited states by penalizing overlap with previously obtained lower-energy eigenstates, then repeating the procedure for higher states.The updated Hamiltonian makes the first excited state the ground state, while overlap terms can be measured with circuits such as the SWAP test (Endo et al., 2018a; Higgott et al., 2018).Source: §2 · p. 38

3. Contraction VQE methods

  • Contraction VQE methods prepare orthonormal initial states, preserve orthogonality under a shared unitary ansatz, and optimize a low-energy subspace rather than sequentially targeting states.SSVQE uses a hybrid ordering procedure, whereas MCVQE classically diagonalizes a Hamiltonian matrix in the optimized subspace (Nakanishi et al., 2018; Parrish et al., 2019b).Source: §3 · p. 38
  • Contraction VQE obtains eigenstates simultaneously with similar expected accuracy, but finding one unitary circuit that prepares many states equally well may hinder practical realization.Overlap-based methods instead obtain states sequentially, while overlap calculations add circuit and qubit resources and are sensitive to errors in lower-state approximations (Higgott et al., 2018).Source: §3 · p. 38, §2 · p. 38

VI. ERROR MITIGATION FOR CHEMISTRY

  • Near-term error-mitigation methods seek accurate chemistry results with low-depth circuits and substantially less overhead than full error correction, but they are effective only when circuit error rates remain sufficiently low.VQE can be robust to some coherent errors, yet experiments show that noise can still prevent the desired accuracy; mitigation generally adds a multiplicative measurement overhead and is not scalable to arbitrarily large computations.Source: §VI · pp. 38–39

A. Extrapolation

B. Probabilistic error cancellation

  • Probabilistic error cancellation effectively applies the inverse noise channel by randomly inserting operators, combining signed expectation values, and recovering an estimate of the noiseless measurement.For depolarizing noise, the procedure measures operator-transformed observables and linearly combines them; its variance increases by γ^G for a circuit with G gates.Source: B. Probabilistic error cancellation · pp. 39–40, Figure 9 · p. 40
  • Because exhaustive correction is impractical for many gates, probabilistic cancellation samples the most important low-error circuit variants, applies Pauli operators with prescribed probabilities, and combines their parity-weighted results.The construction extends to multi-qubit gates by inserting specified Pauli pairs after two-qubit gates.Source: B. Probabilistic error cancellation · p. 40
  • Probabilistic error cancellation applies to general Markovian noise and extensions of correlated or low-frequency noise, but requires a detailed per-gate noise model obtained through process or gate-set tomography.It has been experimentally demonstrated on superconducting and trapped-ion systems (Endo et al., 2017; Huo and Li, 2018; Song et al., 2018; Zhang et al., 2019).Source: B. Probabilistic error cancellation · p. 40

C. Quantum subspace expansion … A. Hydrogen

The section surveys QSE, symmetry-based, and other error-mitigation methods, then illustrates quantum-chemistry mappings and shows how basis-set choice affects H2 ground-state energies. It emphasizes subspace measurements, conserved-quantity checks, and explicit hydrogen examples as practical routes to more accurate calculations.

C. Quantum subspace expansion

  • QSE mitigates VQE errors and can also estimate excited states by constructing a Pauli-generated subspace around an approximate ground state.It is most effective against systematic errors but can suppress some stochastic errors as well.Source: C. Quantum subspace expansion · p. 40
  • QSE measures Hamiltonian and overlap matrices in the nonorthogonal Pauli subspace, then solves a generalized eigenvalue problem to obtain an error-mitigated spectrum.Using only a small number of Pauli operators limits the measurement cost; the method was demonstrated on a two-qubit superconducting system for H2 energies.Source: C. Quantum subspace expansion · pp. 40–41

D. Symmetry based methods

  • Symmetry checks mitigate errors by rejecting outcomes that violate conserved quantities such as electron number or s_z, which is important because particle-number errors increase energy variance.Implementations include ancilla stabilizer checks, additional measurements with post-processing, n-representability constraints, and low-rank plus orbital-rotation measurements.Source: D. Symmetry based methods · p. 41
  • The low-rank plus orbital-rotation method appears especially powerful because it uses modest circuit depth and limited connectivity while enabling post-selection on electron number and s_z.Symmetry mitigation can also be combined with extrapolation and related techniques.Source: D. Symmetry based methods · p. 41
  • Error-correcting extensions engineer additional symmetries, using repeated measurements and stochastic stabilizer sampling to detect some errors while retaining polynomial cost.Fermion-to-qubit mappings can likewise add constrained qubits whose stabilizer checks detect and correct single-qubit errors (McClean et al., 2019b; Setia et al., 2018; Jiang et al., 2018a).Source: D. Symmetry based methods · p. 41

