Enhance feature classification using projected quantum kernels

Usage estimate: 80 minutes on a Heron r3 processor (NOTE: This is an estimate only. Your runtime might vary.)

In this tutorial, we demonstrate how to run a projected quantum kernel (PQK) with Qiskit on a real-world biological dataset, based on the paper Enhanced Prediction of CAR T-Cell Cytotoxicity with Quantum-Kernel Methods [1].

PQK is a method used in quantum machine learning (QML) to encode classical data into a quantum feature space and project them back into the classical domain, by using quantum computers to enhance feature selection. It involves encoding classical data into quantum states using a quantum circuit, typically through a process called feature mapping, where the data is transformed into a high-dimensional Hilbert space. The "projected" aspect refers to extracting classical information from the quantum states, by measuring specific observables, to construct a kernel matrix that can be used in classical kernel-based algorithms like support vector machines. This approach leverages the computational advantages of quantum systems to potentially achieve better performance on certain tasks compared to classical methods.

This tutorial also assumes general familiarity with QML methods. For further exploration of QML, refer to the Quantum machine learning course in IBM Quantum Learning.

Requirements

Before starting this tutorial, ensure that you have the following installed:

Qiskit SDK v2.0 or later with visualization support (pip install 'qiskit[visualization]')
Qiskit Runtime v0.40 or later (pip install qiskit-ibm-runtime)
Category encoders 2.8.1 (pip install category-encoders)
NumPy 2.3.2 (pip install numpy)
Pandas 2.3.2 (pip install pandas)
Scikit-learn 1.7.1 (pip install scikit-learn)
Tqdm 4.67.1 (pip install tqdm)

Setup

import warnings
 
# Standard libraries
import os
import numpy as np
import pandas as pd
 
# Machine learning and data processing
import category_encoders as ce
from scipy.linalg import inv, sqrtm
from sklearn.metrics.pairwise import rbf_kernel
from sklearn.model_selection import GridSearchCV, StratifiedKFold
from sklearn.svm import SVC
 
# Qiskit and Qiskit Runtime
from qiskit import QuantumCircuit
from qiskit.circuit import ParameterVector
from qiskit.circuit.library import UnitaryGate, ZZFeatureMap
from qiskit.quantum_info import SparsePauliOp, random_unitary
from qiskit.transpiler import generate_preset_pass_manager
from qiskit_ibm_runtime import (
    Batch,
    EstimatorOptions,
    EstimatorV2 as Estimator,
    QiskitRuntimeService,
)
 
# Progress bar
import tqdm
 
warnings.filterwarnings("ignore")

Step 1: Map classical inputs to a quantum problem

Dataset preparation

In this tutorial we use a real-world biological dataset for a binary classification task, which is generated by Daniels et al. (2022) and can be downloaded from the supplementary material included with the paper. The data consists of CAR T-cells, which are genetically engineered T-cells used in immunotherapy to treat certain cancers. T-cells, a type of immune cell, are modified in a lab to express chimeric antigen receptors (CARs) that target specific proteins on cancer cells. These modified T-cells can recognize and destroy cancer cells more effectively. The data features are the CAR T-cell motifs, which refer to the specific structural or functional component of the CAR engineered into T-cells. Based on these motifs, our task is to predict the cytotoxicity of a given CAR T-cell, labeling it as either toxic or non-toxic.

The following shows the helper functions to preprocess this dataset.

def preprocess_data(dir_root, args):
    """
    Preprocess the training and test data.
    """
    # Read from the csv files
    train_data = pd.read_csv(
        os.path.join(dir_root, args["file_train_data"]),
        encoding="unicode_escape",
        sep=",",
    )
    test_data = pd.read_csv(
        os.path.join(dir_root, args["file_test_data"]),
        encoding="unicode_escape",
        sep=",",
    )
 
    # Fix the last motif ID
    train_data[train_data == 17] = 14
    train_data.columns = [
        "Cell Number",
        "motif",
        "motif.1",
        "motif.2",
        "motif.3",
        "motif.4",
        "Nalm 6 Cytotoxicity",
    ]
    test_data[test_data == 17] = 14
    test_data.columns = [
        "Cell Number",
        "motif",
        "motif.1",
        "motif.2",
        "motif.3",
        "motif.4",
        "Nalm 6 Cytotoxicity",
    ]
 
    # Adjust motif at the third position
    if args["filter_for_spacer_motif_third_position"]:
        train_data = train_data[
            (train_data["motif.2"] == 14) | (train_data["motif.2"] == 0)
        ]
        test_data = test_data[
            (test_data["motif.2"] == 14) | (test_data["motif.2"] == 0)
        ]
 
    train_data = train_data[
        args["motifs_to_use"] + [args["label_name"], "Cell Number"]
    ]
    test_data = test_data[
        args["motifs_to_use"] + [args["label_name"], "Cell Number"]
    ]
 
