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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 targets chemically important systems that are difficult to simulate classically, while available quantum hardware remains too limited for many such problems. This review synthesizes computational chemistry and quantum computing, explains mappings and algorithms with emphasis on near-term methods, and surveys resource requirements and future directions.

  • Problem

    Strongly correlated chemical systems, including transition-metal catalysts, can be difficult to study with classical methods, while quantum computers require substantial resources to address them.

  • Method

    The review combines introductions to computational chemistry and quantum computing with mappings, algorithms, resource-reduction techniques, and worked quantum-circuit examples.

  • Results

    Current estimates indicate that surpassing classical techniques still requires on the order of 100,000 physical qubits, while current quantum computers possess around 100.

  • Takeaways & Limitations

    Near-term research should test variational algorithms on systems of around 30 to 40 qubits, with and without noise, before migrating promising methods to real hardware.

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

Quantum computational chemistry uses quantum computers to address chemistry problems whose wavefunctions are exponentially difficult to simulate classically. This review bridges quantum computing and computational chemistry, emphasizing resource-efficient and near-term approaches.

  • I. INTRODUCTION: Wavefunction complexity grows exponentially with particle number, preventing classical computers from exactly simulating general quantum systems efficiently.Feynman proposed quantum simulation as a response to this limitation.
  • I. INTRODUCTION: Quantum simulation may address classically intractable problems in chemistry, physics, and materials science, including transition-metal catalysis and high-temperature superconductivity.Classical simulations remain important for rationalising experiments, testing models, and understanding system properties.
  • I. INTRODUCTION: Existing quantum algorithms are expected to use polynomial runtime and physical resources, but experiments remain limited to small systems already simulable classically.Current gate counts and hardware capabilities motivate methods requiring fewer quantum resources.
  • I. INTRODUCTION: The review focuses on low-lying energy levels, or the electronic structure problem, because these determine reaction rates, stable structures, and optical properties.The review also develops the terminology and methods needed to map chemistry problems onto quantum computers.
  • I. INTRODUCTION: The field has expanded rapidly, so the review prioritizes general chemistry-simulation methods and tractability techniques rather than covering every related approach.Quantum machine learning and problem-decomposition approaches are explicitly outside the review’s scope.
  • I. INTRODUCTION: The review adopts the qubit-based circuit model while distinguishing it from alternative paradigms such as adiabatic, measurement-based, and continuous-variable quantum computing.It introduces qubits, quantum gates, superposition, measurement, and circuits as foundational concepts.

B. Quantum simulation

Quantum simulation maps a physical Hamiltonian and initial state onto qubits, implements time evolution with quantum circuits, and extracts observables through measurement. The review distinguishes digital and analog simulation and discusses static, dynamics, and electronic-structure problems.

  • B. Quantum simulation: Digital quantum simulation maps a target problem onto implementable quantum gates, whereas analog simulation emulates a specific real system and is generally more robust to noise.Digital simulation is considered necessary for accurate large chemical simulations, although it is more vulnerable to device imperfections.
  • B. Quantum simulation: Dynamics simulations evolve wavefunctions in time and measure changing observables, while static simulations estimate observables from a mapped target wavefunction.Examples include charge densities, particle correlation functions, and energy eigenvalues.
  • B. Quantum simulation: Hamiltonian simulation maps Hs to Hq, maps the initial wavefunction to qubits, and approximates time evolution with a Lie-Trotter-Suzuki circuit.The Hamiltonian is decomposed into local terms whose short-time evolutions are implemented as gates.
  • B. Quantum simulation: Polynomially many Hamiltonian terms arise for chemical systems because Coulomb interactions are two-body, and each exponential term can be implemented efficiently on a quantum computer.This places the dynamics problem in BQP, although BPP ≠ BQP remains unproven.
  • B. Quantum simulation: Variational approaches use parametrized gates and updated parameters, potentially reducing gate counts relative to Trotterization for open and closed-system time evolution.The update rule is algorithm-specific.
  • B. Quantum simulation: Electronic-structure calculations seek low-lying energy levels, a task generally exponentially difficult classically but widely believed efficiently solvable quantumly for physically relevant chemical systems.The review develops the classical background before presenting quantum mappings and algorithms.

