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Consider the following quantum circuit that consists of a three qubit quantum register and an ancilla qubit: enter image description here

Let $W$ and $V$ be unitary operators. $U(t)$ is an implementation of unitary time evolution. Followed by measurement in the computational basis, the two alternate paths on the end supposedly allow for extraction of the expectation of $\sigma_x$ and $\sigma_y$. So, consider $\sigma_x$. Suppose that I perform a measurement after applying the Hadamard. How do I obtain the expectation of $\sigma_x$ from that measurement? Namely, is there a way to do this using a function in qiskit?

Note: If it is helpful, this circuit is actually an ancilla qubit measurement algorithm to compute dynamic spin correlation functions. Namely, the expectation of $\sigma_x$ and $\sigma_y$ are proportional to the real and imaginary parts of the correlation function, respectively. See Section 3F. of the following paper:

https://arxiv.org/pdf/1907.03505.pdf

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2 Answers 2

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When measuring in qiskit you are always measuring in the $Z$ base ($|0\rangle,|1\rangle$). So you just need to change the base by inserting the proper gates before measurement. No need to call special functions.

Let me explain. If you want to get $\langle \sigma_z \rangle$, you know that you can get two results: +1 (associated to measuring $|0\rangle$) and $-1$ (associated to measuring $|1\rangle$). So if you measure $N$ times, where $n_0$ times gives $|0\rangle$ and $n_1$ times $|1\rangle$ then $$\langle \sigma_z \rangle=\frac{n_0(+1)+n_1(-1)}{N}=\frac{n_0-n_1}{N}$$.

The same applies to the eigenstates of $\sigma_x$ named $|+\rangle$ and $|-\rangle$:

$$\langle \sigma_x \rangle=\frac{n_+(+1)+n_-(-1)}{N}=\frac{n_+-n_-}{N}$$, where $n_\pm $ is the number of times that $|\pm\rangle$ is measured.

However as we cannot measure in the $|\pm\rangle$ basis, we need to change basis. One way to do that is to add a Hadamard gate at the end of the measurement. Notice that the Hadamard gate transforms states and operators related to $x$ into operators of $z$, most importantly $H|0\rangle=|+\rangle$ and $H|1\rangle=|-\rangle$. So if instead of measuring directly, you apply a Hadamard gate first and then measure so you are extracting the counts in the $|\pm\rangle$ basis (with the new $n_0$ for $n_+$, and $n_1$ for $n_-$). This works because $n_+=\langle +|\psi\rangle=\langle0|H|\psi\rangle$ and $n_-=\langle -|\psi\rangle=\langle 1|H|\psi\rangle$

For a similar gate for $\sigma_y$, you need a more complicated gate that is given by $HS^\dagger$ or $R_x(\pi/2)$ (be careful as these gates are not Hermitian). So in order to measure $\sigma_y$ you need to apply $HS^\dagger$ to your state before measuring.

This is well hinted already in the picture of the circuit.

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I don't know how efficient it is to do using qiskit, but you can easily find expectation values using 'tequila'.

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