    # Adjust motif at the last position
    if not args["allow_spacer_motif_last_position"]:
        last_motif = args["motifs_to_use"][len(args["motifs_to_use"]) - 1]
        train_data = train_data[
            (train_data[last_motif] != 14) & (train_data[last_motif] != 0)
        ]
        test_data = test_data[
            (test_data[last_motif] != 14) & (test_data[last_motif] != 0)
        ]
 
    # Get the labels
    train_labels = np.array(train_data[args["label_name"]])
    test_labels = np.array(test_data[args["label_name"]])
 
    # For the classification task use the threshold to binarize labels
    train_labels[train_labels > args["label_binarization_threshold"]] = 1
    train_labels[train_labels < 1] = args["min_label_value"]
    test_labels[test_labels > args["label_binarization_threshold"]] = 1
    test_labels[test_labels < 1] = args["min_label_value"]
 
    # Reduce data to just the motifs of interest
    train_data = train_data[args["motifs_to_use"]]
    test_data = test_data[args["motifs_to_use"]]
 
    # Get the class and motif counts
    min_class = np.min(np.unique(np.concatenate([train_data, test_data])))
    max_class = np.max(np.unique(np.concatenate([train_data, test_data])))
 
    num_class = max_class - min_class + 1
    num_motifs = len(args["motifs_to_use"])
    print(str(max_class) + ":" + str(min_class) + ":" + str(num_class))
 
    train_data = train_data - min_class
    test_data = test_data - min_class
 
    return (
        train_data,
        test_data,
        train_labels,
        test_labels,
        num_class,
        num_motifs,
    )
 
 
def data_encoder(args, train_data, test_data, num_class, num_motifs):
    """
    Use one-hot or binary encoding for classical data representation.
    """
    if args["encoder"] == "one-hot":
        # Transform to one-hot encoding
        train_data = np.eye(num_class)[train_data]
        test_data = np.eye(num_class)[test_data]
 
        train_data = train_data.reshape(
            train_data.shape[0], train_data.shape[1] * train_data.shape[2]
        )
        test_data = test_data.reshape(
            test_data.shape[0], test_data.shape[1] * test_data.shape[2]
        )
 
    elif args["encoder"] == "binary":
        # Transform to binary encoding
        encoder = ce.BinaryEncoder()
 
        base_array = np.unique(np.concatenate([train_data, test_data]))
        base = pd.DataFrame(base_array).astype("category")
        base.columns = ["motif"]
        for motif_name in args["motifs_to_use"][1:]:
            base[motif_name] = base.loc[:, "motif"]
        encoder.fit(base)
 
        train_data = encoder.transform(train_data.astype("category"))
        test_data = encoder.transform(test_data.astype("category"))
 
        train_data = np.reshape(
            train_data.values, (train_data.shape[0], num_motifs, -1)
        )
        test_data = np.reshape(
            test_data.values, (test_data.shape[0], num_motifs, -1)
        )
 
        train_data = train_data.reshape(
            train_data.shape[0], train_data.shape[1] * train_data.shape[2]
        )
        test_data = test_data.reshape(
            test_data.shape[0], test_data.shape[1] * test_data.shape[2]
        )
 
    else:
        raise ValueError("Invalid encoding type.")
 
    return train_data, test_data

You can run this tutorial by running the following cell, which automatically creates the required folder structure and downloads both the training and test files directly into your environment. If you already have these files locally, this step will safely overwrite them to ensure version consistency.

## Download dataset
 
# Create data directory if it doesn't exist
!mkdir -p data_tutorial/pqk
 
# Download the training and test sets from the official Qiskit documentation repo
!wget -q --show-progress -O data_tutorial/pqk/train_data.csv \
  https://raw.githubusercontent.com/Qiskit/documentation/main/datasets/tutorials/pqk/train_data.csv
 
!wget -q --show-progress -O data_tutorial/pqk/test_data.csv \
  https://raw.githubusercontent.com/Qiskit/documentation/main/datasets/tutorials/pqk/test_data.csv
 
!wget -q --show-progress -O data_tutorial/pqk/projections_train.csv \
  https://raw.githubusercontent.com/Qiskit/documentation/main/datasets/tutorials/pqk/projections_train.csv
 
!wget -q --show-progress -O data_tutorial/pqk/projections_test.csv \
  https://raw.githubusercontent.com/Qiskit/documentation/main/datasets/tutorials/pqk/projections_test.csv
 
# Check the files have been downloaded
!echo "Dataset files downloaded:"
!ls -lh data_tutorial/pqk/*.csv

args = {
    "file_train_data": "train_data.csv",
    "file_test_data": "test_data.csv",
    "motifs_to_use": ["motif", "motif.1", "motif.2", "motif.3"],
    "label_name": "Nalm 6 Cytotoxicity",
    "label_binarization_threshold": 0.62,
    "filter_for_spacer_motif_third_position": False,
    "allow_spacer_motif_last_position": True,
    "min_label_value": -1,
    "encoder": "one-hot",
}
dir_root = "./"
 
# Preprocess data
train_data, test_data, train_labels, test_labels, num_class, num_motifs = (
    preprocess_data(dir_root=dir_root, args=args)
)
 
# Encode the data
train_data, test_data = data_encoder(
    args, train_data, test_data, num_class, num_motifs
)

Output:

14:0:15

We also transform the dataset such that $1$ is represented as $\pi/2$ for scaling purposes.

