Simulation of Quantum Chemistry

Quantum simulation of chemistry represents a finite molecular Hamiltonian on a quantum processor and estimates an energy, state, or dynamical response. The central challenge is connecting the chemical quantity to an executable operator: a circuit can solve its encoded Hamiltonian accurately while its basis or active space remains inadequate for the molecule.

This page follows that connection from electronic integrals to qubits, state preparation, measurement, and chemical interpretation. Geometry, charge, nuclear treatment, relativity, environment, basis, and target state determine the model before the quantum algorithm starts. A reaction barrier also needs consistent calculations at several geometries; a force needs derivatives, not merely a precise energy at one point.

Required background. Electronic Structure Overview explains basis and correlation choices; Many-Particle Hamiltonians provides creation and annihilation operators. Digital Quantum Simulation and Algorithmic Primitives introduce encoded evolution and measurement.

From Molecular Model to Finite Hamiltonian

Fixed-geometry electronic problem

For a standard nonrelativistic, field-free, clamped-nuclei calculation in atomic units, the electronic Hamiltonian is

$$\hat H_{\mathrm e}(\mathbf R) ={} -\frac12\sum_i\nabla_i^2 -\sum_{iA}\frac{Z_A}{r_{iA}} +\sum_{i<j}\frac{1}{r_{ij}} + E_{\mathrm{NN}}(\mathbf R).$$

This equation already encodes assumptions. The nuclei are parameters rather than quantum degrees of freedom, the interaction is instantaneous Coulomb, and relativistic, radiative, solvent, and external-field effects are absent unless added deliberately. The scalar nuclear repulsion $E_{\mathrm{NN}}$ may be stored inside the qubit Hamiltonian or restored classically. The convention must remain consistent when comparing geometries.

Choose $N$ orthonormal spin-orbitals $\{\phi_p\}$. For a combined spatial and spin coordinate $x$, use the physicists’ convention

$$\langle pq\vert rs\rangle =\int dx_1dx_2\, \phi_p^*(x_1)\phi_q^*(x_2)\frac{1}{r_{12}} \phi_r(x_1)\phi_s(x_2), \qquad \langle pq\Vert rs\rangle =\langle pq\vert rs\rangle-\langle pq\vert sr\rangle.$$

The spin integrals include sums over spin labels. With these antisymmetrized integrals, the finite operator is

$$H_{\mathrm{orb}} ={} E_{\mathrm{NN}} +\sum_{pq}h_{pq}a_p^\dagger a_q + \frac14\sum_{pqrs} \langle pq\Vert rs\rangle a_p^\dagger a_q^\dagger a_s a_r.$$

The finite Fock space has dimension $2^N$, while a fixed $\eta$-electron sector has dimension

$$\dim\mathcal H_{N,\eta} = \binom{N}{\eta}.$$

Quantum storage can represent amplitudes in this sector compactly, but that fact alone gives neither an efficient state-preparation algorithm nor an efficient route to every observable.

Integral conventions must be executable

Chemistry software also commonly exports unantisymmetrized two-electron integrals and writes

$$H_2 = \frac12 \sum_{pqrs} \langle pq\vert rs\rangle a_p^\dagger a_q^\dagger a_s a_r.$$

For spatial orbitals, the chemists’ tensor obeys $\langle pq\vert rs\rangle=(pr\vert qs)$; explicit spin Kronecker deltas must be included when lifting it to spin-orbitals. This is equivalent to the $1/4$ expression only when the integral definition, index order, spin convention, and antisymmetrization are matched. Spatial- orbital integrals with implicit spin labels are not the same data tensor as spin-orbital integrals. Before mapping to qubits, an implementation should test:

• Hermiticity of the one-body matrix;
• permutation and conjugation symmetries of the two-body tensor;
• agreement between direct matrix elements and the assembled Fock-space operator on a small sector;
• whether nuclear repulsion and frozen-core constants are included;
• the units and orbital ordering used by every file boundary.

A factor-of-two or index-permutation error can preserve a superficially reasonable spectrum while changing the chemistry.

