Level 7 · Doctoral & postdoctoral
Objective. Evaluate a kernel claim with its complete access cost.
Before this lesson. L7-01, L5-04
Hands-on lab. Qubits and the Bloch sphere
Learn¶
A feature map embeds an input x into a quantum state |phi(x)>. An overlap can define a positive semidefinite kernel. A classical learner can then use the kernel matrix. The quantum circuit supplies a similarity estimate rather than a causal interpretation of the data.
The total procedure includes data encoding, circuit execution, repeated measurements, kernel storage, optimization, and prediction. A short overlap circuit does not make arbitrary classical data loading free. The full cost depends on the input representation and the accuracy required for every estimated matrix entry.
A meaningful advantage claim compares strong classical methods with comparable data and resources. Hardness of a feature-map calculation alone does not establish learning advantage. Training examples can make prediction easier for classical methods. Report generalization and resource scaling separately.
Practice¶
Solution
The angle difference is -2pi/3, so the half-angle cosine squared is 1/4. Include data encoding and measurement repetitions. Also report classical training and kernel storage costs when they contribute.
Check your understanding¶
Does a hard feature calculation alone prove a learning advantage?
A. Yes, for any data set
B. Only when a kernel is normalized
C. No, the learning task and classical data access also matter
Show the answer
C. No, the learning task and classical data access also matter
The comparison concerns a complete learning task. Training data, approximation, and classical alternatives can change its difficulty relative to computing one feature exactly.
Sources¶
This lesson draws on Havlicek et al., 2019Huang et al., 2021.
Next: L7-04 · Barren plateaus and trainability
- Havlicek, V., Córcoles, A. D., Temme, K., Harrow, A. W., Kandala, A., Chow, J. M., & Gambetta, J. M. (2019). Supervised learning with quantum enhanced feature spaces. Nature, 567(7747), 209–212. 10.1038/s41586-019-0980-2
- Huang, H.-Y., Broughton, M., Mohseni, M., Babbush, R., Boixo, S., Neven, H., & McClean, J. R. (2021). Power of data in quantum machine learning. Nature Communications, 12(1). 10.1038/s41467-021-22539-9