In [1]:

```
from qiskit import *
from qiskit.visualization import plot_histogram
```

You now know something about bits, and about how our familiar digital computers work. All the complex variables, objects and data structures used in modern software are basically all just big piles of bits. Those of us who work on quantum computing call these *classical variables.* The computers that use them, like the one you are using to read this article, we call *classical computers*.

In quantum computers, our basic variable is the *qubit*: a quantum variant of the bit. These are quantum objects, obeying the laws of quantum mechanics. Unlike any classical variable, these cannot be represented by some number of classical bits. They are fundamentally different.

The purpose of this section is to give you your first taste of what a qubit is, and how they are unique. We'll do this in a way that requires essentially no math. This means leaving terms like 'superposition' and 'entanglement' until future sections, since it is difficult to properly convey their meaning without pointing at an equation.

Instead, we will use another well-known feature of quantum mechanics: the uncertainty principle.

The most common formulation of the uncertainty principle refers to the position and momentum of a particle: the more precisely its position is defined, the more uncertainty there is in its momentum, and vice-versa.

This is a common feature of quantum objects, though it need not always refer to position and momentum. There are many possible sets of parameters for different quantum objects, where certain knowledge of one means that our observations of the others will be completely random.

To see how the uncertainty principle affects qubits, we need to look at measurement. As we saw in the last section, this is the method by which we extract a bit from a qubit.

In [2]:

```
measure_z = QuantumCircuit(1,1)
measure_z.measure(0,0)
measure_z.draw(output='mpl')
```

Out[2]:

On the Circuit Composer, the same operation looks like this.

This version has a small ‘z’ written in the box that represents the operation. This hints at the fact that this kind of measurement is not the only one. In fact, it is only one of an infinite number of possible ways to extract a bit from a qubit. Specifically, it is known as a *z measurement*.

Another commonly used measurement is the *x measurement*. It can be performed using the following sequence of gates.

In [3]:

```
measure_x = QuantumCircuit(1,1)
measure_x.h(0)
measure_x.measure(0,0)
measure_x.draw(output='mpl')
```

Out[3]:

Later chapters will explain why this sequence of operations performs a new kind of measurement. For now, you'll need to trust us.

Like the position and momentum of a quantum particle, the z and x measurements of a qubit are governed by the uncertainty principle. Below we'll look at results from a few different circuits to see this effect in action.

The easiest way to see an example is to take a freshly initialized qubit.

In [4]:

```
qc_0 = QuantumCircuit(1)
qc_0.draw(output='mpl')
```

Out[4]:

Qubits are always initialized such that they are certain to give the result `0`

for a z measurement. The resulting histogram will therefore simply have a single column, showing the 100% probability of getting a `0`

.

In [5]:

```
qc = qc_0 + measure_z
print('Results for z measurement:')
counts = execute(qc,Aer.get_backend('qasm_simulator')).result().get_counts()
plot_histogram(counts)
```

Out[5]:

If we instead do an x measurement, the results will be completely random.

In [6]:

```
qc = qc_0 + measure_x
print('Results for x measurement:')
counts = execute(qc,Aer.get_backend('qasm_simulator')).result().get_counts()
plot_histogram(counts)
```

Out[6]:

Note that the reason why the results are not split exactly 50/50 here is because we take samples by repeating the circuit a finite number of times, and so there will always be statistical noise. In this case, the default of `shots=1024`

was used.

Now we'll try a different circuit. This has a single gate called a Hadamard, which we will learn more about in future sections.

In [7]:

```
qc_plus = QuantumCircuit(1)
qc_plus.h(0)
qc_plus.draw(output='mpl')
```

Out[7]:

To see what effect it has, let's first try the z measurement.

In [8]:

```
qc = qc_plus + measure_z
qc.draw()
print('Results for z measurement:')
counts = execute(qc,Aer.get_backend('qasm_simulator')).result().get_counts()
plot_histogram(counts)
```

Out[8]:

Here we see that it is the results of the z measurement that are random for this circuit.

Now let's see what happens for an x measurement.

In [9]:

```
qc = qc_plus + measure_x
print('Results for x measurement:')
counts = execute(qc,Aer.get_backend('qasm_simulator')).result().get_counts()
plot_histogram(counts)
```

Out[9]:

For the x measurement, it is certain that the output for this circuit is `0`

. The results here are therefore very different to what we saw for the empty circuit. The Hadamard has lead to an entirely opposite set of outcomes.