Frozen cores and active spaces

If a set $C$ of spin-orbitals is constrained to remain occupied, normal ordering with respect to that determinant gives an active-space Hamiltonian

$$H_{\mathrm{act}} ={}E_{\mathrm{core}} +\sum_{p,q\in A} h^{\mathrm{eff}}_{pq}a_p^\dagger a_q + \frac14 \sum_{p,q,r,s\in A} \langle pq\Vert rs\rangle a_p^\dagger a_q^\dagger a_s a_r,$$

where, for occupied spin-orbitals $i,j\in C$,

$$E_{\mathrm{core}} = E_{\mathrm{NN}} +\sum_{i\in C}h_{ii} +\frac12\sum_{i,j\in C} \langle ij\Vert ij\rangle,$$

and

$$h^{\mathrm{eff}}_{pq} = h_{pq} +\sum_{i\in C} \langle pi\Vert qi\rangle.$$

Orbitals outside $A\cup C$ are constrained to be empty. The active indices $p,q,r,s$ run only over $A$. This reduction is exact for the constrained frozen-core model, not for the original molecular Hamiltonian. Orbitals excluded from $A$ cannot acquire arbitrary correlated occupations. An active space must therefore be justified by occupations, entanglement or orbital diagnostics, state character, convergence tests, and the property being predicted.

Changing geometry can change which orbitals are chemically important. Potential-energy curves and reaction paths require a consistent orbital and state-tracking protocol; independently choosing the easiest active space at each point can manufacture discontinuities or favorable error cancellation.

Representation Choices

Second quantization

In an occupation encoding, one qubit records whether each spin-orbital is empty or occupied. Before exact symmetry reduction, $N$ spin-orbitals require $N$ qubits even when only $\eta\ll N$ electrons are present. The computational basis state

$$\lvert n_0n_1\cdots n_{N-1}\rangle, \qquad n_p\in\{0,1\},$$

represents a Slater determinant with the declared fermionic mode order. Second quantization makes determinant preparation and number-conserving excitation operators natural, and it is the standard language for many molecular-orbital algorithms.

First quantization

A first-quantized encoding stores the orbital or grid label of each of the $\eta$ electrons. A compact orbital encoding can use approximately

$$\eta\left\lceil\log_2N\right\rceil$$

data qubits before ancillas. The register must nevertheless represent an antisymmetric state and implement kinetic, potential, permutation, and data- access operations coherently. Plane-wave and real-space structures can make these operations favorable; arbitrary molecular-orbital problems may favor second quantization instead.

Question	Second quantization	First quantization
stored labels	one occupation bit per mode	one orbital or grid label per electron
data-qubit scale	$N$	roughly $\eta\log_2N$
antisymmetry	encoded by operator algebra and mode mapping	must be built into the state or algorithm
natural inputs	molecular orbitals and fermionic excitations	plane waves, grids, compact particle registers
common bottleneck	Pauli strings, measurements, PREPARE/SELECT	antisymmetric preparation, arithmetic, interaction oracles

The smaller data register is not automatically the cheaper algorithm. Ancillas, arithmetic width, oracle depth, normalization, state preparation, and fault-tolerant non-Clifford cost decide the comparison.

Fermions on Qubits

Jordan–Wigner convention

Taking $\lvert0\rangle$ to mean empty and $\lvert1\rangle$ to mean occupied, the Jordan–Wigner map is

$$\begin{aligned} a_p &= \left(\prod_{j<p}Z_j\right) \frac{X_p+iY_p}{2}, \\ a_p^\dagger &= \left(\prod_{j<p}Z_j\right) \frac{X_p-iY_p}{2}. \end{aligned}$$

Consequently,

$$n_p = a_p^\dagger a_p = \frac{I-Z_p}{2}.$$

The $Z$ string records the parity of earlier modes and enforces the anticommutation relations. Jordan–Wigner Transformation derives the parity signs and explains changes of mode order.

After substitution and collection, the molecular Hamiltonian becomes

$$H_q = c_0I + \sum_{\ell=1}^{L}c_\ell P_\ell,$$

where each $P_\ell$ is a tensor product of Pauli operators. A dense generic two-body tensor has $O(N^4)$ entries, but integral symmetry, locality, screening, low-rank structure, and exact collection can materially change the number and cost of distinct Pauli terms.

Bravyi–Kitaev and parity encodings distribute occupation and parity information differently. They can shorten some strings or expose useful symmetries, but they do not change the spectrum of the correctly encoded fermionic operator. A benchmark must compare compiled circuits under the same mode order, hardware graph, state preparation, and observable protocol.

