Carcará

Carcará is a lightweight, high-performance Python framework for fermionic quantum simulations based on variational quantum algorithms (VQAs). Developed with an end-to-end physical simulation pipeline, it targets both noise-free research validation and real NISQ-era quantum computing execution (driven via IBM Qiskit).

From molecular geometry inputs, Carcará constructs real-space grids, evaluates one- and two-body integrals, performs Hartree-Fock reference calculations, maps operators to qubit systems, and executes variational eigensolving through both standard VQE and adaptive growth algorithms (ADAPT-VQE) with multiple operator pools. Beyond ground states, it computes excited states (variational deflation and subspace-search / SSVQE), offers a stochastic adaptive solver (VASQE) with temperature annealing, and handles periodic crystals (Bloch band structure and Born–von Kármán total energies). All solvers share a common, extensible driver architecture.

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Key Features

1. Localized Basis Sets (Generated Native)

All basis set functions are generated from scratch mathematically rather than relying on tabulated basis databases. Supported localized single-particle basis sets include:

• FAO (Full Atomic Orbital): Analytic hydrogen-like orbitals built from the actual atomic number (bare nuclear charge, no Slater screening), one per occupied subshell.
• NAO (Numerical Atomic Orbital): Confined Sankey/SIESTA-type atomic orbitals solved numerically on radial grids within a hard-wall sphere boundary dictated by a user-specified energy shift.
• GTO (Gaussian-Type Orbital): Minimal STO-nG bases generated via scale-covariant least-squares fitting of primitives to Slater-type orbitals.
• Pople Split-Valence: Contracted GTO split-valence bases (e.g., 6-31G and 6-31G(d)), featuring native polarization d-shells.

2. High-Performance C-Accelerated Integral Engine

A basis-agnostic integration engine handles the heavy lifting of one-body (kinetic $T$, nuclear attraction $V$) and two-body electron-repulsion integrals (ERI, $\langle ab|cd \rangle$ in physicists' notation) in real space:

• Geometry-Agnostic Grids: Supports cubic, anisotropic (orthorhombic), and non-orthogonal grids (sampling skewed crystal lattices directly).
• Fast ERI Solver: Features an $O(N \log N)$ FFT-based Poisson solver alongside a direct real-space double-sum method.
• C Backend Acceleration: An OpenMP-parallelized C backend (libcarcara_integrals) built with ctypes zero-copy pointer passing.
• Graceful Fallback: Automatically falls back to a vectorized NumPy reference implementation if the C shared library is not compiled.

3. Second Quantization & Fermion-to-Qubit Mappings

A robust second-quantized algebra layer implements:

• Fermion Operator: Full creation/annihilation operator algebra, including helper methods to construct Hamiltonians directly from molecular integrals.
• PauliSum Output: Clean qubit Pauli operator representation wrapping Qiskit's sparse Pauli operators.
• Fermion-to-Qubit Mappings: Jordan-Wigner (default), Parity (with optional two-qubit reduction), and Bravyi-Kitaev mappings.

