Theory¶
Theory and Methodology¶
This document summarizes the main algorithms, physical assumptions, and implementation choices used in this project. It covers:
molecular systems and shared chemistry infrastructure
the Variational Quantum Eigensolver (VQE)
ansatz families and optimizers
fermion-to-qubit mappings
excited-state methods
ADAPT-VQE
Quantum Phase Estimation (QPE)
variational imaginary-time evolution (VarQITE)
variational real-time evolution (VarQRTE)
noise models
For practical workflows and CLI usage, see USAGE.md.
For overview and installation, see README.md.
Detailed implementation references:
Contents¶
Background¶
Quantum chemistry algorithms in this repository approximate eigenvalues and eigenstates of electronic Hamiltonians for small molecular systems.
Workflow:
choose molecule and geometry
construct electronic Hamiltonian
map fermionic operators to qubit operators
apply quantum algorithm to estimate spectral properties
Centralizing steps 1–3 ensures consistent physical inputs across algorithm families.
Variational Principle¶
For normalized states:
$$ E_0 \le \langle \psi | H | \psi \rangle $$
Minimizing the expectation value over a parameterized state family
$$ |\psi(\theta)\rangle $$
produces the best approximation to the ground state accessible within the chosen ansatz manifold.
Variational methods rely on:
differentiable parameterized circuits
classical optimization loops
expectation-value estimation
Why qubit mappings are needed¶
Electronic Hamiltonians are expressed using fermionic creation and annihilation operators. Quantum circuits act on qubits, so the Hamiltonian must be mapped to qubit operators.
Common mappings:
Jordan–Wigner
Bravyi–Kitaev
parity mapping
These mappings preserve physical observables but affect:
Pauli-string locality
circuit depth
measurement grouping
optimizer conditioning
Hartree-Fock Reference State¶
Many workflows begin from a Hartree–Fock (HF) reference determinant.
$$ |HF\rangle = |\phi_1 \phi_2 \cdots \phi_n\rangle $$
In qubit form, this becomes an occupation-number bitstring.
Example:
|111100000000⟩
Roles of the HF reference:
initialization for chemistry-inspired ansätze
reference state for ADAPT-VQE
approximate eigenstate input for QPE
starting state for imaginary-time evolution
From classical chemistry to quantum algorithms¶
Once the qubit Hamiltonian is constructed, algorithm families differ in how they extract spectral information:
method |
mechanism |
|---|---|
VQE |
variational energy minimization |
post-VQE methods |
linear algebra in reduced manifolds |
QPE |
phase extraction from unitary evolution |
VarQITE |
imaginary-time filtering |
VarQRTE |
projected real-time dynamics |
All share identical Hamiltonians, enabling consistent algorithm comparisons.
VQE Overview¶
The Variational Quantum Eigensolver couples:
parameterized quantum circuits
classical optimization
Workflow:
prepare ansatz state
measure expectation value
update parameters
iterate to convergence
optimizer → parameters → circuit → expectation → update
Performance depends on:
ansatz expressibility
optimization landscape
Hamiltonian structure
Ansatz Families¶
Ansätze define the accessible variational manifold.
Tradeoffs:
physical structure
circuit depth
parameter count
trainability
UCCSD¶
Unitary Coupled Cluster Singles and Doubles:
$$ |\psi(\theta)\rangle¶
e^{T(\theta)-T^\dagger(\theta)} |HF\rangle $$
with
$$ T = T_1 + T_2 $$
Properties:
chemistry motivated
interpretable excitation structure
strong performance for small molecules
Hardware-efficient ansätze¶
Example: RY–CZ layered circuits.
Motivations:
shallow depth
tunable expressibility
hardware compatibility
useful for benchmarking optimizer behaviour
Minimal ansätze¶
Low-parameter circuits used for:
visualization
pedagogical examples
landscape analysis
Optimizers¶
Optimization minimizes:
$$ E(\theta)¶
\langle \psi(\theta) | H | \psi(\theta) \rangle $$
Supported optimizers:
Adam
Gradient Descent
RMSProp
Adagrad
Momentum / NesterovMomentum
Differences:
adaptive learning-rate scaling
momentum accumulation
noise robustness
Implementation uses a unified optimizer interface shared across ansätze.
