Quick Start Guide¶

This guide demonstrates basic usage of BowtieQGT for computing quantum geometric tensors and gradients.

Basic Usage¶

Computing QGT and Gradients¶

from qiskit import QuantumCircuit
from qiskit.circuit import ParameterVector
from qiskit.quantum_info import SparsePauliOp
from bowtie_qgt.bowtieqgt import BowtieQGT

# Create a parameterized quantum circuit
num_qubits = 4
qc = QuantumCircuit(num_qubits)
params = ParameterVector('θ', 8)

# Build a simple ansatz
for i in range(num_qubits):
    qc.ry(params[i], i)

for i in range(num_qubits - 1):
    qc.cx(i, i + 1)

for i in range(num_qubits):
    qc.ry(params[i + 4], i)

# Define a Hamiltonian observable
obs = SparsePauliOp.from_list([
    ("ZIII", -1.0),
    ("IZII", -1.0),
    ("IIZI", -1.0),
    ("IIIZ", -1.0),
    ("XXII", -0.5),
    ("YYII", -0.5),
])

# Initialize BowtieQGT
bowtie = BowtieQGT(qc, obs, phase_fix=True, pbar=1)

# Set parameter values
param_values = {p: 0.1 for p in qc.parameters}

# Compute generalized QGT and energy
gen_qgt, energy = bowtie.get_derivatives(param_values)

# Extract components
qgt = bowtie.extract_qgt(gen_qgt)
gradient = bowtie.extract_gradient(gen_qgt)
variance = bowtie.extract_variance(gen_qgt, energy)

print(f"Energy: {energy}")
print(f"QGT shape: {qgt.shape}")
print(f"Gradient shape: {gradient.shape}")
print(f"Variance: {variance}")

Real vs Imaginary Time Evolution¶

For real-time evolution (VarQITE):

bowtie = BowtieQGT(qc, obs, VarQITE_gradient=True)

For imaginary-time evolution:

bowtie = BowtieQGT(qc, obs, VarQITE_gradient=False)

GPU Acceleration¶

Enable GPU acceleration for faster computation:

bowtie = BowtieQGT(qc, obs, accelerator="GPU")

Phase Fixing¶

Phase fixing improves numerical stability by removing contributions from the |0…0⟩ state:

# With phase fixing (recommended)
bowtie = BowtieQGT(qc, obs, phase_fix=True)

# Without phase fixing
bowtie = BowtieQGT(qc, obs, phase_fix=False)

Progress Bars¶

Control verbosity with the pbar parameter:

# No progress bars
bowtie = BowtieQGT(qc, obs, pbar=0)

# Main computation progress
bowtie = BowtieQGT(qc, obs, pbar=1)

# Detailed progress including QGT computation
bowtie = BowtieQGT(qc, obs, pbar=2)

Understanding the Output¶

Generalized QGT Matrix Structure¶

The generalized QGT matrix has the following block structure:

┌─────────────┬──────────────┐
│     QGT     │   Gradient   │
│  (N×N)      │   (N×M)      │
├─────────────┼──────────────┤
│  Gradient†  │   Variance   │
│  (M×N)      │   (M×M)      │
└─────────────┴──────────────┘

Where: - N = number of circuit parameters - M = number of observable terms

QGT Block¶

The QGT (upper-left N×N block) is the metric tensor for the parameter space:

\[G_{ij} = \langle\partial_i\psi|\partial_j\psi\rangle - \langle\partial_i\psi|\psi\rangle\langle\psi|\partial_j\psi\rangle\]

Gradient Block¶

The gradient (upper-right N×M block) contains energy derivatives:

\[\frac{\partial E }{\partial\theta_i} = \sum_{j=0}^{M-1} c_j \frac{\partial h_j}{\partial\theta_i} = \langle\partial_i\psi|h_j|\psi\rangle - \langle\partial_i\psi|\psi\rangle\langle\psi|h_j|\psi\rangle\]

for a Hamiltonian:

\[H=\sum_{j=0}^{M-1} c_j h_j\]

Variance Block¶

The variance (lower-right M×M block) contains observable variance information.

\[\text{Var}[H] = \sum_{i=0}^{M-1}\sum_{j=0}^{M-1} \langle\psi|h_i h_j|\psi\rangle - \langle\psi|h_i|\psi\rangle\langle\psi|h_j|\psi\rangle\]

Next Steps¶

Explore the API Reference for detailed API documentation
Check Examples for advanced usage patterns
Review the test suite in test/test_bowtie_qgt.py for more examples