5 votes
Accepted

What is the "additive error" of Swap Test?

Additive error $\epsilon$ means that, if the true value of the expectation value we are attempting to measure is $\langle \hat{O} \rangle_{\textrm{True}}$, then our estimate $\langle \hat{O} \rangle_{\...
bm442's user avatar
  • 1,107
4 votes
Accepted

Can we test whether $|\psi\rangle$ is orthogonal to $|\phi\rangle$ without creating a coherent superposition therebetween?

No, you can't do it (except for trivial things). Think about what you're asking for: a map that performs $$ |0\rangle|\psi\rangle|\psi^\perp\rangle\longrightarrow |1\rangle|\psi\rangle|\psi^\perp\...
DaftWullie's user avatar
4 votes
Accepted

Alternatives to the swap test using a smaller number of qubits

Let us say that you want to evaluate $|\langle\psi|\varphi\rangle|^2$. Let us denote $U$ and $V$ the unitaires associated to your parametrized quantum circuits, producing respectively $|\psi\rangle$ ...
Tristan Nemoz's user avatar
  • 6,162
3 votes
Accepted

How to implement the state $|\psi\rangle = \frac{1}{\sqrt{2}}\left[|0\rangle \otimes |X_i\rangle + |1\rangle \otimes |X_j\rangle\right]$

Suppose that $U_i$ prepares the state $\left|X_i\right\rangle$. Start by applying an Hadamard gate on the first qubit: $$\frac{1}{\sqrt{2}}\left(|00\rangle+|10\rangle\right)$$ Then apply $U_j$ ...
Tristan Nemoz's user avatar
  • 6,162
2 votes
Accepted

How to compute the measurement probability in the Hadamard test?

In both cases you compute the probability by computing the square norm of the state after projection on the first qubit outcome you're considering. In the first example, you can rewrite your state as $...
glS's user avatar
  • 24.8k
2 votes

Alternatives to the swap test using a smaller number of qubits

The SWAP test is great when you don't have access to $U$ or $V$ and have no other way to prepare $|\psi\rangle$ or $|\phi\rangle$, and uses $2n+1$ qubits - while Tristan's test uses only $n$ qubits, ...
Mark Spinelli's user avatar
1 vote

How to apply error-correction to the swap test circuit?

How to design a quantum circuit that applies the quantum-error-correction method to handle the errors? Each qubit in the circuit has to be encoded in a quantum error correction code. You need to ...
Peter-Jan's user avatar
  • 1,529
1 vote
Accepted

perform a SWAP measurement using local operations and classical feedback

Measure SWAP from 7 CNOTs, linear connectivity, ancilla outside: (Note: improved this to not need any feedback anymore, and to just generally be cleaner.)
Craig Gidney's user avatar
  • 36.7k
1 vote

How do you test a pair of unknown qubits for orthogonality with certainty?

Though gIS' answer concerns deterministic measurements, its conclusion doesn't change when considering probabilistic measurements. Though it's not a formal proof, we can use ...
Tristan Nemoz's user avatar
  • 6,162

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