Exact symmetry reduction

The spin-independent, field-free electronic Hamiltonian commutes with particle number, $S_z$, and $S^2$; spin–orbit coupling or external fields can remove some of these symmetries. In a qubit representation, choose mutually commuting, independent Hermitian Pauli generators $S_j$, with no nonempty product proportional to the identity, satisfying

$$[H_q,S_j]=0, \qquad S_j\lvert\psi\rangle=s_j\lvert\psi\rangle, \qquad s_j\in\{+1,-1\}.$$

A Clifford change of basis can map each $S_j$ to a single-qubit $Z$ operator. Fixing its known eigenvalue then removes that qubit from the target block. This tapering is exact only for the selected sector (Bravyi and colleagues, 2017). Commutation with $H_q$ alone is insufficient: for $H_q=I$, both $X$ and $Z$ commute with $H_q$, but they have no common eigenvector and cannot both be replaced by fixed eigenvalues.

The input and state-preparation circuit must remain in the chosen sector. An observable that changes a symmetry label needs matrix elements between sectors; replacing it by its within-sector projection can erase a nonzero transition amplitude. Every geometry in a comparison needs compatible symmetry and state labels.

A penalty such as

$$H_{\mu} = H_q + \mu \left( \hat N-\eta \right)^2$$

is different. It discourages the wrong particle-number sector but does not remove it exactly, and it can enlarge coefficient norms, measurement variance, or simulation normalization. Symmetry-preserving ansatzes and exact tapering are preferable when their assumptions hold.

Total spin deserves separate attention. Fixing $N_\alpha-N_\beta$ fixes $S_z$, not $S^2$. A trial state may therefore contain unwanted total-spin components even while particle number and spin projection are correct.

State Preparation Is Part of the Algorithm

A Hartree–Fock determinant maps to a computational-basis bitstring and is usually inexpensive to prepare after the orbital basis is fixed. Its overlap with a target eigenstate,

$$p_0 = \left| \langle E_0\vert\psi_{\mathrm{in}}\rangle \right|^2,$$

can nevertheless be poor for bond breaking, open shells, transition-metal complexes, conical intersections, and other multireference regimes.

For $0<p_0<1$, if an ideal spectral algorithm samples the desired eigenvalue with probability $p_0$, the chance of seeing it at least once in $r$ independent runs is

$$1-(1-p_0)^r.$$

Reaching success probability at least $1-\delta$ requires

$$r \geq \frac{\log\delta}{\log(1-p_0)}.$$

For $p_0=1$, one ideal run suffices; for $p_0=0$, repetition cannot help. For small $p_0$, this scales approximately as $\log(1/\delta)/p_0$. State preparation can therefore erase a favorable Hamiltonian-simulation query count.

Correlated inputs include short configuration-interaction expansions, multiconfigurational states, adiabatically prepared states, tensor-network states compiled into circuits, or states produced by a variational routine. Their classical preprocessing, amplitude loading, normalization, circuit depth, and verification contribute to the total cost. Quoting the cost of phase estimation while assuming an uncosted exact eigenstate is not an end-to-end estimate.

Choosing the Quantum Route

Variational energy estimation

For a parameterized state $\lvert\psi(\boldsymbol\theta)\rangle$, VQE estimates

$$E(\boldsymbol\theta) = c_0 + \sum_{\ell=1}^{L} c_\ell \langle P_\ell\rangle_{\boldsymbol\theta}$$

and uses a classical optimizer to reduce it. This replaces long coherent evolution with many state preparations, measurements, and adaptive quantum–classical iterations. For any normalized physical density matrix, including a noisy or postselected state, $\operatorname{Tr}(\rho H_q)$ is bounded below by the ground energy of $H_q$. A sector-specific bound additionally requires the state to lie in that sector. A finite-sample estimate, optimizer-selected minimum, or mitigated combination need not preserve the bound. Postselection also changes success probability and shot cost.

If Pauli terms are measured independently with $m_\ell$ shots, their contribution to the estimator variance is

$$\operatorname{Var}(\widehat E) = \sum_\ell \frac{c_\ell^2v_\ell}{m_\ell}, \qquad v_\ell = \operatorname{Var}(P_\ell).$$

For fixed target variance $\epsilon_{\mathrm{stat}}^2$, the ideal continuous shot allocation obeys

$$m_\ell \propto |c_\ell|\sqrt{v_\ell},$$

with total

$$M_{\min} = \frac{ \left( \sum_\ell |c_\ell|\sqrt{v_\ell} \right)^2 }{ \epsilon_{\mathrm{stat}}^2 }.$$

Commuting-group measurements, basis rotations, low-rank measurement schemes, classical shadows, and covariance-aware allocation can change this cost. Counting groups alone is insufficient: groups have different circuit depths, variances, covariances, and shot allocations. The VQE page develops ansatz design, optimization, variational error, mitigation, and independent validation. A term with exactly known zero variance needs no shots in this ideal allocation; a zero variance inferred from a small pilot sample is not a reliable certificate.