E. Other methods of error mitigation

VII. ILLUSTRATIVE EXAMPLES

  • The illustrative examples map electronic-structure problems for H2 and LiH using JW, BK, and BK-tree mappings, active-orbital reduction, symmetry-based qubit reduction, and the unitary coupled-cluster ansatz.The examples cover H2 in STO-3G, 6-31G, and cc-PVDZ and LiH in STO-3G, demonstrating workflows applicable to ground-state and general chemical problems.Source: §VII · p. 41

A. Hydrogen

  • 1.6 mHartree: energy differences among H2 minima across STO-3G, 6-31G, and cc-PVDZ exceed chemical accuracy, showing that sufficiently large basis sets are crucial.Larger orbital sets recover more correlation energy and produce more accurate ground-state estimates.Source: Figure 10 · p. 42, A. Hydrogen · p. 42

1. STO-3G basis

The STO-3G treatment of H2 reduces each hydrogen atom to one 1s orbital, yielding four spin-orbitals and a four-qubit Jordan–Wigner Hamiltonian. It then constructs Hartree–Fock and correlated ground-state descriptions and derives a UCCSD circuit whose simplified form reproduces the ground state.

1. STO-3G basis

  • STO-3G H2 has one 1s orbital per hydrogen, represented by three Gaussian functions, producing four spin-orbitals and a four-qubit Jordan–Wigner Hamiltonian.Molecular spin-orbitals are formed from bonding and antibonding combinations of the atomic orbitals, with normalization determined by their overlap.Source: §1 · p. 42
  • At equilibrium separation, the Hartree–Fock determinant dominates H2’s ground state, while the antibonding determinant supplies dynamical correlation.At large separation, the bonding and antibonding determinants contribute equally because they become degenerate, producing static correlation and requiring both configurations.Source: §1 · p. 43
  • For H2, UCCSD retains the single and double excitation operators that conserve electron number and spin projection, with variational parameters optimized to prepare the correlated state.Jordan–Wigner encoding maps these excitation operators to Pauli-string exponentials, and commuting terms can be separated using Trotterization (Romero et al., 2019).Source: §1 · p. 43
  • Hempel et al. (2018) simplify H2’s UCCSD implementation by realizing single excitations as basis rotations and combining double-excitation terms, a strategy that is not generally scalable.The simplification relies on H2 having only one double excitation operator and is implemented by the circuit in Fig. 11.Source: §1 · p. 43
  • The simplified UCCSD operator applied to the Hartree–Fock state reproduces the H2 ground state.This connects the compact circuit construction to the correlated ground-state wavefunction described above.Source: §1 · p. 43

2. 6-31G basis … VIII. DISCUSSION AND CONCLUSIONS

The section demonstrates basis-specific fermion-to-qubit reductions for H2 and LiH, then outlines the review’s comparison of quantum and classical methods, resource demands, and routes toward future calculations.

2. 6-31G basis

  • The 6-31G H2 example also illustrates UCCSD circuit construction through basis rotations implementing X- and Y-type exponentiated operators.Single-excitation terms are implemented with a change of basis.Source: Figure 11 · p. 44, §2 · p. 44
  • For 6-31G H2, the eight-spin-orbital Bravyi-Kitaev representation removes two conserved qubits, reducing the Hartree–Fock state from eight to six qubits.The fourth and eighth encoded entries store conserved spin-up and total electron counts; the reduced BK state is |011011⟩.Source: §2 · pp. 43–44

