Simulating Chemistry

A cricket ball, once Rohit Sharma has finished with it, is obeying Newton's laws. The air around it is obeying the Navier-Stokes equation. But the leather of the ball, the enzymes in the grass that break down after the match, the catalytic converter in the car that drives the team home — those are obeying the Schrödinger equation, and they are doing it with many electrons at once. The electrons are not independent; they push each other around through the Coulomb force, they obey the Pauli exclusion principle, and their joint wavefunction lives in a Hilbert space exponentially larger than any classical computer can store exactly.

This is the electronic structure problem. Given a molecule — a fixed arrangement of positively charged nuclei — what is the ground state of its electron cloud, and what is the energy of that state? From the ground-state energy you get binding energies, reaction rates, spectra, bond lengths, catalytic selectivity, and drug-target affinity. From the ground-state wavefunction you get electron density, chemical reactivity, everything a chemist means by "how the molecule behaves." The whole edifice of computational chemistry is built on answering this one question as accurately as possible, for molecules as large as possible.

The previous chapter gave you Feynman's three tasks: time evolution, ground states, and thermal states. This chapter specialises to the ground-state task for molecular electrons — the sub-problem with the deepest industrial stakes, the clearest classical failure mode, and the most heavily instrumented quantum algorithms. By the end you will know what Hamiltonian gets encoded, how fermion anticommutation becomes qubit arithmetic, where VQE ends and phase estimation begins, and what "useful quantum advantage in chemistry" concretely means.

Born-Oppenheimer — freeze the nuclei

A real molecule has both electrons and nuclei, all of them quantum mechanical. But nuclei are at least 1836 times heavier than electrons (the proton-to-electron mass ratio), which means on the timescale of electronic motion, the nuclei are essentially standing still. This is the Born-Oppenheimer approximation (Born and Oppenheimer, 1927): solve the electronic Schrödinger equation at fixed nuclear positions, then treat the nuclear motion as a separate, slower problem in the potential landscape the electrons have drawn.

Concretely, write the full molecular Hamiltonian as

and throw away T_{\text{nuc}} for the moment. What remains is the electronic Hamiltonian

with V_{\text{nuc-nuc}} now a constant (the nuclei don't move, so the nuclear-nuclear Coulomb energy is just a number you add at the end). The problem is to diagonalise H_{\text{el}} on the Hilbert space of N electrons and find the ground state |\psi_0\rangle and its energy E_0.

Why Born-Oppenheimer is a good approximation and not a crude one: the timescale separation is big enough that corrections (non-adiabatic couplings, vibronic mixing) are small for ground-state chemistry. Where it breaks is at conical intersections in excited-state photochemistry and in light atoms like hydrogen where nuclear quantum effects matter. For ground states of heavy-atom molecules, it is typically accurate to better than 1 kcal/mol.

From here on, "the Hamiltonian" means H_{\text{el}}.

The molecular electronic Hamiltonian

Pick a basis of one-electron molecular orbitals — spatial functions \phi_p(\vec r), each paired with a spin (up or down) to make a spin-orbital. Chemists typically build these from Gaussian basis sets (6-31G, cc-pVDZ, cc-pVTZ, etc.), which are tabulated and standard. If the molecule has M spatial orbitals, there are 2M spin-orbitals; N of them are filled in any given electronic configuration.

In this basis, the electronic Hamiltonian takes the second-quantised form

where c_p^\dagger creates an electron in spin-orbital p, c_p destroys one, and the coefficients are integrals:

Why second quantisation and not first quantisation: first-quantised wavefunctions \psi(\vec r_1, \vec r_2, \ldots) need explicit antisymmetrisation over electron labels ("electron 1 in orbital p, electron 2 in orbital q, minus the swap"). Second quantisation bakes the antisymmetry into the operators via the anticommutation relations \{c_p, c_q^\dagger\} = \delta_{pq}, and the state is described by occupation numbers (is orbital p filled or not?) rather than by which labelled electron sits where. The Hamiltonian looks identical for any number of electrons — a huge technical simplification.

The one-electron integrals t_{pq} are an M \times M matrix; the two-electron integrals V_{pqrs} are a rank-four tensor with O(M^4) entries. Chemists compute these integrals once, in a classical pre-processing step, using software like Gaussian, Psi4, or PySCF. They are then the input to whatever quantum algorithm you are going to run.