III. CLASSICAL COMPUTATIONAL CHEMISTRY

Classical computational chemistry formulates molecular electronic structure using antisymmetric many-electron wavefunctions and complementary basis-set or grid representations. The review explains why exact descriptions become intractable and motivates quantum approaches for strongly correlated systems.

  • III. CLASSICAL COMPUTATIONAL CHEMISTRY: The Born-Oppenheimer approximation treats nuclei as classical point charges, leaving an electronic Hamiltonian to solve for each fixed nuclear configuration.Chemical accuracy is defined here as 1.6 × 10^-3 Hartree.
  • III. CLASSICAL COMPUTATIONAL CHEMISTRY: The electronic structure problem seeks energy eigenstates and eigenvalues, especially ground and low-lying excited states, across nuclear configurations.These energies define potential-energy surfaces and support chemical predictions.
  • III. CLASSICAL COMPUTATIONAL CHEMISTRY: Strongly correlated systems such as transition-metal catalysts and high-temperature superconductors remain difficult for classical methods, motivating quantum-computational study.Nitrogenase mechanisms remain incompletely understood, while competing orders complicate Fermi-Hubbard ground-state calculations.
  • III. CLASSICAL COMPUTATIONAL CHEMISTRY: Electronic wavefunctions must be antisymmetric under electron exchange, represented explicitly in first quantisation or through operator properties in second quantisation.These representations affect how physical systems are simulated on quantum computers.
  • III. CLASSICAL COMPUTATIONAL CHEMISTRY: Basis-set methods project the Hamiltonian onto M basis wavefunctions that approximate electron spin-orbitals, reducing resources by exploiting known orbital structure.First and second quantisation are distinct from grid-based and basis-set discretisation.
  • III. CLASSICAL COMPUTATIONAL CHEMISTRY: Grid-based first-quantised simulation stores the wavefunction on a spatial grid, requiring P^3N × 2^N complex amplitudes for N electrons.This memory requirement scales exponentially and makes more than a few particles classically intractable on a grid.

b. Basis set methods

Basis-set methods encode antisymmetric many-electron states with Slater determinants and, more compactly, occupation-number and second-quantised representations. Exact and approximate classical methods trade computational cost against accuracy and size-related properties.

  • b. Basis set methods: Slater determinants enforce fermionic exchange antisymmetry by changing sign when two electrons are exchanged.Only N of the M available spin-orbitals are occupied in a determinant.
  • b. Basis set methods: Second quantisation represents determinants with occupation-number vectors in Fock space and uses creation and annihilation operators to change orbital occupations.The occupation operator counts electrons in a specified spin-orbital.
  • b. Basis set methods: The electronic Hamiltonian contains one- and two-body terms, including electron kinetic, nuclear-attraction, electron-repulsion, and nuclear-repulsion contributions.Two-body Coulomb interactions limit the operator structure to at most four creation and annihilation operators and up to M^4 terms.
  • b. Basis set methods: Full configuration interaction is exact in a complete orbital basis but contains exponentially many determinants, making large calculations classically intractable.Truncated configuration interaction avoids some cost but is neither size extensive nor size consistent.

b. Grid based methods

The review presents second-quantised basis-set methods and classical approximations for electronic wavefunctions, emphasizing how correlation and basis restrictions affect tractability.

  • Hartree–Fock: Hartree–Fock iteratively optimises orbitals in an average electron-charge potential to obtain a dominant Slater determinant.The procedure constructs and diagonalises the Fock operator until the orbitals converge.
  • Correlation: Hartree–Fock neglects dynamic and static correlation, limiting its performance for strongly correlated systems such as transition metals and dissociation limits.Static correlation arises when multiple Slater determinants are equally dominant, while dynamic correlation reflects electron–electron repulsion corrections.
  • Multiconfigurational methods: MCSCF optimises determinant amplitudes and orbital rotations, providing an effective approximation for strong static correlation but scaling exponentially with electron number.Chemical intuition can restrict the determinant space, and MCSCF/CASSCF are among the most effective classical approaches for strong static correlation.
  • Configuration interaction: Configuration interaction adds excitations above a reference state, recovering full configuration interaction only when all determinants are included.Truncated CI converges slowly and is neither size extensive nor size consistent, affecting calculations involving non-interacting reactants.