4. Variational Quantum Algorithms (VQAs)

• VQE (Variational Quantum Eigensolver): High-precision state-vector simulator employing parameterized quantum circuits (e.g., UCCSD) and classical optimizers (SPSA, COBYLA, Nelder-Mead, SLSQP, Adam, L-BFGS-B).
• ADAPT-VQE: Adaptive grows-then-reoptimizes ansatz builder utilizing energy gradients to grow ansätze one operator at a time. It supports four distinct operator pools:
  • fermionic (spin-adapted fermionic excitations, Jordan-Wigner mapped).
  • qubit (individual JW Pauli strings, providing the shallowest individual operators).
  • qeb (qubit-excitation generators with Jordan-Wigner Z-strings dropped).
  • ceo (coupled-exchange operators sharing entangling structures, yielding the highest accuracy per CNOT).
• Excited States: Both VQE and ADAPTVQE expose energy_levels(num_states=...), computing the ground state and low-lying excited states by variational quantum deflation (VQD); every returned level is a true eigenvalue within the ansatz's reachable sector. SubspaceVQE / SubspaceADAPTVQE implement subspace-search VQE (SSVQE), finding the ground state and several excited states simultaneously in one optimization (one shared unitary over orthogonal references, weighted-energy cost) — returning variational upper bounds (Hylleraas–Undheim).
• VASQE (Variational Adaptive Stochastic Quantum Eigensolver): ADAPT-VQE with stochastic operator selection — the next operator is sampled from a softmax of the pool gradients, P(i,τ) = exp(|gᵢ|/τ) / Σⱼ exp(|gⱼ|/τ). Low temperature reduces exactly to greedy ADAPT-VQE; a high (optionally annealed) temperature explores the ansatz space, with constant, exponential, linear, and logarithmic cooling schedules. A SubspaceVASQE combines it with subspace search.
• Periodic Crystals (BlochADAPTVQE): A general 1-/2-/3-D crystal driver from an ASE primitive cell. Solves the single-particle Bloch Hamiltonian H(k)c = ε(k)S(k)c for the band structure, and computes a correlated total energy over all k-points via the Born–von Kármán supercell equivalence.
• Hartree-Fock Reference Drivers: Restricted Hartree-Fock (RHF) and Unrestricted Hartree-Fock (UHF) models to supply stable molecular-orbital bases and stationary reference states.
• Frozen-Core Approximation: Both VQE and ADAPTVQE accept frozen_core (True/"auto" for the chemical noble-gas core, or an integer count of lowest MOs) and frozen_orbitals (an explicit list of core spatial-MO indices). Frozen core orbitals are removed from the active space and replaced by their mean-field contribution (a constant core energy plus an effective one-body potential), shrinking the qubit count.
• Spin-Polarized References: The initial spin state is set the ASE way, through the atoms' initial magnetic moments (Atoms(..., magmoms=[1, 1]) for a triplet); the calculators read it and build a reference with n_alpha - n_beta unpaired electrons.
• Sparse Large Active Spaces: For 12+ qubits ADAPTVQE automatically switches to a sparse operator pool (sparse="auto") that screens with the exact analytic gradient and applies each excitation with a closed-form exp(θA), keeping frozen-core problems such as water tractable on an exact state-vector backend.
• Expressibility & Profiling Analysis: Evaluates parameterized quantum circuit expressibility (KL-divergence vs. Haar distribution within symmetry-conserving subspaces) and tracks circuit complexity (CNOT counts and depth compilation).

5. ASE Calculator Integration

The molecular drivers (VQE, ADAPTVQE, VASQE, and their subspace variants) act as standard calculators for the Atomic Simulation Environment (ASE):

atoms.calc = VQE(basis="FAO", optimizer="COBYLA", h=0.20)
# Asking ASE for the energy executes the entire quantum simulation pipeline!
energy_ev = atoms.get_total_energy()

6. Extensible Driver Architecture

Every variational solver subclasses a single VariationalDriver base that owns the shared machinery — the ASE-calculator surface (basis / grid / k-points / spin / frozen core), Hamiltonian materialization (dense or sparse), the state-vector expectation energy(psi), and timing/profiling. Concrete algorithms implement only their optimization loop, and cross-cutting capabilities are composable mixins: excited-state deflation (energy_levels) and subspace search plug into any driver. Adding a new method (a new operator-selection rule, ansatz, or excited-state technique) requires no changes to the setup code.

