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 hardware — running on IBM Qiskit, Amazon Braket (including real QPUs) and Google Cirq through one unchanged API.

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, the default method) with multiple operator pools. Beyond ground states, it computes excited states (variational deflation and subspace-search / SSVQE) and handles periodic systems (Bloch band structure and Born–von Kármán total energies). All algorithms are reached through a single ASE calculator, Carcara, which selects the solver with a method= argument, and share a common, extensible driver architecture underneath.

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. A size argument selects multiple-zeta and polarized variants (SZ, DZ, DZP, TZP, TZ2P, QZP, ...): extra zetas are built by the SIESTA split-valence construction, polarization by an $l+1$ shell solved in the same confining sphere.
• Named Gaussian families (Pople, Dunning, Karlsruhe): the standard basis-set names — 3-21G … 6-311+G(2df,2p), cc-pVDZ … cc-pV5Z, aug-cc-pVDZ, cc-pCVDZ, def2-SV(P) … def2-QZVPPD, plus STO-3G … STO-6G — are parsed into their structure (core contraction, valence split, polarization, diffuse and tight functions) and generated natively with the published shell structure and function count (cc-pVTZ C = [4s3p2d1f]), exponents from Carcará's own Slater-orbital fits rather than a basis-set table. basis="cc-pVDZ", basis="def2-TZVP".
• Per-element basis mappings: basis={"O": {"name": "NAO", "size": "DZP"}, "H": "6-31G", "*": "FAO"} puts a different family on different elements — a polarized basis where the chemistry happens and a minimal one on a spectator ion — accepted by every driver, the dry run and BasisSet.build.
• NAO-AE (All-Electron Numerical Atomic Orbital): every occupied shell of the self-consistent LDA atom — core included — re-solved under a smooth exponential-wall confinement (zero up to an onset radius, divergent at onset + width), plus hydrogen-like tiers: a polarization shell at $l_\max+1$ and diffuse / contracted functions per valence channel, whose effective charges are derived from the atom's own valence radius rather than tabulated. Each $l$ channel is Gram–Schmidt orthonormalized. basis={"name": "NAO-AE", "tier": 1}.

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, checked and compiled on demand before any integration; the FFT two-body step runs in memory-bounded blocks over the Hermitian pair densities (CARCARA_ERI_MEMORY_MB).
• 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: Every Carcara exposes 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. method="subspace-vqe" / method="subspace-adapt-vqe" 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).
• Periodic Systems (BlochCalculator): A general 1-/2-/3-D crystal calculator from an ASE primitive cell, with the correlated solver selected by method= ("adapt-vqe" by default, or "vqe"). 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) supplies the molecular-orbital basis and stationary reference for closed shells; open-shell systems — any odd electron count, or a high-spin state set through magmoms — are built in the natural-orbital basis of the unrestricted (UHF) solution, one spatial basis shared by both spins, so radicals and doublets run through the same solvers, pools and mappings (Carcara on an OH radical gives a (5, 4) reference on 12 qubits).
• Interaction energies on a shared grid: interaction_energy(complex, fragments=[[0,1,2],[3]], charges=[0,1], charge=1, method="adapt-vqe", ...) evaluates the complex and every fragment on one grid at identical coordinates — the only way a real-space difference of total energies is meaningful — with method="rhf" for a circuit-free mean-field check; also Carcara(...).interaction_energy(...).
• Frozen-Core Approximation: Every method accepts 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 States & Open Shells: The initial spin state is set the ASE way, through the atoms' initial magnetic moments (Atoms(..., magmoms=[1, 1]) for a triplet, [1, 1, 1] for a quartet); an odd electron count is a doublet by default. Odd-electron systems — radicals, ions, single atoms — are built in the UHF natural-orbital basis in every basis family, plane waves included, and run through the same solvers, pools, mappings, frozen core and dry run.
• Sparse Large Active Spaces: For 12+ qubits the adaptive methods automatically switch 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).
• Dynamic Parametrization (quenching): True (default) re-optimizes every variational parameter at each iteration — standard ADAPT-VQE. False freezes previously optimized angles and varies only the newest one, turning each growth step into a cheap one-dimensional line search.