4. Coupled cluster

The review introduces coupled cluster as a product-parametrised correlation method and discusses basis-set choices that trade accuracy, resource requirements, and integral complexity.

  • Coupled cluster: Coupled cluster includes additional determinants through a product parametrisation, which gives faster convergence than configuration interaction.The product form includes all possible determinants in the trial wavefunction, although with an incorrect parametrisation.
  • Coupled cluster: CCSD solves coupled nonlinear equations instead of storing its exponentially large wavefunction, with time scaling as O((M − N)^4N^2) and integral memory as O(M^4).CCSD(T) adds perturbative triple excitations for greater accuracy, with higher computational cost.
  • Coupled cluster: Coupled cluster ensures size extensivity and size consistency but does not obey the Rayleigh–Ritz variational principle.Its conventional single-determinant reference also performs poorly for multireference states with strong static correlation.
  • Orbital basis sets: Gaussian-type orbitals simplify two-electron integral evaluation but require more functions than Slater-type orbitals to describe a given orbital.GTOs are more localised because their exponent depends on r^2, while STOs better approximate atomic charge distributions.
  • Orbital basis sets: Plane-wave basis sets reduce Hamiltonian-term counts from O(M^4) to O(M^3), while the plane-wave dual basis reduces them to O(M^2).These bases diagonalise kinetic and potential operators, respectively; efficient basis construction remains an open research problem.

E. Reduction of orbitals

The review describes orbital reduction and first-quantised encodings as ways to represent chemistry problems with fewer resources, while identifying accuracy and scaling limits of grid methods.

  • E. Reduction of orbitals: Orbitals with occupation numbers close to 0 or 1 can be removed, leaving only ambiguously occupied spin-orbitals in the calculation.Reduced density matrices from an approximate ground state provide the information used to identify these orbitals.
  • First quantised encoding methods: First-quantised methods encode each particle in a discrete basis or spatial grid and must explicitly antisymmetrise the resulting wavefunction.A single-particle basis with M functions can encode an orbital index using ⌈log2(M)⌉ qubits, followed by antisymmetrisation.
  • First quantised encoding methods: Grid-based representations store an N-particle wavefunction over P grid points per spatial axis, with wavefunction amplitudes assigned to grid configurations.The review illustrates this representation using one- and two-electron one-dimensional examples.
  • First quantised encoding methods: Grid resolution cannot exponentially improve accuracy because known grid algorithms have gate counts polynomial in the inverse grid spacing.Some systems require exponentially finer spacing with particle number to maintain constant accuracy, making them inefficient for this method.
  • First quantised encoding methods: A grid representation can require considerably more qubits than a basis-set approach; storing one spinless particle to 32-bit positional accuracy requires 96 logical qubits.Basis-set simulations of interesting molecules or Fermi–Hubbard models are described as requiring around 100 logical qubits.

B. Second quantised basis set encoding methods

Second-quantised basis-set encodings map fermionic Fock states and operators onto qubits, enabling chemistry Hamiltonians to be expressed as measurable Pauli strings.

  • B. Second quantised basis set encoding methods: An encoding maps fermionic Fock space to qubit Hilbert space so that every fermionic state has a corresponding qubit state.The review focuses on second-quantised basis-set mappings and their resource-reduction techniques.
  • B. Second quantised basis set encoding methods: Jordan–Wigner encoding stores each spin-orbital’s occupation number in one qubit, using |0⟩ for unoccupied and |1⟩ for occupied.Its main advantage is simplicity, but parity is stored non-locally.
  • B. Second quantised basis set encoding methods: Under Jordan–Wigner encoding, the second-quantised Hamiltonian becomes a linear combination of products of single-qubit Pauli operators.Each product is a Pauli string, and its Pauli weight counts the non-identity single-qubit operators it contains.
  • B. Second quantised basis set encoding methods: Jordan–Wigner fermionic operators generally map to Pauli strings with O(M) non-identity operators because of the Z-operator parity string.The Z string reproduces the exchange phase factor required by fermionic antisymmetry.