# Change 1 to pi/2
angle = np.pi / 2
 
tmp = pd.DataFrame(train_data).astype("float64")
tmp[tmp == 1] = angle
train_data = tmp.values
 
tmp = pd.DataFrame(test_data).astype("float64")
tmp[tmp == 1] = angle
test_data = tmp.values

We verify sizes and shapes of the training and test datasets.

print(train_data.shape, train_labels.shape)
print(test_data.shape, test_labels.shape)

Output:

(172, 60) (172,)
(74, 60) (74,)

Step 2: Optimize problem for quantum hardware execution

Quantum circuit

We now construct the feature map that embeds our classical dataset into a higher-dimensional feature space. For this embedding, we use the ZZFeatureMap from Qiskit.

feature_dimension = train_data.shape[1]
reps = 24
insert_barriers = True
entanglement = "pairwise"
 
# ZZFeatureMap with linear entanglement and a repetition of 2
embed = ZZFeatureMap(
    feature_dimension=feature_dimension,
    reps=reps,
    entanglement=entanglement,
    insert_barriers=insert_barriers,
    name="ZZFeatureMap",
)
embed.decompose().draw(output="mpl", style="iqp", fold=-1)

Output:

Another quantum embedding option is the 1D-Heisenberg Hamiltonian evolution ansatz. You can skip running this section if you would like to continue with the ZZFeatureMap.

feature_dimension = train_data.shape[1]
num_qubits = feature_dimension + 1
embed2 = QuantumCircuit(num_qubits)
num_trotter_steps = 6
pv_length = feature_dimension * num_trotter_steps
pv = ParameterVector("theta", pv_length)
 
# Add Haar random single qubit unitary to each qubit as initial state
np.random.seed(42)
seeds_unitary = np.random.randint(0, 100, num_qubits)
for i in range(num_qubits):
    rand_gate = UnitaryGate(random_unitary(2, seed=seeds_unitary[i]))
    embed2.append(rand_gate, [i])
 
 
def trotter_circ(feature_dimension, num_trotter_steps):
    num_qubits = feature_dimension + 1
    circ = QuantumCircuit(num_qubits)
    # Even
    for i in range(0, feature_dimension, 2):
        circ.rzz(2 * pv[i] / num_trotter_steps, i, i + 1)
    for i in range(0, feature_dimension, 2):
        circ.rxx(2 * pv[i] / num_trotter_steps, i, i + 1)
    for i in range(0, feature_dimension, 2):
        circ.ryy(2 * pv[i] / num_trotter_steps, i, i + 1)
    # Odd
    for i in range(1, feature_dimension, 2):
        circ.rzz(2 * pv[i] / num_trotter_steps, i, i + 1)
    for i in range(1, feature_dimension, 2):
        circ.rxx(2 * pv[i] / num_trotter_steps, i, i + 1)
    for i in range(1, feature_dimension, 2):
        circ.ryy(2 * pv[i] / num_trotter_steps, i, i + 1)
    return circ
 
 
# Hamiltonian evolution ansatz
for step in range(num_trotter_steps):
    circ = trotter_circ(feature_dimension, num_trotter_steps)
    if step % 2 == 0:
        embed2 = embed2.compose(circ)
    else:
        reverse_circ = circ.reverse_ops()
        embed2 = embed2.compose(reverse_circ)
 
 
embed2.draw(output="mpl", style="iqp", fold=-1)

Output:

Step 3: Execute using Qiskit primitives

Measure 1-RDMs

The main building blocks of projected quantum kernels are the reduced density matrices (RDMs), which are obtained though projective measurements of the quantum feature map. In this step, we obtain all single-qubit reduced density matrices (1-RDMs), which will later be provided into the classical exponential kernel function.

Let's look at how to compute the 1-RDM given a single data point from the dataset before we run over all data. The 1-RDMs are a collection of single-qubit measurements of Pauli X, Y and Z operators on all qubits. This is because a single-qubit RDM can be fully expressed as: $\rho = \frac{1}{2} \big( I + \braket \sigma_x \sigma_x + \braket \sigma_y \sigma_y + \braket \sigma_z \sigma_z \big)$

First we select the backend to use.

service = QiskitRuntimeService()
backend = service.least_busy(
    operational=True, simulator=False, min_num_qubits=133
)
target = backend.target

Then we run the quantum circuit and measure the projections. Note that we turn on error mitigation, including Zero Noise Extrapolation (ZNE).