5. Multi-Backend Execution (Qiskit / Braket / Cirq)

backend_provider selects which quantum SDK builds — and, with execute_circuits=True, runs — the ansatz circuits. Each generator is an anti-Hermitian PauliSum whose terms commute, so exp(θA) factorizes exactly into Pauli rotations (no Trotter error). One shared gate stream (X, H, S, S†, CNOT, Rz) is translated per SDK, so all three reproduce the internal NumPy state vector to machine precision:

provider            E (Ha)   err vs FCI   ops  cnots  depth
(matrix)       -6.88824276     1.34e-07     8    208    273
qiskit         -6.88824283     6.27e-08     8    208    273
braket         -6.88824279     1.09e-07     8    208    359
cirq           -6.88824281     8.17e-08     8    208    358

6. Real Quantum Hardware: IBM Quantum and Amazon Braket

A QPU never returns a state vector — Braket rejects the StateVector result type whenever shots > 0, and every QPU requires it. Carcará therefore implements the shot-based protocol hardware actually supports:

• Qubit-wise commuting (QWC) grouping partitions H = Σ cⱼ Pⱼ into simultaneously measurable sets — 118 Pauli terms collapse to 29 measurement circuits for LiH — and ⟨H⟩ is assembled from the returned bit-string counts, converging as 1/√shots.
• Device registry: IBM Quantum processors through Qiskit Runtime (device="ibm-quantum" for the least-busy QPU of your account, or a name such as "ibm_torino"; "fake_torino" rehearses the same path locally), the Braket local simulator, the AWS managed simulators (SV1/DM1/TN1), and the IonQ / IQM / Rigetti QPUs — or any Braket ARN. Naming a QPU without shots is rejected up front rather than at submission.

atoms.calc = Carcara(method="vqe", basis="FAO",
                               device="braket-ionq-aria", shots=8192)
atoms.get_total_energy()          # measured on a trapped-ion QPU

Tip: mapping="parity", two_qubit_reduction=True (with the fermionic pool) removes two qubits and their gates — H₂ runs on 2 qubits — which is what makes a hardware curve usable.

Scope: the energy evaluation is hardware-native. ADAPT-VQE's pool-gradient screening is still classical, so fixed-ansatz method="vqe" is the fully hardware-native method today. Run examples/13_braket_aws_compatibility.py for a verified compatibility report (no AWS account needed), and examples/24_ADAPTVQE_LiH_IBM.py for a LiH curve optimized locally and then measured on an IBM processor in a single Estimator job (set its HARDWARE constant).

7. Reusable Hamiltonians (Parquet / JSON Cache)

Building the qubit Hamiltonian — integrals plus the fermion-to-qubit mapping — is the most expensive stage of a run and is independent of the algorithm that follows. It can be serialized and replayed:

Carcara(method="adapt-vqe", basis="FAO",
                  save_hamiltonian="lih.parquet")   # build once
Carcara(method="adapt-vqe", pool="ceo",
                  load_hamiltonian="lih.parquet")   # reload: no geometry,
                                                    # no integrals, no mapping

Two formats, selected with hamiltonian_format: Parquet (compressed, columnar, queryable straight from pandas; ~4× smaller) and JSON (plain text, no native dependency). Loading detects the format automatically — from the extension, else from the file's leading bytes. Because the file also records num_particles and n_spatial_orbitals, a reloaded calculator runs with no Atoms object at all, turning a pool/optimizer/mapping sweep into seconds.

8. ASE Calculator Integration

Carcara is a standard calculator for the Atomic Simulation Environment (ASE); the method argument selects the solver ("adapt-vqe" — the default — "vqe", "subspace-vqe", "subspace-adapt-vqe"):

atoms.calc = Carcara(method="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()

9. Dry Run and the Command Line

A dry run reports the qubit budget of a calculation — one qubit per active spin-orbital, after the frozen core or plane-wave cutoff is accounted for — without computing an integral, mapping a Hamiltonian or executing a circuit, and compares it with the capacity of the target device:

carcara water.xyz --frozen-core --dry-run                  # 12 qubits
carcara H2O --basis NAO --basis-option size=DZP --device braket-ionq-aria --dry-run
carcara --load-hamiltonian lih.parquet --dry-run --json

The same carcara command runs the full calculation without --dry-run. From Python, Carcara(...).dry_run(atoms) returns the QubitEstimate, and dry_run=True makes every driver stop before the Hamiltonian is built.

10. Extensible Driver Architecture

Underneath Carcara, 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.