Notice the structure. The Hamiltonian is a sum of O(M^4) fermionic terms, each of which acts on at most four spin-orbitals. When you translate this to qubits, each term will become a sum of O(n) Pauli strings (for Jordan-Wigner; better for Bravyi-Kitaev), giving total Pauli-term count O(M^4 \cdot n) = O(n^5) for M \sim n. For a 100-qubit molecule, that is around 10^{10} Pauli terms — a big but tractable number.

Why classical methods struggle

Before the quantum routes, look at what classical methods do. Three workhorses dominate computational chemistry, in increasing order of accuracy and cost.

Hartree-Fock (HF). Approximate the true N-electron wavefunction by a single Slater determinant — a particular antisymmetric combination of N occupied spin-orbitals. This is the "mean-field" approximation: each electron sees the average field of the others, but correlations between them are missed. Cost: O(M^4). Accuracy: often qualitatively right for simple molecules; quantitatively wrong by 1–2% of the total energy. That 1–2% is called "correlation energy" and it happens to be where all the interesting chemistry lives.

Coupled cluster (CC). Dress up the HF determinant with excitations — single, double, triple — parameterised by amplitudes. CCSD (singles + doubles) is O(M^6); CCSD(T) (plus perturbative triples) is O(M^7). CCSD(T) is called "the gold standard of computational chemistry" for weakly correlated molecules, and it is. It fails for strongly correlated systems — transition-metal complexes, multi-reference cases, bond-breaking — because a single-reference perturbation on top of HF can't cover those.

Density Functional Theory (DFT). Reformulate the problem in terms of the electron density n(\vec r) instead of the wavefunction — a 3-dimensional function rather than a 3N-dimensional one. DFT is O(M^3) and is the workhorse of modern chemistry for systems of a few hundred atoms. Its accuracy depends on the chosen exchange-correlation functional; it is not systematically improvable, and it fails in roughly the same places CCSD(T) does, sometimes for different reasons.

Quantum Monte Carlo (QMC). Sample the wavefunction stochastically. Exact in principle; limited by the fermion sign problem — probability weights can go negative, and variance grows exponentially. Works for some systems, fails on most strongly correlated ones.

The exponential wall. The exact solution (full configuration interaction, FCI) requires diagonalising a matrix of dimension \binom{2M}{N} — the number of ways to place N electrons in 2M spin-orbitals. For a modest molecule with N = 50 electrons in M = 50 spatial orbitals (100 spin-orbitals), that is \binom{100}{50} \approx 10^{29} basis states. Storing that vector would require 1.6 \times 10^{30} bytes at double precision — more than the mass of the Earth if each byte were a proton. FCI is impossible beyond a handful of electrons.

The honest takeaway. Classical chemistry has an enormous toolkit and gets the right answer for most everyday molecules. It fails, systematically and predictably, on strongly correlated systems — transition metals, multi-reference states, bond-breaking transition states. That is not an accident of current software; it is a structural limitation of approximations that assume weak correlation. Quantum computers do not replace DFT on caffeine. They target the 5% of chemistry where every classical method breaks, and that 5% happens to be where the industrial leverage is: catalysts, drug-target binding, battery chemistry, photosynthesis.

Fermions on qubits — the encoding problem

Qubits, on their own, are not fermions. A qubit has two states |0\rangle and |1\rangle that commute with the qubits next to them. An electron, on the other hand, is a fermion: swap two electrons and the wavefunction changes sign. If you want to run a fermionic Hamiltonian on a qubit computer, you need a map from fermion operators c_p^\dagger, c_p to qubit operators (Paulis) that preserves the anticommutation relations \{c_p, c_q^\dagger\} = \delta_{pq}, \{c_p, c_q\} = 0.

There are several such maps. Two are standard.

Jordan-Wigner (1928)

The oldest encoding, and the most intuitive. Assign one qubit per spin-orbital, labelled 1, 2, \ldots, n. An electron sitting in spin-orbital p corresponds to |1\rangle on qubit p; an empty orbital is |0\rangle. The creation and annihilation operators become

Why the string of Zs: the minus signs from fermionic antisymmetry have to come from somewhere. Each Z_q returns +1 on |0\rangle and -1 on |1\rangle. Multiplying all Z_q for q < p counts the parity of the occupation of orbitals below p, and that parity contributes the sign needed when you permute electrons past each other. The (X_p \pm i Y_p)/2 factor is the ladder operator \sigma_p^\pm that flips qubit p between |0\rangle and |1\rangle.