3. cc-PVDZ basis

  • For cc-PVDZ H2, natural-orbital occupations identify one effectively empty spatial orbital, reducing 20 spin-orbitals to 18 before Bravyi-Kitaev-tree mapping.The NMO basis is obtained by diagonalizing a CISD-derived 1-RDM; the fifth orbital’s NOON is 20 times smaller than the next smallest.Source: §3 · p. 44
  • The cc-PVDZ H2 mapping then removes conserved spin-up and total-electron qubits, yielding a 16-qubit problem with mapped state |0001011100010111⟩.The corresponding Jordan–Wigner state uses 18 qubits: |000000001000000001⟩.Source: §3 · p. 44

B. Lithium Hydride STO-3G basis

  • For LiH in STO-3G, natural-orbital occupations remove effectively doubly occupied and empty orbitals, reducing the Hamiltonian from 12 to 8 spin-orbitals.The procedure uses a CISD-derived 1-RDM and then applies BK or BK-tree mappings, with BK-tree chosen for an explicit Fenwick-tree example.Source: B. Lithium Hydride STO-3G basis · pp. 44–45
  • LiH’s final Hamiltonian acts on six qubits while differing from the full 12-qubit Hamiltonian by only 0.2 mHartree.Two qubits associated with conservation symmetries are removed after the orbital reduction.Source: B. Lithium Hydride STO-3G basis · p. 45

VIII. DISCUSSION AND CONCLUSIONS

  • The discussion and conclusions compare quantum techniques with classical methods, assess their currently unavailable resource requirements, and propose heuristic large-system and exact small-system routes.The review concludes with a blueprint for future investigations.Source: §VIII · p. 46

A. Classical limits … D. Summary and outlook

The review identifies a classical accuracy-and-size gap where quantum computers may help, but current fault-tolerant and near-term approaches remain resource- and noise-limited. It therefore emphasizes algorithmic development, benchmarking, and collaboration between computational chemists and quantum information scientists.

A. Classical limits

  • Classical methods trade accuracy against system size: DFT and HF handle large systems qualitatively, exact methods scale exponentially, while CC, tensor networks, and Monte Carlo cover intermediate regimes with limitations.CCSD reaches hundreds of spin-orbitals without strong static correlation; tensor networks and Monte Carlo reach larger strongly correlated systems but face scaling or sign-problem constraints.Source: A. Classical limits · pp. 46–47
  • A classical “sweet spot” of roughly 100–200 spin-orbitals is too strongly correlated for HF, DFT, or CCSD, too large for FCI, and a promising target for quantum computers.Around 100 perfect qubits could calculate the FCI energy of a 100-spin-orbital system in polynomial time, while problem decomposition may extend relevance to larger molecules.Source: A. Classical limits · pp. 47–48

B. Quantum resources: medium to long-term

  • Surpassing classical chemistry methods still requires on the order of 100,000 physical qubits, although condensed-phase and Fermi-Hubbard simulations require fewer resources than molecular simulations.Resource estimates include around 10^11 Toffoli gates for FeMo-co and around 2–7.1 × 10^8 T gates for representative condensed-matter problems.Source: B. Quantum resources: medium to… · pp. 48–49

C. Quantum resources: near to medium-term

  • VQE may fit 100–200-qubit near-term devices, but its heuristic optimization is difficult, noise limits circuits to roughly 10,000 gates, and Fermi-Hubbard estimates can reach O(M^2) gates.Error-corrected VQE may still require approximately 4 × 10^5 to 4 × 10^6 T gates, comparable to phase-estimation approaches.Source: C. Quantum resources: near to… · p. 50
  • Known small-molecule energies and benchmark targets such as small Fermi-Hubbard models and the G1 molecular set provide staged tests before pursuing classically intractable chemistry.The review argues that reproducing classically known results should precede claims of quantum advantage and later error-corrected demonstrations.Source: C. Quantum resources: near to… · pp. 50–51

D. Summary and outlook

  • The review bridges classical computational chemistry and quantum algorithms while stressing that high hardware error rates and low qubit counts remain decisive obstacles.Progress depends on larger, better-controlled machines and improved algorithms, mappings, error-correction codes, basis sets, and error-mitigation techniques.Source: D. Summary and outlook · p. 51
  • Future work should systematically compare variational proposals in noisy and noiseless simulations, explore chemistry-specific error correction and classical ideas, and strengthen chemist–quantum-information collaboration.Simulations of roughly 30–40 qubits can help identify promising near-term methods before migration to real hardware.Source: D. Summary and outlook · p. 51

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