Project Structure

carcara/
├── src/
│   └── carcara/
│       ├── algorithms/  # VariationalDriver base; VQE, ADAPT-VQE (default), subspace
│       │                #   (SSVQE) + deflation excited states, Bloch crystals,
│       │                #   HF (RHF/UHF), expressibility, the Carcara calculator, dry run
│       ├── experimental/ # features under development -- not stable API
│       ├── backends/    # hardware.py    device registry (ideal sim, Braket, QPUs)
│       │                # providers.py   Qiskit / Braket / Cirq circuit builders
│       │                # measurement.py QWC grouping, shot-based <H>
│       │                # mitigation.py  error mitigation (stub)
│       ├── basis/       # Localized basis sets (FAO, NAO, NAO-AE, GTO/STO-nG, Pople)
│       ├── circuits/    # Ansatz protocol, UCCSD & AdaptAnsatz, gates, pools, profiling
│       ├── core/        # Fermionic operators, mappings, molecular integrals,
│       │                #   Parquet/JSON Hamiltonian serialization
│       ├── 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/            # 17 runnable walkthroughs (see below)
│   └── data/            #   all generated logs, CSV and plots land here
├── test/                # Comprehensive pytest suite (609 tests)
└── docs/                # Sphinx source files and configuration

Examples

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. The C Integral Backend

Real-space integrals run on an OpenMP C library. Carcará prefers it and compiles it automatically the first time an integral engine is created in a process (CMake, or the system C compiler when CMake is absent; Homebrew libomp on macOS), falling back to the NumPy reference kernels only when that compile fails (with a RuntimeWarning). carcara.integrals.check_backend() reports what is loaded, ensure_backend() runs the same check the engine runs, and CARCARA_BACKEND=auto|c|numpy sets the policy. To compile by hand instead:

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

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 Carcara

# 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 the calculator with the VQE method
atoms.calc = Carcara(method="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.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 Carcara

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 the calculator with the ADAPT-VQE method
atoms.calc = Carcara(
              method="adapt-vqe",
              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.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 Carcara

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 = Carcara(method="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 = Carcara(method="subspace-vqe", basis="FAO", h=0.20,
                               num_states=2)
atoms.get_potential_energy()
print("levels (eV):", atoms.calc.result.in_units("eV"))

Example 6: Cache the Hamiltonian, then sweep

Build the expensive part once and replay it — the reload needs no geometry, no integrals and no fermion-to-qubit mapping:

from ase import Atoms
from carcara.algorithms import Carcara

atoms = Atoms("LiH", positions=[[7.5, 7.5, 6.7], [7.5, 7.5, 8.3]],
              cell=[[15, 0, 0], [0, 15, 0], [0, 0, 15]], pbc=True)

# Build once (use hamiltonian_format="json" for a plain-text cache).
atoms.calc = Carcara(method="adapt-vqe", basis="FAO", h=0.25,
                               save_hamiltonian="lih.parquet")
atoms.get_total_energy()

# Compare every pool against the *same* operator, in seconds.
for pool in ("fermionic", "qubit", "qeb", "ceo"):
    result = Carcara(method="adapt-vqe", pool=pool,
                               load_hamiltonian="lih.parquet",
                               verbose=False).run()
    print(f"{pool:<10} {result.optimal_energy:.8f} Ha  "
          f"{result.num_operators} ops  {result.metrics.cnot_count} CNOTs")

Example 7: Choose a backend — or a real QPU

The calculator API does not change; only the device does:

from carcara.algorithms import Carcara

# Build and execute the circuits with Cirq (or "braket", or "qiskit").
Carcara(method="adapt-vqe", basis="FAO",
                  backend_provider="cirq", execute_circuits=True)

# Braket's local simulator, shot-based -- the same protocol a QPU uses.
Carcara(method="vqe", basis="FAO", device="braket-local", shots=8192)

# The AWS managed simulator, or a real trapped-ion QPU.
Carcara(method="vqe", basis="FAO", device="braket-sv1", shots=8192)
Carcara(method="vqe", basis="FAO", device="braket-ionq-aria",
                  shots=8192)   # needs AWS credentials

Testing

Carcará features a comprehensive unit testing suite (900+ tests) verifying integrals, basis definitions, operators, Hartree-Fock solvers, VQE/ADAPT-VQE, the Hamiltonian cache, backend-provider equivalence, and the Braket shot-based measurement path.

To run the complete test suite:

# From the project root directory
pytest

Documentation

Documentation is built using Sphinx:

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

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).