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I am wondering if it is possible to generalize the Hadamard test for computing $\text{Re} \langle \phi | U | \psi \rangle$ (different states for left and right operands).

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Yes, it is possible to adapt the Hadamard test to compute the real and imaginary parts of expectation values of the form $\langle \phi | U | \psi \rangle$, as shown in the figure below, where $|\psi \rangle \langle 0|$ corresponds to a unitary operation that prepares the state $|\psi \rangle$ starting from the fiducial state $|0\rangle$ and, as usual, the filled (empty) circle denotes a controlled-operation where the nontrivial action on the target-qubit(s) is triggered when the control-qubit is in state $|1\rangle$ ($|0\rangle$). As in the standard version of the Hadamard test, upon setting $b = 0$ ($b = 1$), the real (imaginary) part is given by $2P_0-1$, where $P_0$ is the probability of measuring the ancillary qubit in state $|0\rangle$. This is valid for an arbitrary number of qubits $n$ in the main register.

enter image description here

P.S.: As @gIS noted in a comment below before I made this update, this circuit does not compute $\textrm{Re}\langle\phi|U|\psi\rangle$ in the same sense that the SWAP test computes $|\langle\phi|\psi\rangle|^2$ or the original Hadamard test computes the real and imaginary parts of $\langle\psi|U|\psi\rangle$. Indeed, this adaptation of the Hadamard test does not take $|\psi\rangle$ and $|\phi\rangle$ as inputs, but rather encodes them via unitaries within the circuit itself. In fact, there cannot be a circuit taking $|\psi\rangle$ and $|\phi\rangle$ and giving as output the real part of $\langle\phi|U|\psi\rangle$, as that quantity would depend on the unphysical global phases of the ket states.

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  • $\begingroup$ Thanks! What is the gate $S^\dagger$? The $b$ parameter is litterally an exponent providing the identity for $b=0$ and $S^\dagger$ for $b=1$, right? $\endgroup$
    – francler
    Jan 25 at 16:00
  • $\begingroup$ $S$ usually represents the square root of the Pauli $Z$ operator. Its unitary matrix is $\begin{equation}\begin{pmatrix}1 & 0 \\ 0 & i \end{pmatrix}\end{equation}$ Therefore $S^\dagger$ is the conjugate of this $S$ gate namely $\begin{equation} S^\dagger = \begin{pmatrix}1 & 0 \\ 0 & -i \end{pmatrix}\end{equation}$ $\endgroup$
    – Callum
    Jan 25 at 16:51
  • $\begingroup$ You use $S^\dagger$ for the imaginary part. If you want the real part use $(S^\dagger)^{0} = I$. $\endgroup$
    – Callum
    Jan 25 at 18:13
  • $\begingroup$ Thanks! Concerning the controlled operation with the empty circle, can this be achieved by putting an X on the ancilla and then the standard $|1\rangle$ controlled gate right after? Apart from this, do you have any reference about that test? $\endgroup$
    – francler
    Jan 26 at 7:56
  • $\begingroup$ The gate with the empty circle is a $|0\rangle$ controlled gate. Yes you could use $X$ gates on either side of the gate on the control qubit of a standard "filled circle" gate. Some quantum libraries like qiskit and pytket support $|0\rangle$ controlled gates directly. $\endgroup$
    – Callum
    Jan 26 at 9:16

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