2. Bravyi–Kitaev encoding

The Bravyi–Kitaev encoding balances local occupation and parity information through partial sums, producing logarithmic Pauli weight while offering practical efficiency advantages over Jordan–Wigner in many systems.

  • Encoding structure: BK stores partial sums of orbital occupation numbers, balancing the locality of occupation and parity information.The included occupation numbers are determined by the BK matrix β_pq.
  • Encoding structure: Applying BK to a fermionic operator produces a qubit operator with Pauli weight O(log2 M).
  • Performance and trade-offs: Across 86 molecular systems, BK was at least as efficient as Jordan–Wigner for finding ground-state energies and was considerably more efficient in many cases.
  • Resource reduction: The review emphasizes qubit-removal symmetries and low-rank decompositions as general strategies for reducing quantum-chemistry simulation resources.
  • Resource reduction: Symmetry reduction can remove two qubits by exploiting conserved electron number and spin, but restricts accessible excited states to matching quantum numbers.

V. QUANTUM COMPUTATIONAL CHEMISTRY ALGORITHMS

The review presents phase estimation and variational methods for electronic-structure calculations, alongside state-preparation and noise-related considerations. It emphasizes that phase estimation requires suitable eigenstate overlap and coherent Hamiltonian evolution, while adiabatic preparation depends on the spectral gap and can benefit from better initial states.

  • Algorithm scope: Both phase estimation and VQE are discussed for ground-state energies, while related methods address excited states and energy derivatives.
  • Phase estimation: Quantum phase estimation can find ground and excited energy eigenstates after encoding a fermionic Hamiltonian as a qubit Hamiltonian.
  • Phase estimation: The canonical circuit applies controlled Hamiltonian evolutions, an inverse QFT, and measurement to estimate energy eigenvalues and collapse the register to corresponding eigenstates.
  • Phase estimation: For 0.5 success probability, phase estimation requires ω = n + 2 ancilla qubits and total coherent evolution time T = 8π/ϵPE.
  • Phase estimation: The phase-estimation energy error is bounded by ϵPE + ϵU + ϵCS, although optimally allocating resources among these error budgets is difficult.
  • State preparation: Phase estimation needs nonzero target-state overlap and coherent implementation of a suitable Hamiltonian-dependent unitary.
  • State preparation: A randomly chosen initial state can have exponentially vanishing probability of collapsing to the ground state as system size increases.
  • State preparation: Adiabatic state preparation interpolates slowly from a simple Hamiltonian to the target Hamiltonian, with efficiency depending on the minimum spectral gap along the path.

b. Advanced Hamiltonian simulation methods

Advanced Hamiltonian simulation methods reduce the asymptotic cost of quantum chemistry algorithms through techniques such as Taylor series expansions, qubitization, and improved chemical representations. Their relative practical performance remains unsettled because gate counts, time dependence, and classical preprocessing impose different trade-offs.

  • Taylor-series and quantum-signal-processing methods can scale exponentially better with simulation accuracy than product formulas.
  • Advanced simulation methods reduce asymptotic chemistry-algorithm scaling, including O(M^5) and O(N^2M^3) algorithms for Gaussian-basis molecules.The O(N^2M^3) result uses oracle circuits accessing nonzero Hamiltonian elements in the Slater-determinant basis.
  • For plane-wave representations, interaction-picture simulation scales as O(M^2), while qubitization achieves O(M^3) gates for condensed-phase electronic structure.The interaction-picture algorithm requires O(Mlog^2(M)) qubits.
  • Alternative representations improve qubitization scaling: discontinuous Galerkin basis sets were estimated at around O(M^2.6), and first-quantized plane waves achieve sublinear scaling in M.The first-quantized qubitization scaling is O(N^4/3M^2/3 + N^8/3M^1/3).
  • For the Fermi-Hubbard model, qubitization has O(M) gate scaling, outperforming a Trotter-based method when targeting intensive energy error.Trotterization was reported to cost effectively between O(1) and O(M^1/2) when targeting extensive energy error.
  • No simulation method can yet be declared best for chemistry because Taylor series may have large T-gate counts, while quantum signal processing lacks a time-dependent variant and requires intensive classical preprocessing.