Check the anticommutation:

For p = q, the Z strings cancel and you get \{c_p, c_p^\dagger\} = \{\sigma_p^-, \sigma_p^+\} = I — correct. For p \neq q, the Z strings do not cancel, and their anticommutation with the intervening X or Y generates the required minus signs to give zero. The encoding works.

The catch. Every fermionic operator becomes a Pauli string of weight O(n) — the long \prod Z_q tail is non-local, even though the original c_p was "local" in the sense of touching just one orbital. A single fermionic hop c_p^\dagger c_q becomes a Pauli string of length |p - q| + 2. This is fine for small molecules but expensive at scale.

Bravyi-Kitaev (2002)

A cleverer encoding designed to reduce the Pauli-string length. Instead of storing occupation numbers directly (as Jordan-Wigner does), Bravyi-Kitaev stores partial sums of occupations along a binary tree. Each qubit holds the parity of a specific subset of orbitals, chosen so that both the occupation of any single orbital and the parity of any prefix can be computed from O(\log n) qubits.

The result: c_p^\dagger and c_p become Pauli strings of weight O(\log n) — exponentially shorter than Jordan-Wigner in the asymptotic regime. The encoding is more intricate to implement (you need a specific bijection from orbital indices to bit positions), but for large molecules it reduces circuit depth significantly.

Parity encoding

A middle-ground variant. Each qubit stores the parity of occupations 1 through p (a cumulative sum). Weight of c_p^\dagger: O(n) for one end but O(1) for the other — useful for specific circuit patterns.

Which encoding to use depends on the hardware. Jordan-Wigner is simpler and often sufficient for 50–100 qubit molecules; Bravyi-Kitaev wins once you scale past a few hundred qubits. Both preserve the exact spectrum of the molecular Hamiltonian — they are coordinate transformations on Hilbert space, not approximations.

VQE versus phase estimation — the two routes to E_0

Once the molecular Hamiltonian is a qubit operator H = \sum_k \alpha_k P_k (a weighted sum of Pauli strings P_k with classical coefficients \alpha_k), the goal becomes to find the smallest eigenvalue E_0 and — optionally — the corresponding eigenstate |\psi_0\rangle. The quantum toolkit offers two quite different routes.

Route A — Variational Quantum Eigensolver (VQE)

VQE, introduced by Peruzzo et al. in 2014, is the NISQ-era workhorse. The idea:

1. Choose a parameterised trial wavefunction |\psi(\vec\theta)\rangle — a short quantum circuit with classical parameters \vec\theta (rotation angles). This is called an ansatz.
2. On the quantum computer, prepare |\psi(\vec\theta)\rangle and measure the expectation value E(\vec\theta) = \langle\psi(\vec\theta)|H|\psi(\vec\theta)\rangle by measuring each Pauli term P_k separately and combining: E(\vec\theta) = \sum_k \alpha_k \langle P_k\rangle_{\psi(\vec\theta)}.
3. Use a classical optimiser (SPSA, COBYLA, L-BFGS) to update \vec\theta toward lower E.
4. Iterate until E converges. The final E^* is an upper bound on the true E_0 by the variational principle; it equals E_0 if and only if the ansatz family contains the true ground state.

The whole loop is hybrid — quantum preparation and measurement, classical optimisation. Each individual circuit is short (tens to hundreds of gates), which is what makes it NISQ-friendly. The accuracy depends entirely on the expressiveness of the ansatz.

Standard ansatze:

• UCCSD (unitary coupled cluster, singles and doubles). A natural quantum generalisation of CCSD. Exact up to double excitations. Deep circuits; accurate.
• Hardware-efficient ansatz. A short alternating layer of single-qubit rotations and entangling gates. Cheap; often inexpressive on highly correlated systems; notorious for "barren plateaus" (flat optimisation landscape at scale).

Route B — Quantum Phase Estimation (QPE)

QPE, covered in chapters 73–76 of this track, extracts the eigenvalue of a unitary operator with precision \epsilon using O(1/\epsilon) ancilla-assisted controlled applications of e^{-iHt}. For chemistry:

1. Prepare a trial state |\phi\rangle with non-negligible overlap with |\psi_0\rangle (say, the Hartree-Fock determinant, or a UCCSD-style state).
2. Apply the QPE circuit with U = e^{-iH\tau} for a chosen \tau. This requires implementing e^{-iH\tau} as a quantum circuit — Trotter-Suzuki or qubitisation handle this.
3. Measure the ancilla. With probability |\langle\phi|\psi_0\rangle|^2, the output is an estimate of E_0 \tau / (2\pi) to precision \epsilon.

