Kernel estimation in QML workflows

Advanced Quantum Machine Learning

Created by Pavel · 17.03.2026 at 07:05 UTC

Once a feature map is fixed, the kernel is an operational quantity: overlaps, fidelities, and shot budgets decide how precisely you fill a Gram matrix. The adjoint trick rewrites kernel entries as survival probability of an all-zero string after inverse encoding, and SWAP-test statistics link to fidelity.

Finite shots add variance; hardware noise biases overlaps. Reporting a kernel value without uncertainty is as risky as reporting a mean without error bars.

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Tasks

Question 1

Which identity underlies the adjoint-trick estimate of the quantum kernel?

Hint

Rewrite the overlap between encoded states using the reference state.

$K_Q(x,y)=\|x-y\|^2$

$K_Q(x,y)=|\langle 0|U_\phi(x)^\dagger U_\phi(y)|0\rangle|^2$

$K_Q(x,y)=\mathrm{Tr}(U_\phi(x)U_\phi(y))$

$K_Q(x,y)=\langle 0|U_\phi(x)U_\phi(y)|0\rangle$

Question 2

After preparing $U_\phi(x)^\dagger U_\phi(y)|0\ldots 0\rangle$, what does the probability of measuring $0\ldots 0$ equal in the ideal setting?

Hint

Use the Born rule.

The Euclidean distance between $x$ and $y$

The number of trainable parameters

The quantum kernel $K_Q(x,y)$

Always $1/2$

Question 3

In the SWAP test for pure states, if $P(\mathrm{anc}=0)=0.9$, what is the fidelity $F$?

Hint

Use $P(\mathrm{anc}=0)=(1+F)/2$.

$0.4$

$0.8$

$0.9$

$1.8$

Question 4

For $N$ samples, how many pairwise kernel evaluations are needed to fill the upper triangle, including the diagonal, of a symmetric Gram matrix?

Hint

Count pairs with $i \leq j$.

$N$

$N^2$

$\frac{N(N-1)}{2}$

$\frac{N(N+1)}{2}$

Question 5

Implement fidelity_from_swap_p0(p0: float) -> float given $p_0 = (1+F)/2$ for pure states. Clamp logically: if p0 not in $[0,1]$, raise ValueError. Return $F=2p_0-1$.

Hint

Rearrange the SWAP-test identity.

def fidelity_from_swap_p0(p0: float) -> float:
    # TODO: validate range, return 2*p0 - 1
    pass

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Card Info

Topic: Quantum Machine Learning
Difficulty: Advanced
Completed: 0 users

Creator

Pavel