Q1c. An analysis of the variational quantum classifier using data

In binary classification, let’s say labelling if someone is likely to have a heart attack or not, we would build a function that takes in the information about the patient and gives results aligned to the reality. E.g,

This probabilistic classification is well suited for quantum computing and we would like to build a quantum state that, when measured and post processed, returns

P(hear attack = YES)

By optimizing the circuits, you then find parameters that will give the closest probability to the reality based on training data.

Given a dataset about patients’ information, can we predict if the patient is likely to have a heart attack or not? This is a binary classification problem, with a real input vector {x} and a binary output {y} in {0, 1}. We want to build a quantum circuit whose output is the quantum state:

2. We use a feature map such as, ZZFeaturemap, ZFeaturemap or PauliFeaturemap and choose the number of qubits based on the input dimension of the data and how many repetitions (i.e. the circuit depth) we want. We use 1, 3, 5.

3. We choose the variational form as RealAmplitudes and specify the number of qubits as well as how many repetitions we want. We use 1, 2, 4 to have models with an increasing number of trainable parameters.

4. We then combine our feature map to the variational circuit. ZZfeaturemap and RealAmplitudes both with a depth of 1

5. We create a function that associates the parameters of the feature map with the data and the parameters of the variational circuit with the parameters passed. This is to ensure in Qiskit that the right variables in the circuit are associated with the right quantities.

6. We create another function that checks the parity of the bit string passed. If the parity is even, it returns a ‘yes’ label and if the parity is odd it returns a ‘no’ label. We chose this since we have 2 classes and parity checks either returns true or false for a given bitstring. There are also other methods e.g for 3 classes you might convert the bistring to a number and pass is through an activation function. Or perhaps interpret the expectation values of a circuit as probabilities. The important thing to note is that there are multiple ways to assign labels from the output of a quantum circuit and you need to justify why or how you do this. In our case, the parity idea was originally motivated in this very nice paper (https://arxiv.org/abs/1804.11326) and the details are contained therein.

7. Now we create a function that returns the probability distribution over the model classes. After measuring the quantum circuit multiple times (i.e. with multiple shots), we aggregate the probabilities associated with ‘yes’ and ‘no’ respectively, to get probabilities for each label.

8. Finally, we create a function that classifies our data. It takes in data and parameters. For every data point in the dataset we assign the parameters to the feature map and the parameters to the variational circuit. We then evolve our system and store the quantum circuit. We store the circuits so as to run them at once at the end. We measure each circuit and return the probabilities based on the bit string and class labels.

Data classification was performed by using the implemented version of VQC in IBM’s framework and executed on the provider simulator

Every combination of the experiments were executed with 1024 shots, using the implemented version of the optimizers. We conducted tests with different feature maps and depths, the RealAmplitudes variational form with differing depths and different optimizers in Qiskit. In each case, we compared the loss values after 50 training iterations on the training data. Our best model configs were

From the results, the ZFeatureMap with a depth of 2, RealAmplitudes variational form with a depth of 5 and the SPSA optimizer achieved the lowest cost. These results seem to indicate that the feature map which resulted in a lower cost function generally was the ZFeatureMap. But does this mean that the ZFeaturemap typically performs better in general?

Iris dataset

Wine dataset

This time, our best model configs are totally different! What’s fascinating about this is that the dataset used seems to demand a particular model structure. This makes sense intuitively right? Because the first step in these quantum machine learning models is to load the data and encode it into a quantum state. If we use different data, perhaps there is a different (or more optimal) data encoding strategy depending on the kind of data you have.

Another thing that surprised me, especially coming from a classical ML background, is the performance of the SPSA optimizer. I would have thought something more state-of-the-at, like ADAM, would be the clear winner. This was not the case at all. It would be cool to understand why SPSA seems to well suited for optimizing these quantum models.

A final remark is that we only looked at the loss values on training data. Ultimately we would like to also see if any of these quantum models are good at generalization. A model is said to have good generalization if it is capable of performing well on new data that it has never seen before. A proxy for this is usually the error we would get on test data. By taking the best configs here and checking their performance on test sets, we could gauge how well these toy models perform and generalize which would be pretty interesting even in these small examples!

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