The circuit depth of QPE is large — dominated by the Hamiltonian-simulation subroutine, which is O(n^5) gates for a molecule of n orbitals at chemical accuracy. This is fault-tolerant territory; NISQ machines cannot run QPE at useful scale. But when fault-tolerance arrives, QPE gives exact ground-state energies (up to the chosen precision), not a variational upper bound.

Which to use? VQE if you have a NISQ machine and a molecule small enough that the ansatz is expressive. QPE if you have a fault-tolerant machine and want the exact answer. In the interim, hybrid schemes (QPE with partial error correction, VQE with error mitigation) blur the line.

The progression — from \text{H}_2 to FeMoco

Quantum chemistry on hardware has a recognisable arc. Each step up the ladder takes roughly a decade and doubles or triples the qubit count.

\text{H}_2 (4 spin-orbitals, 4 qubits). The hydrogen molecule. Two electrons, two nuclei, simplest non-trivial molecule in chemistry. Ground-state energy is known to 13 decimal places from ultra-precise classical calculations. Quantum chemists have run VQE for \text{H}_2 on every major quantum platform — IBM superconducting, Google superconducting, Rigetti, IonQ trapped ions, Quantinuum, Microsoft topological, Pasqal neutral atoms, photonic platforms. The result always reproduces the classical reference. \text{H}_2 is the "hello world" of quantum chemistry.

\text{LiH} (12 qubits). Lithium hydride. Four electrons. Still classically trivial, but exercises the full fermion-to-qubit pipeline at larger scale. Demonstrated on IBM hardware at reasonable accuracy with both VQE and phase-estimation variants.

\text{BeH}_2 (14 qubits), \text{H}_2\text{O} (14 qubits), \text{N}_2 (20 qubits). Small molecules with more orbitals, serving as benchmarks for error mitigation, ansatz design, and fermion-encoding circuit optimisations. Still classically solvable to chemical accuracy, so the goal is not new science — it is hardware validation.

\text{Cr}_2, \text{Fe}_2\text{S}_2, organic photosensitisers (50–100 qubits). The transition region. Chromium dimer is notoriously hard for classical methods because the molecular orbital structure is genuinely multi-reference — several Slater determinants contribute comparably to the ground state, and CCSD(T) fails. Some experimental results on small such systems exist; the quantum advantage frontier lives here.

FeMoco (~150 orbitals, ~200–300 qubits logical). The iron-molybdenum cofactor inside nitrogenase, the enzyme that fixes atmospheric \text{N}_2 into ammonia at room temperature and ambient pressure. Industrial ammonia synthesis (the Haber-Bosch process) uses 2% of global energy to do the same chemistry at 400°C and 200 atmospheres. Understanding the FeMoco mechanism at quantum accuracy might unlock a room-temperature catalyst — a revolution for fertiliser production, which underpins food security for a billion people in India. Reiher, Wiebe, Svore, Wecker, Troyer, and Babbush's 2017 analysis [1] estimated that extracting the FeMoco ground-state energy to chemical accuracy would require \sim 10^{15} T-gates on a fault-tolerant machine of \sim 100 logical qubits. Later analyses — Babbush et al. 2018 using qubitisation, Berry et al. 2019 with tensor hypercontraction — have pushed the T-count down to \sim 10^{10} and the logical qubit count up to \sim 4000 (the latter because qubitisation needs more ancilla but vastly fewer time steps). The target is plausible for fault-tolerant machines of the 2030s.

Hype check. Quantum chemistry is the most likely near-term useful application of quantum computing — more likely than cryptanalysis, optimisation, or machine learning. But "most likely" is not "soon" and it is not "any day now." Small molecules like \text{H}_2 and \text{LiH} are classically trivial; every quantum demonstration of them is a hardware validation, not a scientific breakthrough. Useful quantum advantage in chemistry means a specific molecule whose ground-state energy classical methods cannot compute, and a quantum algorithm that can. As of 2026, that molecule has not been demonstrated. The first such demonstration is likely to come from IBM, Google Quantum AI, Quantinuum, or PsiQuantum, will involve a transition-metal complex, and will be publishable in Nature on its own merits — not buried in a press release. When it happens, it will be news. Until then, treat any "quantum computer designs a drug" headline with scepticism.

Industrial efforts

Most large quantum-computing companies have a chemistry programme.