# Let's select the ZZFeatureMap embedding for this example
qc = embed
num_qubits = feature_dimension
 
# Identity operator on all qubits
id = "I" * num_qubits
 
# Let's select the first training datapoint as an example
parameters = train_data[0]
 
# Bind parameter to the circuit and simplify it
qc_bound = qc.assign_parameters(parameters)
transpiler = generate_preset_pass_manager(
    optimization_level=3, basis_gates=["u3", "cz"]
)
transpiled_circuit = transpiler.run(qc_bound)
 
# Transpile for hardware
transpiler = generate_preset_pass_manager(optimization_level=3, target=target)
transpiled_circuit = transpiler.run(transpiled_circuit)
 
# We group all commuting observables
# These groups are the Pauli X, Y and Z operators on individual qubits
observables_x = [
    SparsePauliOp(id[:i] + "X" + id[(i + 1) :]).apply_layout(
        transpiled_circuit.layout
    )
    for i in range(num_qubits)
]
observables_y = [
    SparsePauliOp(id[:i] + "Y" + id[(i + 1) :]).apply_layout(
        transpiled_circuit.layout
    )
    for i in range(num_qubits)
]
observables_z = [
    SparsePauliOp(id[:i] + "Z" + id[(i + 1) :]).apply_layout(
        transpiled_circuit.layout
    )
    for i in range(num_qubits)
]
 
# We define the primitive unified blocs (PUBs) consisting of the embedding circuit,
# set of observables and the circuit parameters
pub_x = (transpiled_circuit, observables_x)
pub_y = (transpiled_circuit, observables_y)
pub_z = (transpiled_circuit, observables_z)
 
# Experiment options for error mitigation
num_randomizations = 300
shots_per_randomization = 100
noise_factors = [1, 3, 5]
 
experimental_opts = {}
experimental_opts["resilience"] = {
    "measure_mitigation": True,
    "zne_mitigation": True,
    "zne": {
        "noise_factors": noise_factors,
        "amplifier": "gate_folding",
        "return_all_extrapolated": True,
        "return_unextrapolated": True,
        "extrapolated_noise_factors": [0] + noise_factors,
    },
}
experimental_opts["twirling"] = {
    "num_randomizations": num_randomizations,
    "shots_per_randomization": shots_per_randomization,
    "strategy": "active-accum",
}
options = EstimatorOptions(experimental=experimental_opts)
 
# We define and run the estimator to obtain <X>, <Y> and <Z> on all qubits
estimator = Estimator(mode=backend, options=options)
 
job = estimator.run([pub_x, pub_y, pub_z])

Next we retrieve the results.

job_result_x = job.result()[0].data.evs
job_result_y = job.result()[1].data.evs
job_result_z = job.result()[2].data.evs

print(job_result_x)
print(job_result_y)
print(job_result_z)

Output:

[ 3.67865951e-03  1.01158571e-02 -3.95790878e-02  6.33984326e-03
  1.86035759e-02 -2.91533268e-02 -1.06374793e-01  4.48873518e-18
  4.70201764e-02  3.53997968e-02  2.53130819e-02  3.23903401e-02
  6.06327843e-03  1.16313667e-02 -1.12387504e-02 -3.18457725e-02
 -4.16445718e-04 -1.45609602e-03 -4.21737114e-01  2.83705669e-02
  6.91332890e-03 -7.45363001e-02 -1.20139326e-02 -8.85566135e-02
 -3.22648394e-02 -3.24228074e-02  6.20431299e-04  3.04225434e-03
  5.72795792e-03  1.11288428e-02  1.50395861e-01  9.18380197e-02
  1.02553163e-01  2.98312847e-02 -3.30298912e-01 -1.13979648e-01
  4.49159340e-03  8.63861493e-02  3.05666566e-02  2.21463145e-04
  1.45946735e-02  8.54537275e-03 -8.09805979e-02 -2.92608104e-02
 -3.91243644e-02 -3.96632760e-02 -1.41187613e-01 -1.07363243e-01
  1.81089440e-02  2.70778895e-02  1.45139414e-02  2.99480458e-02
  4.99137134e-02  7.08789852e-02  4.30565759e-02  8.71287156e-02
  1.04334798e-01  7.72191962e-02  7.10059720e-02  1.04650403e-01]
[-7.31765102e-05  7.42669174e-03  9.82277344e-03  5.92638249e-02
  4.24120486e-02 -9.06473416e-03  4.55057675e-03  8.43494094e-03
  6.92097339e-02 -6.82234424e-02  6.13509008e-02  3.94200491e-02
 -1.24037979e-02  1.01976642e-01  7.90538600e-03 -7.19726160e-02
 -1.19501703e-16 -1.03796614e-02  7.37382463e-02  1.97238568e-01
 -3.59250635e-02 -2.67554009e-02  3.55010633e-02  7.68877990e-02
  6.50677589e-05 -6.59298767e-03 -1.23719487e-02 -6.41938151e-02
  1.95603072e-02 -2.48448551e-02  5.17784810e-02 -5.93767100e-02
  3.11897681e-02 -3.91959720e-18 -4.47769148e-03  1.39202197e-01
 -6.56387523e-02 -5.85665483e-02  9.52905894e-03 -8.61460731e-02
  3.91790656e-02 -1.27544375e-01  1.63712244e-01  3.36816934e-04
  2.26230028e-02 -2.45023393e-05  4.95635588e-03  1.44779564e-01
  3.71625177e-02  3.65675948e-03  2.83694017e-02 -7.10500602e-02
 -1.15467702e-01  6.21712129e-03 -4.80958959e-02  2.21021066e-02
  7.99062499e-02 -1.87164076e-02 -3.67100369e-02 -2.38923731e-02]
[ 6.85871605e-01  5.07725024e-01  8.71024642e-03  3.34823455e-02
  4.58684961e-02  9.44384189e-17 -4.46829296e-02 -2.91296778e-02
  4.15466461e-02  2.89628330e-02  1.88624017e-03  5.37110446e-02
  2.59579053e-03  1.39327071e-02 -2.90781778e-02  5.07209866e-03
  5.83403000e-02  2.60764440e-02  4.45999706e-17 -6.66701417e-03
  3.03215873e-01  2.26172533e-02  2.43105960e-02  4.98861041e-18
 -2.45530791e-02  6.26940708e-02  1.21058073e-02  2.76675948e-04
  2.63980996e-02  2.58302364e-02  7.47856723e-02  8.42728943e-02
  5.70989097e-02  6.92955086e-02 -5.68313712e-03  1.32199452e-01
  8.90511238e-02 -3.45204621e-02 -1.05445836e-01  6.03864150e-03
  2.16291384e-02  8.22303162e-03  1.00856715e-02  6.28973151e-02
  6.26727169e-02  6.15399206e-02  9.67320897e-02  1.03045269e-16
  1.79688783e-01 -1.59960520e-02 -1.15422952e-02  9.60200470e-03
  6.58396672e-02  7.78329830e-03  6.53226955e-02  2.45778685e-03
  4.36694753e-03  5.75098762e-03 -2.48896201e-02  8.33740755e-05]