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Project Structure

carcara/
├── src/
│   └── carcara/
│       ├── algorithms/  # VariationalDriver base; VQE, ADAPT-VQE, VASQE, subspace
│       │                #   (SSVQE) + deflation excited states, Bloch crystals,
│       │                #   HF (RHF/UHF), expressibility
│       ├── backends/    # Hardware and simulation devices, error mitigation
│       ├── basis/       # Localized basis sets (FAO, NAO, GTO/STO-nG, Pople)
│       ├── circuits/    # Ansatz protocol, UCCSD & AdaptAnsatz, gates, pools, profiling
│       ├── core/        # Fermionic operators, mappings, molecular integrals
│       ├── integrals/   # Real-space grid and Poisson engine, C backend
│       │   └── csrc/    # C implementation and CMake build files
│       ├── optimizers/  # Classical optimizers for hybrid loops
│       ├── utils/       # Profiling (timing/memory), logging, start-up banner
│       ├── units.py     # Unified conversion factors (Angstrom/eV <-> Bohr/Hartree)
│       └── version.py   # Package versioning (CalVer YY.M.patch)
├── examples/            # Walkthroughs: ADAPT-VQE (H2, LiH, H2O frozen core, BeH2,
│                        #   mappings, O2 triplet), H-chain bands, energy levels
│                        #   (deflation), Subspace-VQE (SSVQE), and VASQE
├── test/                # Comprehensive pytest suite
└── docs/                # Sphinx source files and configuration

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Installation & Build

1. Prerequisites

• Python $\ge 3.11$
• C compiler with OpenMP support (e.g., GCC, Clang)
• CMake $\ge 3.15$

2. Installation via pip

You can install the stable release of Carcará directly from PyPI:

pip install carcara

3. Installation from Source (Developer Setup)

The package can be used directly from source via PYTHONPATH or installed in editable mode:

# Clone the repository
git clone https://github.com/seixas-research/carcara.git
cd carcara

# Install in editable mode
pip install -e .

4. Compile the C Integral Backend (Recommended)

Compile the C shared library to enable multi-threaded OpenMP acceleration. The compiled artifact will automatically be detected by _backend.py.

On macOS (requires Homebrew libomp):

cd src/carcara/integrals/csrc
cmake -S . -B build -DCMAKE_BUILD_TYPE=Release -DOpenMP_ROOT=$(brew --prefix libomp)
cmake --build build

On Linux:

cd src/carcara/integrals/csrc
cmake -S . -B build -DCMAKE_BUILD_TYPE=Release
cmake --build build

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Quickstart Examples

Example 1: Evaluating Real-Space Integrals (H₂)

Build a minimal basis of Hydrogen 1s orbitals and compute core Hamiltonian matrices:

import numpy as np
from carcara.basis import FullAtomicOrbital
from carcara.integrals import Grid, IntegralEngine, Potentials

# Geometry setup (H2 bond length R = 0.74 A)
R = 0.74
proton_a = np.array([0.0, 0.0, -R / 2])
proton_b = np.array([0.0, 0.0, +R / 2])

# Potential and Grid (spacing h = 0.10 A)
potentials = Potentials([(1.0, proton_a), (1.0, proton_b)])
grid = Grid(center=[0.0, 0.0, 0.0], box_size=5.0, h=0.10)

# Minimal Full Atomic Orbital basis
basis = [FullAtomicOrbital(1, 0, 0, Z=1.0, center=proton_a),
         FullAtomicOrbital(1, 0, 0, Z=1.0, center=proton_b)]

engine = IntegralEngine(basis, grid)

# Compute kinetic T, potential V, and electron-repulsion tensor
T, V = engine.one_body(potentials.nuclear_potential)
h_core = T + V
eri = engine.two_body(method="fft")

print("Core Hamiltonian (eV):\n", h_core.real)
print(f"On-site repulsion <00|00> (eV): {eri[0,0,0,0].real:.3f}")

Example 2: ASE-Driven VQE Simulation

Use the Atomic Simulation Environment (ASE) to run a standard VQE simulation with UCCSD ansatz:

from ase import Atoms
from carcara.algorithms import VQE

# Define H2 molecule in a unit cell
atoms = Atoms("H2",
              positions=[[4.0, 4.0, 3.63], [4.0, 4.0, 4.37]],
              cell=[[8.0, 0.0, 0.0], [0.0, 8.0, 0.0], [0.0, 0.0, 8.0]],
              pbc=True)