Phase estimation of molecular energies

Let

$$U(t_0) = \exp\!\left[ -\frac{i}{\hbar} \left(H_q-E_{\mathrm{ref}}I\right)t_0 \right].$$

For an eigenstate $\lvert E_k\rangle$,

$$U(t_0)\lvert E_k\rangle = e^{-i(E_k-E_{\mathrm{ref}})t_0/\hbar} \lvert E_k\rangle.$$

Phase estimation resolves this phase modulo $2\pi$. If the allowed energy window is closed and has width $W$, choosing

$$t_0 < \frac{2\pi\hbar}{W}$$

keeps the phase map injective within that window after the offset convention is fixed. Equality is usable for a half-open interval with a fixed endpoint convention; practical implementations usually leave a guard margin. With $m$ binary phase bits, the grid scale in energy is approximately

$$\Delta E_{\mathrm{grid}} \sim \frac{2\pi\hbar}{2^m t_0}.$$

The longest controlled evolution and total interrogation time grow as $O(2^m t_0)$, hence as $O(\hbar/\Delta E)$ up to algorithm- and confidence- dependent factors. Reporting only the number of phase ancillas hides this coherent-time cost.

The output is a spectral sample weighted by the input overlaps. Phase estimation does not create the ground state from an arbitrary trial state, and it does not by itself identify which sampled root has the desired chemical character. Quantum Phase Estimation derives the exact finite-bit distribution, confidence amplification, iterative variants, Fourier conventions, and total controlled-power accounting.

Product formulas, LCU, and qubitization

Phase estimation requires controlled access to molecular time evolution or a related signal unitary. The implementation can use:

Method family	Chemistry-facing input	Important chemistry-dependent cost
Trotter–Suzuki	exponentials of fermionic or Pauli terms	ordering, commutators, basis rotations, routed Pauli strings
truncated Taylor or LCU	coefficient preparation and SELECT	coefficient one-norm, state preparation, amplification, arithmetic
qubitization and QSP	controlled block encoding and inverse	normalization $\alpha$, PREPARE/SELECT, phase synthesis, work registers
interaction picture	a split $H=A+B$ with one easy part	cost of the easy frame and integrated norm of the residual interaction

For an LCU decomposition

$$H_q = \sum_j w_jU_j,$$

a direct block encoding often has normalization

$$\lambda = \sum_j|w_j|.$$

The qubitized time parameter is $\lambda|t|/\hbar$, not merely $\|H_q\||t|/\hbar$. Orbital rotations, integral factorization, energy shifts, and decomposition strategy can change both $\lambda$ and the gate cost of one PREPARE/SELECT query. Hamiltonian Simulation compares the methods, while Trotter–Suzuki Methods and Qubitization and Quantum Signal Processing give the executable constructions.

Dynamics and response

Chemistry is not only a ground-energy problem. Real-time simulation can target

$$C_{AB}(t) = \langle\psi_0\vert A(t)B(0) \vert\psi_0\rangle,$$

transition amplitudes, dipole response, charge migration, scattering, or nonadiabatic dynamics in an enlarged model. A frequency-domain spectrum may require a windowed transform of many time samples, so maximum evolution time, time spacing, window bias, and shot noise jointly set spectral resolution.

Forces, reduced density matrices, and response properties require additional observables. For a differentiable normalized state in a fixed Hilbert space,

$$\frac{dE}{dR} =\langle\psi|\partial_R H|\psi\rangle +2\operatorname{Re}\langle\partial_R\psi|(H-E)|\psi\rangle.$$

The second term vanishes for an exact eigenstate. Stationarity also cancels the response of fully optimized variational parameters, but not arbitrary state, orbital, or basis response. Geometry-dependent basis functions generate Pulay terms; incomplete optimization can leave further response terms. See Potential Energy Surfaces and Helgaker, Jørgensen, and Olsen (2000).