We print out the circuit size and the two-qubit gate depth.

print(f"qubits: {qc.num_qubits}")
print(
    f"2q-depth: {transpiled_circuit.depth(lambda x: x.operation.num_qubits==2)}"
)
print(
    f"2q-size: {transpiled_circuit.size(lambda x: x.operation.num_qubits==2)}"
)
print(f"Operator counts: {transpiled_circuit.count_ops()}")
transpiled_circuit.draw("mpl", fold=-1, style="clifford", idle_wires=False)

Output:

qubits: 60
2q-depth: 64
2q-size: 1888
Operator counts: OrderedDict({'rz': 6016, 'sx': 4576, 'cz': 1888, 'x': 896, 'barrier': 31})

We can now loop over the entire training dataset to obtain all 1-RDMs.

We also provide the results from an experiment that we ran on quantum hardware. You can either run the training yourself by setting the flag below to True, or use the projection results that we provide.

# Set this to True if you want to run the training on hardware
run_experiment = False

# Identity operator on all qubits
id = "I" * num_qubits
 
# projections_train[i][j][k] will be the expectation value of the j-th Pauli operator (0: X, 1: Y, 2: Z)
# of datapoint i on qubit k
projections_train = []
jobs_train = []
 
# Experiment options for error mitigation
num_randomizations = 300
shots_per_randomization = 100
noise_factors = [1, 3, 5]
 
experimental_opts = {}
experimental_opts["resilience"] = {
    "measure_mitigation": True,
    "zne_mitigation": True,
    "zne": {
        "noise_factors": noise_factors,
        "amplifier": "gate_folding",
        "return_all_extrapolated": True,
        "return_unextrapolated": True,
        "extrapolated_noise_factors": [0] + noise_factors,
    },
}
experimental_opts["twirling"] = {
    "num_randomizations": num_randomizations,
    "shots_per_randomization": shots_per_randomization,
    "strategy": "active-accum",
}
options = EstimatorOptions(experimental=experimental_opts)
 
if run_experiment:
    with Batch(backend=backend):
        for i in tqdm.tqdm(
            range(len(train_data)), desc="Training data progress"
        ):
            # Get training sample
            parameters = train_data[i]
 
            # Bind parameter to the circuit and simplify it
            qc_bound = qc.assign_parameters(parameters)
            transpiler = generate_preset_pass_manager(
                optimization_level=3, basis_gates=["u3", "cz"]
            )
            transpiled_circuit = transpiler.run(qc_bound)
 
            # Transpile for hardware
            transpiler = generate_preset_pass_manager(
                optimization_level=3, target=target
            )
            transpiled_circuit = transpiler.run(transpiled_circuit)
 