IBM. Qiskit Nature library; VQE demonstrations on their superconducting hardware for \text{H}_2, \text{LiH}, \text{BeH}_2. Collaborations with Boehringer Ingelheim (drug discovery), ExxonMobil (chemistry), Daimler (battery chemistry). IBM Research India is a node in this work.

Google Quantum AI. Phase estimation and VQE demonstrations; \text{H}_4 bond dissociation curves; published work on the Hartree-Fock + UCCSD + QPE pipeline for small molecules.

Quantinuum. Trapped-ion hardware with high fidelity; chemistry work on InQuanto software platform. Partnerships with Roche and Insilico Medicine for drug discovery. Also absorbed Cambridge Quantum, which had a long-standing chemistry focus.

PsiQuantum. Building a photonic fault-tolerant machine; chemistry is their stated flagship use case. Roadmap targets million-qubit devices for chemistry and materials.

Zapata, QunaSys, Classiq, Pasqal, IonQ. Various collaborations with Daimler, Mitsubishi, Sumitomo, Amgen, and others. The map is crowded.

India. TIFR (Tata Institute of Fundamental Research) has a long tradition of quantum chemistry, including Anil Kumar's NMR-based quantum algorithm demonstrations (some of the earliest in the world). IISc Bangalore hosts VQE and chemistry algorithm research. TCS, HCL, Infosys, and Wipro have internal quantum practices; Tata Consultancy Services announced partnerships with IBM Quantum. The National Quantum Mission (₹6000 crore, 2023) lists chemistry and drug discovery as priority application areas.

Worked examples

Common confusions

• "Small molecules need a quantum computer." No — classical methods (CCSD(T), DMRG, FCI for very small systems) handle everything up to about 50 correlated electrons. \text{H}_2 on a quantum computer is a hardware demo; the chemistry answer was known classically to 13 decimal places before any qubit was ever cooled. Quantum advantage lives on molecules where classical methods fail, not on molecules small enough to fit in a laptop's RAM.

• "Quantum chemistry is solved now." No. The classical side is enormously successful — DFT is a Nobel-winning achievement that computes the ground state of most chemistry correctly and cheaply. What is not solved: strongly correlated systems (transition-metal catalysts, bond breaking, photoexcited multi-reference states). Those are the quantum-computing targets. "Chemistry is solved" is a pop-science overclaim on either side.

• "VQE has already beaten classical computing." No convincing demonstration of this exists as of 2026. VQE on \text{H}_2 reproduces classical answers; VQE on molecules beyond the classical reach hits barren plateaus and ansatz expressiveness limits. The best evidence for useful VQE is on small transition-metal systems where it roughly matches DMRG at moderate cost; no one has demonstrated a VQE result that a well-resourced classical calculation could not also produce.

• "Jordan-Wigner is bad; Bravyi-Kitaev is always better." Bravyi-Kitaev has better asymptotic scaling (O(\log n) Pauli weight versus O(n)), but Jordan-Wigner is simpler to implement, has well-optimised circuit patterns, and is often fine for n \lesssim 100 qubits. At the scale of current hardware, Jordan-Wigner is the practical default.

• "The Schrödinger equation will be solved exactly by a quantum computer." Not quite. A quantum computer diagonalises your chosen Hamiltonian on your chosen basis to your chosen precision. Modelling errors — basis truncation, choice of active space, chosen theory level — persist. "Chemical accuracy" (~1 kcal/mol, which is 1.6 millihartree) is the target because that is the accuracy at which predicted and measured reaction rates agree. Quantum computers will hit chemical accuracy on systems classical methods cannot, not solve nature exactly.

• "Quantum chemistry is just a benchmark." It looks that way today because current hardware is small. But the architectural path is clear: the algorithms are in place, the encodings are standard, the resource estimates are published. Once fault-tolerant hardware reaches the FeMoco scale, chemistry is the natural first application because the theory is mature and the targets are concrete.

Going deeper

UCCSD and ansatz design

The unitary coupled cluster singles-and-doubles (UCCSD) ansatz is the natural quantum generalisation of classical CCSD. Classical CCSD uses a non-unitary exponential e^T|\text{HF}\rangle where T contains single and double excitation operators; this is fine classically but not directly a valid quantum state preparation because e^T is not unitary. The unitary version e^{T - T^\dagger} is unitary by construction and generates the UCCSD ansatz. The circuit depth scales as O(M^4) for M orbitals — deep, but systematic and parameter-efficient.