# Attach VQE calculator
atoms.calc = VQE(basis="FAO", mapping="jordan_wigner", optimizer="COBYLA", h=0.20)

# Run calculation (energy returned in eV)
energy_ev = atoms.get_total_energy()
result = atoms.calc.vqe_result

print(f"VQE Energy: {result.optimal_energy:.6f} Ha ({energy_ev:.6f} eV)")

Example 3: Running ADAPT-VQE

Compute H₂ ground state adaptively using the hardware-optimized Coupled-Exchange Operator ("ceo") pool:

from ase import Atoms
from carcara.algorithms import ADAPTVQE

atoms = Atoms("H2",
              positions=[[4.0, 4.0, 3.63], [4.0, 4.0, 4.37]],
              cell=[[8.0, 0.0, 0.0], [0.0, 8.0, 0.0], [0.0, 0.0, 8.0]],
              pbc=True)

# Attach ADAPT-VQE calculator
atoms.calc = ADAPTVQE(
              pool="ceo",
              basis="FAO",
              optimizer="COBYLA",
              gradient="parameter-shift",
              h=0.20,
              max_iterations=15,
              gradient_tolerance=1e-6
)

# Run adaptive loop
atoms.get_total_energy()
result = atoms.calc.adapt_result

print(f"ADAPT-VQE Converged: {result.converged}")
print(f"Optimal Energy: {result.optimal_energy:.8f} Ha")
print(f"CNOT Count: {result.metrics.cnot_count}")

Example 4: Excited States (deflation and subspace search)

Compute the ground state and the first excited state, either one after another (deflation) or simultaneously (SSVQE):

from ase import Atoms
from carcara.algorithms import VQE, SubspaceVQE

atoms = Atoms("H2", positions=[[4.0, 4.0, 3.63], [4.0, 4.0, 4.37]],
              cell=[[8.0, 0.0, 0.0], [0.0, 8.0, 0.0], [0.0, 0.0, 8.0]], pbc=True)

# (a) Deflation: excited states one after another.
atoms.calc = VQE(basis="FAO", h=0.20)
atoms.get_potential_energy()                     # configures the solver
levels = atoms.calc.energy_levels(num_states=2, restarts=4)
print("levels (eV):", levels.in_units("eV"))

# (b) SSVQE: ground + excited states in a single optimization.
atoms.calc = SubspaceVQE(basis="FAO", h=0.20, num_states=2)
atoms.get_potential_energy()
print("levels (eV):", atoms.calc.subspace_result.in_units("eV"))

Example 5: VASQE (stochastic ADAPT with temperature annealing)

Grow the ansatz by sampling operators from a softmax of the gradients, annealing the selection temperature from exploratory to greedy:

from ase import Atoms
from carcara.algorithms import VASQE

atoms = Atoms("H2", positions=[[4.0, 4.0, 3.63], [4.0, 4.0, 4.37]],
              cell=[[8.0, 0.0, 0.0], [0.0, 8.0, 0.0], [0.0, 0.0, 8.0]], pbc=True)

atoms.calc = VASQE(basis="FAO", pool="fermionic", h=0.20, optimizer="L-BFGS-B",
                   temperature=2.0, final_temperature=0.01, schedule="exponential",
                   max_iterations=12, gradient_tolerance=1e-5)
atoms.get_total_energy()
result = atoms.calc.vasqe_result

print(f"Energy: {result.optimal_energy:.8f} Ha")
print(f"Operators: {result.operators}")
print(f"Selection temperatures: {result.temperatures}")

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Testing

Carcará features a comprehensive unit testing suite verifying integrals, basis definitions, operators, Hartree-Fock solvers, VQE, and ADAPT-VQE algorithms.

To run the complete test suite:

# From the project root directory
pytest

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Documentation

Documentation is built using Sphinx:

cd docs
make html
# Output will be located in docs/build/html/index.html

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License & Development

Carcará is released under the MIT License.

Developer: Leandro Seixas Rocha (leandro.rocha@ilum.cnpem.br)

Website/Code: seixas-research/carcara

Documentation: carcara.readthedocs.io

Acknowledgements

We thank financial support from INCT Materials Informatics (Grant No. 406447/2022-5).