            # We group all commuting observables
            # These groups are the Pauli X, Y and Z operators on individual qubits
            observables_x = [
                SparsePauliOp(id[:i] + "X" + id[(i + 1) :]).apply_layout(
                    transpiled_circuit.layout
                )
                for i in range(num_qubits)
            ]
            observables_y = [
                SparsePauliOp(id[:i] + "Y" + id[(i + 1) :]).apply_layout(
                    transpiled_circuit.layout
                )
                for i in range(num_qubits)
            ]
            observables_z = [
                SparsePauliOp(id[:i] + "Z" + id[(i + 1) :]).apply_layout(
                    transpiled_circuit.layout
                )
                for i in range(num_qubits)
            ]
 
            # We define the primitive unified blocs (PUBs) consisting of the embedding circuit,
            # set of observables and the circuit parameters
            pub_x = (transpiled_circuit, observables_x)
            pub_y = (transpiled_circuit, observables_y)
            pub_z = (transpiled_circuit, observables_z)
 
            # We define and run the estimator to obtain <X>, <Y> and <Z> on all qubits
            estimator = Estimator(options=options)
 
            job = estimator.run([pub_x, pub_y, pub_z])
            jobs_train.append(job)

Output:

Training data progress: 100%|██████████| 172/172 [13:03<00:00,  4.55s/it]

Once the jobs are complete, we can retrieve the results.

if run_experiment:
    for i in tqdm.tqdm(
        range(len(train_data)), desc="Retrieving training data results"
    ):
        # Completed job
        job = jobs_train[i]
 
        # Job results
        job_result_x = job.result()[0].data.evs
        job_result_y = job.result()[1].data.evs
        job_result_z = job.result()[2].data.evs
 
        # Record <X>, <Y> and <Z> on all qubits for the current datapoint
        projections_train.append([job_result_x, job_result_y, job_result_z])

We repeat this for the test set.

# Identity operator on all qubits
id = "I" * num_qubits
 
# projections_test[i][j][k] will be the expectation value of the j-th Pauli operator (0: X, 1: Y, 2: Z)
# of datapoint i on qubit k
projections_test = []
jobs_test = []
 
# Experiment options for error mitigation
num_randomizations = 300
shots_per_randomization = 100
noise_factors = [1, 3, 5]
 
experimental_opts = {}
experimental_opts["resilience"] = {
    "measure_mitigation": True,
    "zne_mitigation": True,
    "zne": {
        "noise_factors": noise_factors,
        "amplifier": "gate_folding",
        "return_all_extrapolated": True,
        "return_unextrapolated": True,
        "extrapolated_noise_factors": [0] + noise_factors,
    },
}
experimental_opts["twirling"] = {
    "num_randomizations": num_randomizations,
    "shots_per_randomization": shots_per_randomization,
    "strategy": "active-accum",
}
options = EstimatorOptions(experimental=experimental_opts)
 
if run_experiment:
    with Batch(backend=backend):
        for i in tqdm.tqdm(range(len(test_data)), desc="Test data progress"):
            # Get test sample
            parameters = test_data[i]
 
            # Bind parameter to the circuit and simplify it
            qc_bound = qc.assign_parameters(parameters)
            transpiler = generate_preset_pass_manager(
                optimization_level=3, basis_gates=["u3", "cz"]
            )
            transpiled_circuit = transpiler.run(qc_bound)
 
            # Transpile for hardware
            transpiler = generate_preset_pass_manager(
                optimization_level=3, target=target
            )
            transpiled_circuit = transpiler.run(transpiled_circuit)
 
            # We group all commuting observables
            # These groups are the Pauli X, Y and Z operators on individual qubits
            observables_x = [
                SparsePauliOp(id[:i] + "X" + id[(i + 1) :]).apply_layout(
                    transpiled_circuit.layout
                )
                for i in range(num_qubits)
            ]
            observables_y = [
                SparsePauliOp(id[:i] + "Y" + id[(i + 1) :]).apply_layout(
                    transpiled_circuit.layout
                )
                for i in range(num_qubits)
            ]
            observables_z = [
                SparsePauliOp(id[:i] + "Z" + id[(i + 1) :]).apply_layout(
                    transpiled_circuit.layout
                )
                for i in range(num_qubits)
            ]
 
            # We define the primitive unified blocs (PUBs) consisting of the embedding circuit,
            # set of observables and the circuit parameters
            pub_x = (transpiled_circuit, observables_x)
            pub_y = (transpiled_circuit, observables_y)
            pub_z = (transpiled_circuit, observables_z)
 
            # We define and run the estimator to obtain <X>, <Y> and <Z> on all qubits
            estimator = Estimator(options=options)
 
            job = estimator.run([pub_x, pub_y, pub_z])
            jobs_test.append(job)

Output:

Test data progress: 100%|██████████| 74/74 [00:13<00:00,  5.56it/s]

We can retrieve the results as before.

if run_experiment:
    for i in tqdm.tqdm(
        range(len(test_data)), desc="Retrieving test data results"
    ):
        # Completed job
        job = jobs_test[i]
 
        # Job results
        job_result_x = job.result()[0].data.evs
        job_result_y = job.result()[1].data.evs
        job_result_z = job.result()[2].data.evs
 
