core.converters.fqaoa

core.converters.fqaoa#

This module implements the Fermionic QAOA (FQAOA) converter for the Qamomile framework [Yoshioka et al., 2023]. FQAOA translates the Hamiltonians into the representation of fermion systems, and the equality constraint is naturally incorporated as a constant number of particles condition.

The parameterized state \(|\vec{\beta},\vec{\gamma}\rangle\) of \(p\)-layer QAOA is defined as:

\[|\vec{\beta},\vec{\gamma}\rangle = U(\vec{\beta},\vec{\gamma})|0\rangle^{\otimes n} = e^{-i\beta_{p-1} H_M}e^{-i\gamma_{p-1} H_P} \cdots e^{-i\beta_0 H_M}e^{-i\gamma_0 H_P} U_{init}|0\rangle^{\otimes n}\]

where \(H_P\) is the cost Hamiltonian, \(H_M\) is the mixer Hamiltonian and \(\gamma_l\) and \(\beta_l\) are the variational parameters. The \(2p\) variational parameters are optimized classically to minimize the expectation value \(\langle \vec{\beta},\vec{\gamma}|H_P|\vec{\beta},\vec{\gamma}\rangle\). \(U_{init}\) prepares the initial state using Givens rotation gates [Jiang et al., 2018].

This module provides functionality to convert optimization problems which written by jijmodeling into FQAOA circuits (\(U(\vec{\beta}, \vec{\gamma})\)), construct cost Hamiltonians (\(H_P\)), and decode quantum computation results.

The QAOAConverter class extends the QuantumConverter base class, specializing in FQAOA-specific operations such as ansatz circuit generation and result decoding.

Key Features:

  • Generation of FQAOA ansatz circuits

  • Construction of cost Hamiltonians for QAOA

  • Decoding of quantum computation results into classical optimization solutions

Note

This module requires jijmodeling and jijmodeling_transpiler for problem representation.

[JSK+18]

Zhang Jiang, Kevin J Sung, Kostyantyn Kechedzhi, Vadim N Smelyanskiy, and Sergio Boixo. Quantum algorithms to simulate many-body physics of correlated fermions. Physical Review Applied, 9(4):044036, 2018.

[YSNF23]

Takuya Yoshioka, Keita Sasada, Yuichiro Nakano, and Keisuke Fujii. Fermionic quantum approximate optimization algorithm. Physical Review Research, 5(2):023071, 2023.

Classes#

FQAOAConverter

FQAOA (Fermionic Quantum Approximate Optimization Algorithm) converter class.

Module Contents#

class FQAOAConverter(instance: ommx.v1.Instance, num_fermions: int, normalize_model: bool = False, normalize_ising: Literal['abs_max', 'rms'] | None = None)#

Bases: qamomile.core.converters.converter.QuantumConverter

FQAOA (Fermionic Quantum Approximate Optimization Algorithm) converter class.

This class provides methods to convert optimization problems into FQAOA circuits, construct cost Hamiltonians, and decode quantum computation results.

Examples:

from qamomile.core.qaoa.fqaoa import FQAOAConverter

# Initialize with a compiled optimization problem instance
fqaoa_converter = FQAOAConverter(compiled_instance, num_fermion=4)

# Generate QAOA circuit and cost Hamiltonian
p = 2  # Number of QAOA layers
fqaoa_circuit = fqaoa_converter.get_ansatz_circuit(p)
cost_hamiltonian = fqaoa_converter.get_cost_hamiltonian()

Initialize the FQAOAConverter.

This method initializes the converter with the compiled instance of the optimization problem.

Parameters:

  • compiled_instance – ommx.v1.Instance.

  • num_fermions (int) – Number of fermions. This means the constraint \(M = \sum_{l,d} x_{l,d}\).

  • normalize_model (bool) – The objective function and the constraints are normalized using the maximum absolute value of the coefficients contained in each. Defaults to False

  • normalize_ising (Literal["abs_max", "rms"] | None) – The normalization method for the Ising Hamiltonian. Available options: - “abs_max”: Normalize by absolute maximum value - “rms”: Normalize by root mean square Defaults to None.

num_fermions#
var_map#
ising#
num_qubits#
fqaoa_instance_to_qubo() → tuple[dict[tuple[int, int], float], float]#

Convert the instance to QUBO format.

This method converts the optimization problem instance into a QUBO (Quadratic Unconstrained Binary Optimization) representation, which is suitable for quantum computation.

Returns:

A tuple containing the QUBO dictionary and the constant term.

Return type:

tuple[dict[int, float], float]

fqaoa_get_ising() → qamomile.core.ising_qubo.IsingModel#

Get the Ising model representation of the problem.

Returns:

The Ising model representation.

Return type:

IsingModel

fqaoa_ising_encode() → qamomile.core.ising_qubo.IsingModel#
cyclic_mapping() → dict[tuple[int, int], int]#

Get variable maps between decision variable indices \((l,d)\) and qubit index \(i\).

Returns:

A variable map for ring driver.

Return type:

dict[tuple[int, int], int]

get_fermi_orbital() → numpy.ndarray#

Compute the single-particle wave functions of the occupied spin orbitals.

Returns:

A 2D numpy array of shape (num_fermions, num_qubits)

Return type:

numpy.ndarray

givens_rotation(givens_angles) → qamomile.core.circuit.QuantumCircuit#

Generate givens rotation gates for the initial state preparation of FQAOA.

Parameters:

givens_angles (list[tuple[int, int], float]) – Parameters which represents a givens rotation of coordinates \((i,j)\) by angles \(theta\).

Returns:

A givens rotation gate.

Return type:

qm_c.QuantumCircuit

get_init_state() → qamomile.core.circuit.QuantumCircuit#

Generate the initial state preparation for FQAOA.

get_mixer_ansatz(beta: qamomile.core.circuit.Parameter, hopping: float = 1.0, name: str = 'Mixer') → qamomile.core.circuit.QuantumCircuit#

Generate the fermionic mixer ansatz circuit (\(e^{-\beta H_d}\)) for FQAOA.

Parameters:

  • beta (qm_c.Parameter) – The beta parameter for the mixer ansatz.

  • hopping (float) – The hopping integral. Defaults to 1.0.

  • name (str, optional) – Name of the circuit. Defaults to “Mixer”.

Returns:

The fermionic mixer ansatz circuit.

Return type:

qm_c.QuantumCircuit

get_cost_ansatz(gamma: qamomile.core.circuit.Parameter, name: str = 'Cost') → qamomile.core.circuit.QuantumCircuit#

Generate the cost ansatz circuit (\(e^{-\gamma H_P}\)) for QAOA. This function is mainly used when you have designed your own mixer Hamiltonian and only need the cost Hamiltonian.

Parameters:

  • gamma (qm_c.Parameter) – The gamma parameter for the cost ansatz.

  • name (str, optional) – Name of the circuit. Defaults to “Cost”.

Returns:

The cost ansatz circuit.

Return type:

qm_c.QuantumCircuit

get_fqaoa_ansatz(p: int, hopping: float = 1.0) → qamomile.core.circuit.QuantumCircuit#

Generate the FQAOA ansatz circuit.

Parameters:

  • p (int) – Number of QAOA layers.

  • hopping (float) – The hopping integral.

Returns:

The FQAOA ansatz circuit.

Return type:

qm_c.QuantumCircuit

get_cost_hamiltonian() → qamomile.core.operator.Hamiltonian#

Construct the cost Hamiltonian (\(H_P\)) for FQAOA.

Returns:

The cost Hamiltonian.

Return type:

qm_o.Hamiltonian

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