Factorization and Data Access

The repulsive Coulomb tensor in a real orbital basis has a positive semidefinite pair-density form. Factoring that form gives a representation of its operator (Motta and colleagues, 2021) of the form

$$H_2 \approx H_{1,\mathrm{corr}} + \frac12 \sum_{\ell=1}^{R} \left( \sum_{pq} W_{pq}^{(\ell)}a_p^\dagger a_q \right)^2,$$

where the $W^{(\ell)}$ are real symmetric spin-orbital matrices, with the spin-conserving structure inherited from the spatial factors. For $B_\ell=\sum_{pq}W^{(\ell)}_{pq}a_p^\dagger a_q$, anticommutation gives

$$B_\ell^2 =\sum_{ps}(W^{(\ell)2})_{ps}a_p^\dagger a_s +\sum_{pqrs}W^{(\ell)}_{pq}W^{(\ell)}_{rs} a_p^\dagger a_r^\dagger a_s a_q.$$

Thus $H_{1,\mathrm{corr}}=-\tfrac12\sum_{\ell,ps} (W^{(\ell)2})_{ps}a_p^\dagger a_s$ cancels the contraction. Omitting it changes the Hamiltonian, even at full factorization rank. A general effective interaction need not admit an all-positive sum of Hermitian squares; signed factors or a more general decomposition may be required. Diagonalizing each matrix $W^{(\ell)}$ turns a dense collection of quartic terms into orbital rotations and diagonal number interactions. Double factorization refines this structure by truncating eigencomponents within each factor.

Tensor hypercontraction uses a different approximation, schematically

$$(pq\vert rs) \approx \sum_{\mu\nu} X_{p\mu}X_{q\mu} Z_{\mu\nu} X_{r\nu}X_{s\nu},$$

in the chemists’ spatial-orbital convention (Lee and colleagues, 2021). Such factorizations can reduce storage, measurement settings, basis-rotation cost, or block-encoding complexity. They also introduce a threshold-dependent operator error. To judge a factorization, compare:

• the factorization family and rank-selection rule;
• the norm or observable used to certify truncation;
• coefficient precision and data-layout assumptions;
• the resulting Pauli count or block-encoding normalization;
• PREPARE, SELECT, QROM, arithmetic, and basis-rotation costs;
• validation against the unfactorized finite Hamiltonian on tractable cases.

An asymptotically compact tensor is not useful if loading its factors dominates the calculation. Conversely, treating every two-electron integral as an unstructured independent Pauli term can miss the principal source of modern resource reductions.

Worked Example: The Minimal Hydrogen Symmetry Block

Consider field-free $\mathrm H_2$ with equivalent nuclei in a minimal spatial basis. Bonding and antibonding orbitals each carry two spin states, so the second-quantized model has four spin-orbitals. A direct occupation encoding therefore begins with four qubits, not two electrons’ worth of qubits.

Order the spin-orbitals as

$$g\alpha, \quad g\beta, \quad u\alpha, \quad u\beta.$$

In the two-electron, $M_S=0$, even spatial-symmetry singlet block, the paired determinants

$$\lvert g^2\rangle = \lvert1100\rangle, \qquad \lvert u^2\rangle = \lvert0011\rangle$$

span a two-dimensional invariant subspace for this minimal model. In that basis, write

$$H_{\mathrm{sub}} = \begin{pmatrix} A & B\\ B & D \end{pmatrix},$$

where real orbitals have been chosen. Identifying $\lvert g^2\rangle\leftrightarrow\lvert0_L\rangle$ and $\lvert u^2\rangle\leftrightarrow\lvert1_L\rangle$ gives

$$H_{\mathrm{sub}} = c_0I+c_xX+c_zZ,$$

with

$$c_0 = \frac{A+D}{2}, \qquad c_x=B, \qquad c_z = \frac{A-D}{2}.$$

The two eigenvalues are

$$E_\pm = c_0 \pm \sqrt{c_x^2+c_z^2}.$$

This reduction provides a concrete check: construct the four-qubit Hamiltonian, project it into the declared symmetry block, and verify the same $E_\pm$ from the logical one-qubit form. It does not establish that a chemically converged hydrogen calculation generally needs one qubit. The result relies on a minimal basis, fixed particle number, selected spin and spatial symmetry, and a two-dimensional invariant block. Enlarging the basis or changing the target state changes the model.

From Encoded Energy Error to Chemical Accuracy

Let $E_{\mathrm{phys}}$ be the desired value for the intended physical system. Introduce a chain of exact values for successive approximations:

$$\begin{aligned} E_{\mathrm{phys}} &\longrightarrow E_{\mathrm{Ham}} \longrightarrow E_{\mathrm{basis}} \longrightarrow E_{\mathrm{act}} \\ &\longrightarrow E_{\mathrm{data}} \longrightarrow E_{\mathrm{alg}} \longrightarrow \widehat E. \end{aligned}$$