        # Record <X>, <Y> and <Z> on all qubits for the current datapoint
        projections_test.append([job_result_x, job_result_y, job_result_z])

Step 4: Post-process and return result in desired classical format

Define the projected quantum kernel

The projected quantum kernel is defined with the following kernel function: $k^{\textrm{PQ}}(x_i, x_j) = \textrm{exp} \Big(-\gamma \sum_k \sum_{P \in \{ X,Y,Z \}} (\textrm{Tr}[P \rho_k(x_i)] - \textrm{Tr}[P \rho_k(x_j)])^2 \Big)$ In the above equation, $\gamma>0$ is a tunable hyperparameter. The $K^{\textrm{PQ}}_{ij} = k^{\textrm{PQ}}(x_i, x_j)$ are the entries of the kernel matrix $K^{\textrm{PQ}}$ .

Using the definition of 1-RDMs, we can see that the individual terms within the kernel function can be evaluated as $\textrm{Tr}[P \rho_k (x_i)] = \braket P$ , where $P \in \{ X,Y,Z \}$ . These expectation values are precisely what we measured above.

By using scikit-learn, we can in fact compute the kernel even more easily. This is due to the readily available radial basis function ('rbf') kernel: $\textrm{exp} (-\gamma \lVert x - x' \rVert^2)$ . First, we simply need to reshape the new projected training and test datasets into two-dimensional arrays.

Note that going over the entire dataset can take about 80 minutes on the QPU. To make sure that the rest of the tutorial is easily executable, we additionally provide projections from a previously run experiment (which are included in the files you downloaded in the Download dataset code block). If you performed the training yourself, you can continue the tutorial with your own results.

if run_experiment:
    projections_train = np.array(projections_train).reshape(
        len(projections_train), -1
    )
    projections_test = np.array(projections_test).reshape(
        len(projections_test), -1
    )
else:
    projections_train = np.loadtxt("projections_train.txt")
    projections_test = np.loadtxt("projections_test.txt")

Support Vector Machine (SVM)

We can now run a classical SVM on this precomputed kernel, and use the kernel between test and training sets for prediction.

# Range of 'C' and 'gamma' values as SVC hyperparameters
C_range = [0.001, 0.005, 0.007]
C_range.extend([x * 0.01 for x in range(1, 11)])
C_range.extend([x * 0.25 for x in range(1, 60)])
C_range.extend(
    [
        20,
        50,
        100,
        200,
        500,
        700,
        1000,
        1100,
        1200,
        1300,
        1400,
        1500,
        1700,
        2000,
    ]
)
 
gamma_range = ["auto", "scale", 0.001, 0.005, 0.007]
gamma_range.extend([x * 0.01 for x in range(1, 11)])
gamma_range.extend([x * 0.25 for x in range(1, 60)])
gamma_range.extend([20, 50, 100])
 
param_grid = dict(C=C_range, gamma=gamma_range)
 
# Support vector classifier
svc = SVC(kernel="rbf")
 
# Define the cross validation
cv = StratifiedKFold(n_splits=10)
 
# Grid search for hyperparameter tuning (q: quantum)
grid_search_q = GridSearchCV(
    svc, param_grid, cv=cv, verbose=1, n_jobs=-1, scoring="f1_weighted"
)
grid_search_q.fit(projections_train, train_labels)
 
# Best model with best parameters
best_svc_q = grid_search_q.best_estimator_
print(
    f"The best parameters are {grid_search_q.best_params_} with a score of {grid_search_q.best_score_:.4f}"
)
 
# Test accuracy
accuracy_q = best_svc_q.score(projections_test, test_labels)
print(f"Test accuracy with best model: {accuracy_q:.4f}")

Output:

Fitting 10 folds for each of 6622 candidates, totalling 66220 fits
The best parameters are {'C': 8.5, 'gamma': 0.01} with a score of 0.6980
Test accuracy with best model: 0.8108

Classical benchmarking

We can run a classical SVM with the radial basis function as the kernel without doing a quantum projection. This result is our classical benchmark.

# Support vector classifier
svc = SVC(kernel="rbf")
 
# Grid search for hyperparameter tuning (c: classical)
grid_search_c = GridSearchCV(
    svc, param_grid, cv=cv, verbose=1, n_jobs=-1, scoring="f1_weighted"
)
grid_search_c.fit(train_data, train_labels)
 
# Best model with best parameters
best_svc_c = grid_search_c.best_estimator_
print(
    f"The best parameters are {grid_search_c.best_params_} with a score of {grid_search_c.best_score_:.4f}"
)
 
# Test accuracy
accuracy_c = best_svc_c.score(test_data, test_labels)
print(f"Test accuracy with best model: {accuracy_c:.4f}")

Output:

Fitting 10 folds for each of 6622 candidates, totalling 66220 fits
The best parameters are {'C': 10.75, 'gamma': 0.04} with a score of 0.7830
Test accuracy with best model: 0.7432

Appendix: Verify the dataset's potential quantum advantage in learning tasks

Not all datasets offer potential advantage from the use of PQKs. There are some theoretical bounds that one can use as a preliminary test to see if a particular dataset can benefit from PQKs. To quantify this, authors of Power of data in quantum machine learning [2] define quantities referred to as classical and quantum model complexities and geometric separation of the classical and quantum models. To expect a potential quantum advantage from PQKs, the geometric separation between the classical and quantum-projected kernels should be approximately on the order of $\sqrt{N}$ , where $N$ is the number of training samples. If this condition is satisfied, we move on to checking the model complexities. If the classical model complexity is on the order of $N$ while the quantum-projected model complexity is substantially smaller than $N$ , we can expect potential advantage from the PQK.

