Expectation Values from Circuits

Track: Quantum Gates & Circuits · Difficulty: Beginner · Est: 15 min

Expectation Values from Circuits

Overview

When you run a circuit, you typically get raw measurement samples (bitstrings). But many quantum workflows care about a different quantity:

an expectation value like $\langle Z \rangle$ or $\langle X \rangle$ .

This page addresses the practical problem:

how to turn raw samples into a meaningful numeric summary of the state.

Expectation values are central because they compress many samples into a single number that can be compared, optimized, or tracked across circuit variations.

Intuition

A measurement outcome is discrete (0 or 1). An expectation value is the average value of a quantity over many shots.

In classical probability, if you flip a biased coin and record 1 for heads and 0 for tails, then:

the average of many trials estimates the probability of heads.

In quantum measurement, we often want averages of observables like Z or X. These are “two-outcome measurements” with outcomes usually treated as $+1$ or $-1$ .

So the workflow is:

run the circuit many times,
map each measured bit to a number ( $+1$ or $-1$ ),
average.

That average is the estimated expectation value.

Formal Description

Raw samples vs expectation values

Raw samples: a list of measured outcomes, like 0, 1, 0, 0, 1.
Expectation value: a single number summarizing the average of a measurement.

Computing $\langle Z \rangle$ conceptually

For a single qubit, measuring in the computational basis corresponds to measuring the Z observable. We use the mapping:

outcome 0 corresponds to eigenvalue $+1$
outcome 1 corresponds to eigenvalue $-1$

If you take $N$ shots and obtain outcomes $b_1,\dots,b_N$ with $b_i\in\{0,1\}$ , define

z_i = \begin{cases} +1 & \text{if } b_i=0\\ -1 & \text{if } b_i=1 \end{cases}

Then the estimated expectation value is

\widehat{\langle Z \rangle} = \frac{1}{N}\sum_{i=1}^{N} z_i.

Interpretation:

if you see mostly 0 outcomes, $\langle Z \rangle$ is near +1,
if you see mostly 1 outcomes, $\langle Z \rangle$ is near −1,
if outcomes are balanced, it is near 0.

Computing $\langle X \rangle$ conceptually

To measure X, you measure in the X basis. A practical way to do this using standard computational-basis measurement is:

apply H before measurement,
then measure in the computational basis.

This works because H maps the X basis to the Z basis. So:

“measure X” can be implemented as “apply H, then measure Z.”

Then you use the same $+1/-1$ mapping and average.

We’re not doing heavy operator math here. The key idea is: change basis, then sample, then average.

Worked Example

Suppose you run a single-qubit circuit and measure in the computational basis. You take $N=8$ shots and get:

0, 0, 1, 0, 1, 0, 0, 0

Counts:

six 0 outcomes
two 1 outcomes

Convert to $+1/-1$ values:

each 0 contributes +1
each 1 contributes −1

So the sum is $6\cdot(+1) + 2\cdot(-1) = 4$ . The average is

\widehat{\langle Z \rangle} = 4/8 = 0.5.

Interpretation:

the circuit produced a state biased toward $|0\rangle$ in the Z basis,
but not perfectly; an expectation value of 0.5 is between +1 and 0.

Turtle Tip

To compute an expectation value from shots: map outcomes to numbers, then average. Histograms are great for intuition, but expectation values are often what algorithms actually consume.

Common Pitfalls

Don’t average raw bits (0/1) when you mean $\langle Z \rangle$ . Use the $+1/-1$ mapping.
Don’t forget basis: computational-basis measurement naturally gives Z information. To estimate X, you must effectively measure in the X basis (for example via a basis-change gate).

Quick Check

If all shots return 0, what is $\langle Z \rangle$ ?
Conceptually, how can you measure X using only computational-basis measurement?

Expectation values are one way quantum computation outputs a useful number. Next we discuss the broader picture: quantum computation is typically hybrid. After measurement, classical post-processing turns samples and expectation values into decisions, updates, and final results.

← Measurement Sampling & Shots Classical Post-Processing →