The difference telescopes exactly:

$$\begin{aligned} \widehat E-E_{\mathrm{phys}} ={}& (\widehat E-E_{\mathrm{alg}}) +(E_{\mathrm{alg}}-E_{\mathrm{data}}) \\ &+(E_{\mathrm{data}}-E_{\mathrm{act}}) +(E_{\mathrm{act}}-E_{\mathrm{basis}}) \\ &+(E_{\mathrm{basis}}-E_{\mathrm{Ham}}) +(E_{\mathrm{Ham}}-E_{\mathrm{phys}}). \end{aligned}$$

These terms can represent:

Difference	Representative sources
$E_{\mathrm{Ham}}-E_{\mathrm{phys}}$	nuclear model, relativity, environment, omitted interactions
$E_{\mathrm{basis}}-E_{\mathrm{Ham}}$	finite orbital or grid representation
$E_{\mathrm{act}}-E_{\mathrm{basis}}$	frozen core, active-space restriction, embedding
$E_{\mathrm{data}}-E_{\mathrm{act}}$	integral rounding, screening, low-rank factorization
$E_{\mathrm{alg}}-E_{\mathrm{data}}$	ansatz, optimization, finite-time simulation, phase resolution
$\widehat E-E_{\mathrm{alg}}$	synthesis, hardware, sampling, readout, mitigation, selection bias

The labels are diagnostic, not a promise that each component can be measured independently. Errors may correlate or cancel, and nonlinear mitigation can invalidate a naive probabilistic interpretation. The triangle inequality still supplies a conservative allocation when certified bounds are available.

The conventional $1\,\mathrm{kcal\,mol^{-1}}$ chemical-accuracy scale is approximately $1.5936\times10^{-3}E_{\mathrm h}$, but it is not a universal success criterion. Spectroscopic splittings, barrier heights, spin gaps, forces, and response properties have different tolerances. Sub-millihartree agreement with exact diagonalization of an inadequate active-space Hamiltonian is an algorithmic result, not automatically chemical accuracy.

For an energy difference

$$\Delta E = E_B-E_A,$$

consistent Hamiltonians, bases, active spaces, state identities, and statistical protocols are essential. Correlated errors may cancel; separately optimized approximations may not. Report uncertainty and covariance for the difference itself rather than adding two favorable absolute-error summaries.

Resource Accounting

An orbital count is only the first line of a resource estimate.

Logical resources

A second-quantized estimate should state:

• spin-orbital count and qubits before reduction;
• every frozen orbital, active electron, and active orbital;
• symmetry generators, chosen eigenvalues, and qubits tapered;
• state-preparation gates and success or overlap assumptions;
• Pauli terms or block-encoding normalization;
• logical Clifford, arbitrary rotation, Toffoli or $T$ counts;
• logical depth, connectivity assumptions, clean and dirty ancillas;
• controlled-operation overhead for phase estimation;
• measurement settings, state preparations, optimizer calls, and retries.

For qubitized phase estimation, a useful schematic is

$$C_{\mathrm{logical}} \sim N_{\mathrm{query}} C_{\mathrm{query}} +C_{\mathrm{state}} +C_{\mathrm{phase}} +C_{\mathrm{readout}},$$

where $C_{\mathrm{query}}$ expands PREPARE, SELECT, their inverses, reflection, controls, QROM, and arithmetic. A theorem-level query count with $C_{\mathrm{query}}=1$ is not a gate estimate.

For VQE, a corresponding wall-clock schematic is

$$T_{\mathrm{VQE}} \sim \sum_{k=1}^{N_{\mathrm{eval}}} \sum_{g=1}^{N_{\mathrm{groups}}} m_{kg} \left( t_{\mathrm{prepare},k} +t_{\mathrm{basis},g} +t_{\mathrm{readout}} \right),$$

plus compilation, queueing, calibration, classical optimization, and validation. Circuit depth and shot throughput can trade against one another.

Physical resources

Fault-tolerant projections must further declare the code, physical error model, cycle time, code distance, logical failure allocation, magic-state factories, routing, measurement latency, parallelism, physical qubits, and runtime. Those quantities are conditional engineering outputs, not delivery dates. Resource Estimation Tools explains how code and hardware assumptions convert logical gates into these physical requirements.

Classical work remains

The quantum stage commonly depends on classical geometry generation, orbital optimization, integral evaluation, localization, active-space selection, factorization, circuit generation, phase finding, decoding, optimization, and validation. A fair comparison counts the complete workflow on both sides and uses the strongest applicable classical baseline for the same Hamiltonian and observable.

Checking the Molecular Result

Check the represented operator. On a tractable instance, assemble the fermionic-sector matrix independently of the qubit map, then compare its matrix elements and spectrum after matching mode order, sector, and energy offset. Eigenvalues alone cannot detect an incorrect observable or state map. The minimal-hydrogen example supplies a small exact test.