Geometric separation is defined as follows (F19 in [2]): $g_{cq} = g(K^c \Vert K^q) = \sqrt{\Vert \sqrt{K^q} \sqrt{K^c} (K^c + \lambda I)^{-2} \sqrt{K^c} \sqrt{K^q}\Vert_{\infty}}$

# Gamma values used in best models above
gamma_c = grid_search_c.best_params_["gamma"]
gamma_q = grid_search_q.best_params_["gamma"]
 
# Regularization parameter used in the best classical model above
C_c = grid_search_c.best_params_["C"]
l_c = 1 / C_c
 
# Classical and quantum kernels used above
K_c = rbf_kernel(train_data, train_data, gamma=gamma_c)
K_q = rbf_kernel(projections_train, projections_train, gamma=gamma_q)
 
# Intermediate matrices in the equation
K_c_sqrt = sqrtm(K_c)
K_q_sqrt = sqrtm(K_q)
K_c_inv = inv(K_c + l_c * np.eye(K_c.shape[0]))
K_multiplication = (
    K_q_sqrt @ K_c_sqrt @ K_c_inv @ K_c_inv @ K_c_sqrt @ K_q_sqrt
)
 
# Geometric separation
norm = np.linalg.norm(K_multiplication, ord=np.inf)
g_cq = np.sqrt(norm)
print(
    f"Geometric separation between classical and quantum kernels is {g_cq:.4f}"
)
 
print(np.sqrt(len(train_data)))

Output:

Geometric separation between classical and quantum kernels is 1.5440
13.114877048604

Model complexity is defined as follows (M1 in [2]): $s_{K, \lambda}(N) = \sqrt{\frac{\lambda^2 \sum_{i=1}^N \sum_{j=1}^N (K+\lambda I)^{-2}_{ij} y_i y_j}{N}} + \sqrt{\frac{\sum_{i=1}^N \sum_{j=1}^N ((K+\lambda I)^{-1}K(K+\lambda I)^{-1})_{ij} y_i y_j}{N}}$

# Model complexity of the classical kernel
 
# Number of training data
N = len(train_data)
 
# Predicted labels
pred_labels = best_svc_c.predict(train_data)
pred_matrix = np.outer(pred_labels, pred_labels)
 
# Intermediate terms
K_c_inv = inv(K_c + l_c * np.eye(K_c.shape[0]))
 
# First term
first_sum = np.sum((K_c_inv @ K_c_inv) * pred_matrix)
first_term = l_c * np.sqrt(first_sum / N)
 
# Second term
second_sum = np.sum((K_c_inv @ K_c @ K_c_inv) * pred_matrix)
second_term = np.sqrt(second_sum / N)
 
# Model complexity
s_c = first_term + second_term
print(f"Classical model complexity is {s_c:.4f}")

Output:

Classical model complexity is 1.3578

# Model complexity of the projected quantum kernel
 
# Number of training data
N = len(projections_train)
 
# Predicted labels
pred_labels = best_svc_q.predict(projections_train)
pred_matrix = np.outer(pred_labels, pred_labels)
 
# Regularization parameter used in the best classical model above
C_q = grid_search_q.best_params_["C"]
l_q = 1 / C_q
 
# Intermediate terms
K_q_inv = inv(K_q + l_q * np.eye(K_q.shape[0]))
 
# First term
first_sum = np.sum((K_q_inv @ K_q_inv) * pred_matrix)
first_term = l_q * np.sqrt(first_sum / N)
 
# Second term
second_sum = np.sum((K_q_inv @ K_q @ K_q_inv) * pred_matrix)
second_term = np.sqrt(second_sum / N)
 
# Model complexity
s_q = first_term + second_term
print(f"Quantum model complexity is {s_q:.4f}")

Output:

Quantum model complexity is 1.5806

References

Utro, Filippo, et al. "Enhanced Prediction of CAR T-Cell Cytotoxicity with Quantum-Kernel Methods." arXiv preprint arXiv:2507.22710 (2025).
Huang, Hsin-Yuan, et al. "Power of data in quantum machine learning." Nature communications 12.1 (2021): 2631.
Daniels, Kyle G., et al. "Decoding CAR T cell phenotype using combinatorial signaling motif libraries and machine learning." Science 378.6625 (2022): 1194-1200.

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