Fermion-to-Qubit Mappings¶
Mapping choice affects circuit structure and optimization behaviour.
Jordan–Wigner¶
direct occupation encoding
simple construction
longer Pauli strings
Bravyi–Kitaev¶
balanced parity/occupation encoding
shorter average Pauli strings
Parity mapping¶
parity-based encoding
can expose symmetries
may reduce circuit depth
Excited-State Methods¶
Excited states require additional structure beyond ground-state minimization.
Two main approaches are implemented.
Post-VQE linear-algebra methods¶
Construct reduced eigenproblems around a converged reference state.
Methods:
QSE
EOM-QSE
LR-VQE
EOM-VQE
These rely on:
high-quality ground-state reference
well-conditioned reduced manifolds
Generally noiseless-only due to reliance on statevector information.
QSE¶
Construct operator-generated subspace:
$$ |\phi_i\rangle = O_i|\psi\rangle $$
Solve:
$$ Hc = ESc $$
where:
$$ H_{ij}¶
\langle \psi|O_i^\dagger H O_j|\psi\rangle $$
$$ S_{ij}¶
\langle \psi|O_i^\dagger O_j|\psi\rangle $$
EOM-QSE¶
Commutator-based reduced problem:
$$ A_{ij}¶
\langle \psi|O_i^\dagger[H,O_j]|\psi\rangle $$
Solve:
$$ Ac = \omega Sc $$
Typically non-Hermitian.
Positive real-dominant roots selected.
LR-VQE¶
Tangent-space linear response around converged parameters:
$$ S_{ij}¶
\langle \partial_i\psi|\partial_j\psi\rangle $$
$$ A_{ij}¶
\langle \partial_i\psi|(H-E_0)|\partial_j\psi\rangle $$
Solve:
$$ Ac = \omega Sc $$
Approximate excited energies:
$$ E_k = E_0 + \omega_k $$
Corresponds to a Tamm–Dancoff style approximation.
EOM-VQE¶
Full-response tangent-space formulation.
Produces paired roots:
$$ \pm \omega $$
Positive solutions correspond to excitation energies.
More expressive but more numerically sensitive.
Variational excited states¶
Solve excited states directly.
SSVQE¶
Shared unitary applied to orthogonal inputs:
$$ |\psi_k(\theta)\rangle = U(\theta)|\phi_k\rangle $$
Minimize:
$$ \sum_k w_k \langle \psi_k|H|\psi_k\rangle $$
VQD¶
Sequential deflation:
$$ L_n = \langle \psi_n|H|\psi_n\rangle + \beta \sum_{k<n} |\langle \psi_k|\psi_n\rangle|^2 $$
Supports noisy evaluation using density matrices.
Excited-state summary¶
method |
category |
noise |
|---|---|---|
QSE |
operator subspace |
noiseless |
EOM-QSE |
operator EOM |
noiseless |
LR-VQE |
tangent response |
noiseless |
EOM-VQE |
full response |
noiseless |
SSVQE |
variational |
supported |
VQD |
deflation |
supported |
ADAPT-VQE¶
Adaptive ansatz construction.
Instead of fixing the ansatz size, operators are added iteratively.
Ansatz:
$$ U_k(\theta)¶
\prod_j e^{\theta_j A_j} $$
Operators selected by gradient magnitude.