Variants: k-UpCCGSD (generalised singles and doubles), ADAPT-VQE (grow the ansatz adaptively based on gradient magnitudes), hardware-efficient ansatze (short alternating rotation-entangler layers — cheap but often inexpressive on hard molecules).

The central tension: deeper ansatze are more expressive but harder on noisy hardware (gate errors accumulate); shallower ansatze are cheaper but may miss strong correlation. The sweet spot depends on hardware fidelity and molecular structure.

Active-space reductions

You cannot put all 100+ electrons of a large molecule onto a quantum computer. You pick an active space: the chemically important orbitals (those near the Fermi level, those participating in bond breaking, those with strong multi-reference character). The rest — the "inert" core and high-energy virtual orbitals — are treated at a lower level of theory (Hartree-Fock or DFT) and folded into effective one- and two-electron integrals for the active-space Hamiltonian.

Active-space selection is an art. Standard choices: CASSCF-style selection based on orbital occupation, entanglement-based selection (orbitals with high mutual information), or chemistry-intuition-based selection. Getting the active space wrong is one of the main sources of error in quantum-chemistry results; a poorly chosen active space misses the correlation that makes the problem interesting in the first place.

Resource estimates and the literature

The resource-estimate literature is where "quantum advantage in chemistry" gets its quantitative muscle. Key papers:

• Reiher, Wiebe, Svore, Wecker, Troyer (2017) — the seminal FeMoco resource estimate, \sim 10^{15} T-gates with Trotter.
• Babbush, Berry, McClean, Neven (2018) — qubitisation-based estimates, \sim 10^{10} T-gates for FeMoco.
• Berry, Gidney, Motta, McClean, Babbush (2019) — tensor hypercontraction for further reductions.
• Lee, Berry, Gidney, Huggins, McClean, Wiebe, Babbush (2021) — "Even more efficient quantum computations of chemistry through tensor hypercontraction."

Each paper reduces the T-count by an order of magnitude through better algorithmic choices. The trajectory: the FeMoco target has moved from "impossible even in theory" to "engineering targets with clear milestones" in less than a decade.

Error mitigation on NISQ

Before fault-tolerance, you can run chemistry on noisy hardware with error mitigation — post-processing to cancel noise effects. Standard techniques:

• Zero-noise extrapolation. Run the circuit at multiple effective noise levels (by stretching gate durations), extrapolate to zero noise.
• Probabilistic error cancellation. Sample from a quasi-probability distribution that inverts the noise channel.
• Symmetry verification. Post-select on measurement outcomes that preserve conserved quantum numbers (electron count, S_z).

These do not scale to arbitrary problem sizes — the overhead grows exponentially with circuit depth — but they extend the reach of NISQ chemistry by roughly an order of magnitude in effective circuit depth.

Indian pharma and the return question

The National Quantum Mission's explicit chemistry targets include catalyst design and drug binding. India's generic pharma industry is among the largest in the world; a 10% reduction in R&D cost per successful drug program would be measured in lakhs of crores per decade. The economic argument, in India, is not hypothetical — it is a direct function of how much the industry spends on DFT-driven catalyst screening today and how much more accurate quantum-enhanced screening would make that process. TIFR and IISc are natural hubs; TCS and Wipro have concrete partnerships; the arithmetic is compelling even before the first "quantum advantage" chemistry result has been published.

Where this leads next

• Simulating Condensed Matter — chapter 136, the sister chapter on Hubbard, Heisenberg, and lattice models.
• Variational Quantum Eigensolver — chapter 138, the detailed VQE algorithm.
• Jordan-Wigner Encoding — the fermion-to-qubit map, in detail.
• Bravyi-Kitaev Encoding — the O(\log n) alternative.
• Trotter-Suzuki Decomposition — chapter 131, the time-evolution primitive that powers QPE.
• Feynman Revisited — chapter 129, the big picture for Part 15.

References

1. Reiher, Wiebe, Svore, Wecker, Troyer, Elucidating reaction mechanisms on quantum computers (2017) — arXiv:1605.03590.
2. McClean, Romero, Babbush, Aspuru-Guzik, The theory of variational hybrid quantum-classical algorithms (2016) — arXiv:1509.04279.
3. John Preskill, Lecture Notes on Quantum Computation, Chapter 7 — theory.caltech.edu/~preskill/ph229.
4. Wikipedia, Quantum chemistry.
5. Wikipedia, Variational quantum eigensolver.
6. Babbush, Berry, McClean, Neven, Quantum simulation of chemistry with sublinear scaling in basis size (2018) — arXiv:1807.09802.