1. Ans¨atze

The review contrasts hardware-efficient, chemically inspired, and Hamiltonian variational ansätze for VQE, emphasizing their implementation trade-offs and resource scaling. It also notes that Trotterized UCCSD costs depend on operator ordering and can lose chemical accuracy for strongly correlated systems.

  • Hardware efficient ansätze: Hardware-efficient ansätze use repeated parametrized gate blocks designed for limited coherence times and constrained gate connectivity.They seek flexible trial states with few implementable gates.
  • Chemically inspired ansätze: Chemically inspired ansätze adapt classical chemistry methods, especially coupled cluster, into quantum-efficient unitary coupled cluster trial states.UCC constructs parametrized states from excitations above a reference state and is typically truncated to UCCSD.
  • Chemically inspired ansätze: UCCSD requires O(M 3N 2) gates with Jordan-Wigner mapping, improved to O(M 3N) gates and depth O(M 2N) using swap networks.A low-rank decomposition gives O(M 3)−O(M 4) gates, while UCC retains variationality and can converge from multireference states.
  • Chemically inspired ansätze: Trotterized UCCSD energies can vary by more than chemical accuracy when excitation operators are reordered in strongly correlated systems.The stated gate counts assume that one Trotter step provides accurate results, an assumption questioned by subsequent work.
  • Hamiltonian variational ansatz: The Hamiltonian variational ansatz uses Hamiltonian terms in a Trotterized structure, with efficiency determined by the number of terms.Its Gaussian-basis scaling is approximately O(M 4), while low-rank decompositions reduce Trotter-step costs to O(M 2log2(M)) for increasing molecular size and O(M 3) for fixed molecular size.
  • Hamiltonian variational ansatz: For plane-wave dual-basis Hamiltonians, Trotter steps can use O(M 2) two-qubit gates at depth O(M).The Hamiltonian contains O(M 2) terms in this basis.
  • Hamiltonian variational ansatz: For the Fermi-Hubbard model, nearest-neighbour layouts support Trotter steps with O(M) depth and O(M 1.5) gates on linear arrays, or O(M) gates on two-dimensional arrays.Locality-preserving mappings can achieve constant depth but require additional qubits and may incur large constant-factor gate overhead.

2. Measurement

Measurement and optimization are central bottlenecks for VQE: energy estimation can require steeply growing shot counts, while noisy, high-dimensional optimization remains difficult. The review surveys grouping, low-rank measurement reduction, gradient methods, initialization, and related strategies.

  • Measurement: Hamiltonian averaging estimates each Hamiltonian term through repeated state preparation and measurement, and can also calculate the 1-RDM and 2-RDM.The required measurements for energy precision ϵ scale as O(M 6/ϵ2) in Gaussian bases and O(M 4/ϵ2) in plane-wave dual bases.
  • Measurement: Around 1013 samples per energy evaluation were estimated for a 112 spin-orbital Fe2S2 molecule, motivating measurement-cost reduction strategies.Commuting-term grouping appears to reduce cost by a constant factor, while transformed grouping can reduce term groups from O(M 4) to O(M 3).
  • Measurement: Low-rank Hamiltonian decompositions and orbital rotations can divide an energy-measurement problem into O(M) groups and reduce measurements for small molecules by several orders of magnitude.This result applies when only the energy expectation value, rather than the 1- and 2-RDMs, is needed.
  • Optimization: VQE optimizers must be fast, accurate, and robust to substantial near-term hardware noise; direct-search methods are generally more noise-robust but may require more function evaluations.The review distinguishes direct-search and gradient-based optimization classes.
  • Optimization: Experimental VQE studies use small parameter spaces, limiting how well their optimizer results predict performance on larger systems, although they reveal noise tolerance.Nelder-Mead was outperformed by other direct-search and gradient-based methods in out-of-the-box comparisons, while all methods could become trapped in local minima.
  • Optimization: SPSA has found small-molecule ground-state energies despite shot noise and physical errors, whereas canonical gradient descent methods have struggled.SPSA updates along randomly chosen directions in parameter space.
  • Optimization: Realistic noise has rarely been included in optimization studies, leaving the practical effectiveness of many methods unclear.Sequential optimization, meta-learning, analytic quantum gradients, and annealed variational methods are among the surveyed approaches.
  • Quantum subspace expansion: Quantum subspace expansion was introduced for excited states and also supports better ground-state estimation and hardware-error mitigation.It was experimentally demonstrated for ground and excited states of H2 in a two-qubit superconducting system.