Converge different approximations separately. Increasing shots tests sampling, increasing simulation order or phase bits tests the algorithm, and enlarging the basis or active space tests the finite model. Agreement with exact diagonalization of the same active space does not test its chemical adequacy. Compare energy differences, densities, forces, and state character as appropriate; cancellation in one total energy can hide other errors.

Track the intended state. A small energy error does not always imply accurate state preparation. In a sector with nondegenerate ground state and gap $\Delta>0$, a normalized state satisfies

$$1-\langle E_0|\rho|E_0\rangle \leq\frac{\operatorname{Tr}(\rho H)-E_0}{\Delta}.$$

The bound follows by assigning at least energy $E_0+\Delta$ to every excited component. It becomes weak near a closing gap; a degenerate ground space requires a projector onto the entire space. For a selected excited chemical state, use overlaps, symmetry labels, and transition properties in addition to its energy.

The best practical combination of state preparation, representation, factorization, algorithm, and hardware depends on the target molecule. Quantum Chemistry Case Studies compares demonstrations and conditional resource estimates; Hilbert-space dimension or a query count alone does not establish practical advantage.

Exercises

1. Reconcile the two-electron conventions

Show that

$$\frac14\sum_{pqrs} \langle pq\Vert rs\rangle a_p^\dagger a_q^\dagger a_s a_r$$

equals

$$\frac12\sum_{pqrs} \langle pq\vert rs\rangle a_p^\dagger a_q^\dagger a_s a_r$$

when $\langle pq\Vert rs\rangle=\langle pq\vert rs\rangle- \langle pq\vert sr\rangle$ and the ordinary integrals use the same index convention.

Solution

Expand the antisymmetrized expression:

$$H_2 ={} \frac14\sum_{pqrs} \langle pq\vert rs\rangle a_p^\dagger a_q^\dagger a_s a_r - \frac14\sum_{pqrs} \langle pq\vert sr\rangle a_p^\dagger a_q^\dagger a_s a_r.$$

In the second sum, interchange the dummy labels $r$ and $s$:

$$-\frac14\sum_{pqrs} \langle pq\vert rs\rangle a_p^\dagger a_q^\dagger a_r a_s.$$

Fermionic anticommutation gives $a_ra_s=-a_sa_r$, so this term equals the first $1/4$ contribution. Their sum has prefactor $1/2$.

2. Map a hopping operator

For $p<q$, use the stated Jordan–Wigner convention to show that

$$a_p^\dagger a_q+a_q^\dagger a_p = \frac12 \left( X_pZ_{p+1}\cdots Z_{q-1}X_q + Y_pZ_{p+1}\cdots Z_{q-1}Y_q \right).$$

What happens for adjacent modes?

Solution

Substitute the creation and annihilation maps. The parity strings before $p$ cancel. Moving the remaining $Z_p$ through the endpoint ladder operators and adding the Hermitian conjugate cancels the mixed $XY$ and $YX$ terms, leaving the displayed $XX+YY$ combination with the intervening parity string.

For $q=p+1$, the product over intervening sites is empty, so

$$a_p^\dagger a_{p+1} +a_{p+1}^\dagger a_p = \frac12 \left( X_pX_{p+1}+Y_pY_{p+1} \right).$$

3. Count the relevant dimensions

A model has $N=20$ spin-orbitals and $\eta=6$ electrons. State the number of qubits in a direct second-quantized encoding and the dimension of the fixed- particle sector. Explain why neither number alone determines the circuit cost.

Solution

The direct occupation encoding uses 20 qubits. The fixed-particle sector has

$$\binom{20}{6} = 38760$$

basis determinants. The circuit cost also depends on symmetry reduction, state preparation, Hamiltonian structure, coefficient norms, simulation or measurement method, precision, hardware connectivity, and output protocol. The sector dimension measures representation size, not executable cost.

4. Derive the optimal independent shot allocation

Minimize $M=\sum_\ell m_\ell$ subject to

$$\sum_\ell\frac{c_\ell^2v_\ell}{m_\ell} = \epsilon_{\mathrm{stat}}^2.$$

Treat $m_\ell$ as positive real numbers.