Workflow:
optimize current parameters
evaluate gradients for operator pool
append best operator
repeat until convergence
Advantages:
compact circuits
interpretable operator growth
convergence diagnostics
Quantum Phase Estimation¶
QPE extracts eigenvalues from phase evolution:
$$ U = e^{-iHt} $$
Eigenstate relation:
$$ U|\psi\rangle¶
e^{-iEt}|\psi\rangle $$
Energy recovered via:
$$ E = -\frac{2\pi \theta}{t} $$
Registers:
ancilla → phase precision
system → approximate eigenstate
Tradeoffs:
ancilla count vs precision
Trotter depth vs error
initial state overlap vs success probability
Quantum Imaginary Time Evolution¶
Imaginary-time evolution:
$$ |\psi(\tau)\rangle¶
e^{-H\tau} |\psi(0)\rangle $$
Suppresses higher-energy components.
VarQITE approximates evolution using McLachlan projection.
VarQRTE uses the same projected-variational machinery, but for real-time unitary dynamics rather than imaginary-time relaxation.
Practical distinction in this repository:
VQE/VarQITEare state-finding workflowsQPEis a spectral / phase-estimation workflowVarQRTEis a dynamics workflow used after a relevant state has already been prepared
For a time-independent Hamiltonian, real-time evolution should conserve energy up to numerical error. For that reason, the most useful VarQRTE diagnostics are usually:
time-dependent observables
fidelity to exact evolution on small systems
overlap with the initial state
trajectory error over time
Quantum Real Time Evolution¶
VarQRTE in this repository is a McLachlan-projected real-time evolution method on a variational state manifold.
Conceptually:
start from a prepared state $|\psi(\theta_0)\rangle$
project Schrödinger evolution onto the ansatz tangent space
solve a linear system for parameter velocities
integrate the parameter trajectory forward in time
Unlike VarQITE, VarQRTE does not try to suppress excited-state components or minimize the energy. Its role is to approximate the time evolution of a chosen initial state under a fixed Hamiltonian.
That means a good correctness question for VarQRTE is:
how well does the variational trajectory track exact real-time evolution of the same initial state?
not:
how low is the final energy?
Linear system:
$$ A(\theta)\dot{\theta}¶
-C(\theta) $$
with:
$$ A_{ij}¶
\Re \langle \partial_i\psi|\partial_j\psi\rangle $$
$$ C_i¶
\Re \langle \partial_i\psi| (H-\langle H\rangle) |\psi\rangle $$
Update:
$$ \theta \leftarrow \theta + \Delta\tau \dot{\theta} $$
Implementation features:
noiseless parameter updates
regularized linear solvers
noise applied only during evaluation
Noise Models¶
Noise channels implemented via PennyLane mixed-state simulation.
Supported channels:
depolarizing noise
amplitude damping
Noise placement:
applied between circuit operations
consistent across VQE, QPE, and evaluation stages
Depolarizing channel¶
$$ \mathcal{E}(\rho)¶
(1-p)\rho + \frac{p}{3} (X\rho X + Y\rho Y + Z\rho Z) $$
Produces isotropic decoherence.
Amplitude damping¶
$$
E_0 =
\begin{pmatrix}
1 & 0
0 & \sqrt{1-p}
\end{pmatrix}
\quad
E_1 =
\begin{pmatrix}
0 & \sqrt{p}
0 & 0
\end{pmatrix}
$$
Models relaxation toward ground state.
Typical evaluation metrics¶
Noise studies examine:
energy deviation
convergence stability
excitation ordering robustness
optimizer sensitivity
References¶
Foundations¶
Aspuru-Guzik et al. Simulated Quantum Computation of Molecular Energies
McArdle et al. Quantum Computational Chemistry
Kitaev Quantum Measurements and the Abelian Stabilizer Problem
Excited states¶
Parrish et al. Quantum Computation of Electronic Transitions using VQE
Higgott et al. Variational Quantum Computation of Excited States
ADAPT-VQE¶
Grimsley et al. Adaptive variational algorithm for molecular simulation
Imaginary-time simulation¶
McLachlan Variational solution of the time-dependent Schrödinger equation
Yuan et al. Theory of variational quantum simulation
Fermion mappings¶
Seeley et al. Bravyi–Kitaev transformation
PennyLane¶
PennyLane documentation for:
templates
optimizers
quantum chemistry tools
License¶
MIT. See LICENSE.