2. Overlap-based methods

Overlap-based methods obtain additional eigenstates by enforcing orthogonality, while error-mitigation methods approximate noiseless observables without full error correction. These approaches reduce near-term limitations but remain resource- or depth-dependent.

  • Overlap-based methods: After VQE finds a ground state, overlap-based methods add an energy penalty for overlap with that state, making the first excited state the new target.Repeating the procedure can obtain higher energy eigenstates.
  • Overlap-based methods: SSVQE and MCVQE generate approximations to a low-energy subspace from orthonormal initial states while enforcing eigenstate orthogonality through variational procedures.MCVQE classically diagonalizes a Hamiltonian matrix in the low-energy subspace, whereas SSVQE uses a VQE-like hybrid approach.
  • Error mitigation: Noise can make near-term chemistry calculations inaccurate, so additional mitigation techniques are needed when full error correction is unavailable.Experiments have shown that noise can prevent VQE from reaching desired accuracy levels.
  • Error mitigation: Error mitigation approximates noiseless measurement results from noisy results at sufficiently low error rates, generally requiring multiplicative measurement overhead rather than full error-correction resources.These methods are effective only for low-depth circuits and are not scalable solutions to hardware noise.
  • Extrapolation: Extrapolation increases the dominant gate-error rate by a factor λ and infers the error-free observable by fitting measurements across error rates.Linear extrapolation uses Richardson extrapolation with two points, while exponential extrapolation is described as more appropriate for large circuits.
  • Extrapolation: Extrapolation improves accuracy but requires additional measurements to maintain the variance of the non-extrapolated observable.It has been demonstrated in molecular chemistry VQE experiments and nuclear physics.
  • Probabilistic error cancellation: Probabilistic error cancellation effectively applies the inverse noise channel by combining measurements from circuits with randomly inserted Pauli or identity operations.For a depolarizing channel, Pauli-conjugated expectation values are subtracted and the identity contribution is added; the variance overhead scales as γ^G.

C. Quantum subspace expansion

Quantum subspace expansion mitigates VQE errors by working in a Pauli-generated subspace and solving a generalized eigenvalue problem, while related checks exploit conserved symmetries.

  • C. Quantum subspace expansion: QSE expands an approximate VQE ground state into a subspace of Pauli-transformed states to address unknown errors.The method measures the Hamiltonian and overlap matrices in this nonorthogonal subspace.
  • C. Quantum subspace expansion: QSE is most effective against systematic errors but can also suppress some stochastic errors.Its experimental demonstration measured ground- and excited-state energies of H2 on a two-qubit superconducting system.
  • C. Quantum subspace expansion: Solving the generalized eigenvalue problem yields an error-mitigated Hamiltonian spectrum from the measured subspace matrices.A small number of Pauli operators is typically used to limit measurement cost.
  • C. Quantum subspace expansion: Symmetry checks mitigate errors by testing conserved quantities such as electron number and spin projection.Implementations include ancilla stabilizer checks, additional measurements with post-processing, and n-representability constraints.
  • C. Quantum subspace expansion: Additional mapping symmetries and engineered error-correcting codes can detect or correct some errors through stabilizer checks and post-processing.The engineered-code approach uses stochastic stabilizer sampling to maintain polynomial cost.

1. STO-3G basis

The STO-3G H2 example develops a Jordan–Wigner-encoded electronic-structure problem and constructs a simplified UCCSD circuit that reproduces the molecular ground state.