Solution

With Lagrange multiplier $\lambda$,

$$\mathcal L = \sum_\ell m_\ell + \lambda \left( \sum_\ell\frac{c_\ell^2v_\ell}{m_\ell} -\epsilon_{\mathrm{stat}}^2 \right).$$

Stationarity gives

$$1 - \lambda\frac{c_\ell^2v_\ell}{m_\ell^2} =0,$$

so $m_\ell=\sqrt\lambda|c_\ell|\sqrt{v_\ell}$. Enforcing the constraint gives

$$\sqrt\lambda = \frac{ \sum_j|c_j|\sqrt{v_j} }{ \epsilon_{\mathrm{stat}}^2 },$$

and therefore

$$M_{\min} = \frac{ \left( \sum_j|c_j|\sqrt{v_j} \right)^2 }{ \epsilon_{\mathrm{stat}}^2 }.$$

Integer shots, unknown variances, grouping, and covariance modify the practical allocation.

5. Choose a phase-estimation window

Suppose the desired molecular eigenvalue is known to lie in a half-open interval of width $W=2E_{\mathrm h}$. Under the ideal endpoint convention, choose a base time that fills one $2\pi$ phase period. Approximately how many phase bits are required for grid spacing below $10^{-3}E_{\mathrm h}$?

Solution

Choose

$$t_0 = \frac{2\pi\hbar}{W} = \frac{\pi\hbar}{E_{\mathrm h}}.$$

Then the energy grid scale is $W/2^m$. Requiring

$$\frac{2E_{\mathrm h}}{2^m} < 10^{-3}E_{\mathrm h}$$

gives $2^m>2000$, so $m=11$ bits is the first integer choice. This idealized calculation uses a half-open energy window. A practical guard margin slightly reduces $t_0$; a complete confidence guarantee may also require extra bits or repetitions, and simulation error needs its own allocation.

6. Freeze an orbital and identify the omitted coupling

For two spinless modes, take

$$H=\epsilon_0 n_0+\epsilon_1 n_1+U n_0n_1 +t(a_0^\dagger a_1+a_1^\dagger a_0).$$

Freeze mode 0 occupied. Find $PHP$ in the basis $|10\rangle,|11\rangle$. Does it reproduce the full one-particle ground energy when $\epsilon_0<\epsilon_1$ and $t\ne0$?

Solution

The constrained Hamiltonian is

$$PHP=\epsilon_0 I+(\epsilon_1+U)n_1.$$

The hopping term either leaves the constrained subspace or vanishes by Pauli exclusion. In the full one-particle sector, however, the matrix is $\begin{pmatrix}\epsilon_0&t\\t&\epsilon_1\end{pmatrix}$, with lower energy

$$E_- =\frac{\epsilon_0+\epsilon_1}{2} -\sqrt{\left(\frac{\epsilon_1-\epsilon_0}{2}\right)^2+t^2} <\epsilon_0.$$

For $|t|\ll\epsilon_1-\epsilon_0$, the missing lowering is $-t^2/(\epsilon_1-\epsilon_0)+O(t^4/(\epsilon_1-\epsilon_0)^3)$. Projection is exact for the constrained model, but the constraint removes virtual transitions that affect the original model’s energy.

7. Solve the projected hydrogen model

For

$$H=c_0I+c_xX+c_zZ,$$

find the ground energy and give a normalized ground-state Bloch vector.

Solution

Let

$$\Omega = \sqrt{c_x^2+c_z^2}.$$

The eigenvalues are $c_0\pm\Omega$, so the ground energy is

$$E_0 = c_0-\Omega.$$

For $\Omega>0$, the ground-state density operator is

$$\rho_0 = \frac12 \left[ I - \frac{c_x}{\Omega}X - \frac{c_z}{\Omega}Z \right].$$

Thus its Bloch vector is $-(c_x,0,c_z)/\Omega$. If $c_x=c_z=0$, the block is degenerate and every normalized logical state is a ground state.

8. Recover a transition between symmetry sectors

Let $H=-\Delta Z_1/2+\omega Z_2/2$, with positive $\Delta,\omega$. The symmetry $S=Z_1$ allows tapering the first qubit. In the $S=+1$ sector, what is the projected operator for $D=X_1$? Does this imply that $D$ has no transition from $|01\rangle$ to $|11\rangle$?

Solution

For $P_+=(I+Z_1)/2$, $P_+DP_+=0$ because $D$ flips the symmetry label. Nevertheless $\langle11|D|01\rangle=1$, and the final energy exceeds the initial energy by $\Delta$. The transition uses $P_-DP_+$, which maps the positive sector into the negative one. A calculation restricted to the positive-sector Hamiltonian and projected dipole misses this line. Tapering must therefore retain the relevant pair of sectors or implement their transition map when computing a spectrum.

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