  • 1. STO-3G basis: STO-3G represents each hydrogen atom using one 1s orbital, giving H2 four spin-orbitals.The molecular orbitals are constructed from atomic orbitals, with spin-up and spin-down possibilities for each orbital.
  • 1. STO-3G basis: The H2 Hartree–Fock state is encoded in the occupation-number basis and used as the reference Slater determinant.At equilibrium separation, the Hartree–Fock determinant dominates the ground-state wavefunction.
  • 1. STO-3G basis: The UCCSD ansatz for H2 retains symmetry-preserving single and double excitations with variational parameters.The operator is split using a single Trotter step, and Jordan–Wigner encoding converts the excitation terms into Pauli operators.
  • 1. STO-3G basis: The simplified UCCSD operator uses basis rotations for single excitations and combines double-excitation terms, enabling the circuit shown in Fig. 11.The combination is specific to H2 because it has only one double excitation and is not generally scalable.
  • 1. STO-3G basis: Applying the simplified UCCSD operator to the Hartree–Fock state can reproduce H2’s ground state.The encoded Hamiltonian and preceding transformations can be generated with quantum computational chemistry packages.

2. 6-31G basis

The examples extend fermion-to-qubit encoding to larger H2 basis sets and LiH, using Bravyi–Kitaev mappings, natural orbitals, and conserved symmetries to reduce qubit requirements.

  • 2. 6-31G basis: The 6-31G H2 example contains eight spin-orbitals and constructs Bravyi–Kitaev encoded states in the canonical orbital basis.The spin-orbitals are ordered with all spin-up orbitals followed by all spin-down orbitals.
  • 2. 6-31G basis: Two Bravyi–Kitaev qubits encode conserved spin-up and total electron counts, so both can be removed from the simulation.These encoded quantities do not change during the simulation.
  • 2. 6-31G basis: For cc-PVDZ H2, changing to the natural molecular orbital basis exposes orbital occupations that support active-space reduction.The basis contains 20 spin-orbitals, while the displayed diagonalized 1-RDM combines spin partners into 10 entries.
  • 2. 6-31G basis: The cc-PVDZ H2 mapping removes two symmetry qubits, reducing the problem from 18 to 16 qubits.The lowest-energy states are shown explicitly in both Jordan–Wigner and BK-tree encodings.
  • 2. 6-31G basis: The LiH STO-3G example reduces 12 spin-orbitals to six qubits using natural-orbital occupation assumptions and BK-tree symmetry removal.The final six-qubit Hamiltonian differs from the full 12-qubit Hamiltonian by only 0.2 mHartree.
  • 2. 6-31G basis: The BK-tree mapping is illustrated by constructing a Fenwick tree whose branches connect spin-orbitals according to the mapping algorithm.The figure places algorithmic steps on the left, corresponding actions on the right, and the completed tree at the bottom.

A. Classical limits

Classical computational chemistry spans efficient approximate methods and rarely used exact calculations, with accuracy and system size driving computational cost. Resource estimates indicate that quantum advantages remain hardware-intensive, while near-term approaches and benchmark demonstrations are still needed.

  • Classical limits: Classical chemistry methods generally become more costly as accuracy and simulated system size increase.This creates a trade-off between computational efficiency and the fidelity of predicted ground-state energies.
  • Classical limits: DFT and Hartree-Fock efficiently treat large systems but can struggle with strongly correlated systems and are not systematically improvable.They are therefore often used to obtain qualitative results for large systems.
  • Classical limits: Exact calculations are rarely performed, and their accuracy depends on modeling choices such as relativistic corrections and the Born-Oppenheimer approximation.Most practical calculations instead use approximate, less costly methods including coupled cluster, tensor networks, and quantum Monte Carlo.
  • Quantum-resource requirements: Around 10^14 T gates were estimated for FCI on FeMo-co with 54 electrons in 108 spin-orbitals, whereas qubitization reduced a related 113-electron, 152-spin-orbital estimate to around 10^11 Toffoli gates.These estimates illustrate how algorithmic choices and problem specifications strongly affect fault-tolerant resource requirements.
  • Quantum-resource requirements: 2-3 million qubits and tens of hours were estimated for a 100-site Fermi-Hubbard model, while later estimates required around 400,000–600,000 physical qubits under stated fault-tolerance assumptions.Condensed-phase and Fermi-Hubbard calculations may be better suited to early quantum computers than individual-molecule simulations, but surpassing classical methods still requires roughly 100,000 physical qubits.
  • Near-term approaches: VQE uses short circuits but remains heuristic, and it is difficult to prove that a chosen circuit can obtain an accurate ground-state estimate when classical optimization is included.Small error-corrected demonstrations and classically known benchmark results are proposed as necessary milestones before claiming an advantage.
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