Using other circuits we can manipulate the results in different ways. Here is an example with an `ry`

gate.

In [10]:

```
qc_y = QuantumCircuit(1)
qc_y.ry( -3.14159/4,0)
qc_y.draw(output='mpl')
```

Out[10]:

We will learn more about `ry`

in future sections. For now, just notice the effect it has for the z and x measurements.

In [11]:

```
qc = qc_y + measure_z
print('Results for z measurement:')
counts = execute(qc,Aer.get_backend('qasm_simulator')).result().get_counts()
plot_histogram(counts)
```

Out[11]:

Here we have a case that we have not seen before. The z measurement is most likely to output `0`

, but it is not completely certain. A similar effect is seen below for the x measurement: it is most likely, but not certain, to output `1`

.

In [12]:

```
qc = qc_y + measure_x
print('\nResults for x measurement:')
counts = execute(qc,Aer.get_backend('qasm_simulator')).result().get_counts()
plot_histogram(counts)
```

Out[12]:

These results hint at an important principle: Qubits have a limited amount of certainty that they can hold. This ensures that, despite the different ways we can extract outputs from a qubit, it can only be used to store a single bit of information. In the case of the blank circuit, this certainty was dedicated entirely to the outcomes of z measurements. For the circuit with a single Hadamard, it was dedicated entirely to x measurements. In this case, it is shared between the two.

We have now played with some of the features of qubits, but we haven't done anything that couldn't be reproduced by a few bits and a random number generator. You can therefore be forgiven for thinking that quantum variables are just classical variables with some randomness bundled in.

This is essentially the claim made by Einstein, Podolsky and Rosen back in 1935. They objected to the uncertainty seen in quantum mechanics, and thought it meant that the theory was incomplete. They thought that a qubit should always know what output it would give for both kinds of measurement, and that it only seems random because some information is hidden from us. As Einstein said: God does not play dice with the universe.

No one spoke of qubits back then, and people hardly spoke of computers. But if we translate their arguments into modern language, they essentially claimed that qubits can indeed be described by some form of classical variable. They didn’t know how to do it, but they were sure it could be done. Then quantum mechanics could be replaced by a much nicer and more sensible theory.

It took until 1964 to show that they were wrong. J. S. Bell proved that quantum variables behaved in a way that was fundamentally unique. Since then, many new ways have been found to prove this, and extensive experiments have been done to show that this is exactly the way the universe works. We'll now consider a simple demonstration, using a variant of *Hardy’s paradox*.

For this we need two qubits, set up in such a way that their results are correlated. Specifically, we want to set them up such that we see the following properties.

- If z measurements are made on both qubits, they never both output
`0`

. - If an x measurement of one qubit outputs
`1`

, a z measurement of the other will output`0`

.

If we have qubits that satisfy these properties, what can we infer about the remaining case: an x measurement of both?

For example, let's think about the case where both qubits output `1`

for an x measurement. By applying property 2 we can deduce what the result would have been if we had made z measurements instead: We would have gotten an output of `0`

for both. However, this result is impossible according to property 1. We can therefore conclude that an output of `1`

for x measurements of both qubits must also be impossible.

The paragraph you just read contains all the math in this section. Don't feel bad if you need to read it a couple more times!

Now let's see what actually happens. Here is a circuit, composed of gates you will learn about in later sections. It prepares a pair of qubits that will satisfy the above properties.

In [13]:

```
qc_hardy = QuantumCircuit(2)
qc_hardy.ry(1.911,1)
qc_hardy.cx(1,0)
qc_hardy.ry(0.785,0)
qc_hardy.cx(1,0)
qc_hardy.ry(2.356,0)
qc_hardy.draw(output='mpl')
```

Out[13]:

Let's see it in action. First a z measurement of both qubits.

In [14]:

```
measurements = QuantumCircuit(2,2)
# z measurement on both qubits
measurements.measure(0,0)
measurements.measure(1,1)
qc = qc_hardy + measurements
print('\nResults for two z measurements:')
counts = execute(qc,Aer.get_backend('qasm_simulator')).result().get_counts()
plot_histogram(counts)
```

Out[14]:

The probability of `00`

is zero, and so these qubits do indeed satisfy property 1.

Next, let's see the results of an x measurement of one and a z measurement of the other.

In [15]:

```
measurements = QuantumCircuit(2,2)
# x measurement on qubit 0
measurements.h(0)
measurements.measure(0,0)
# z measurement on qubit 1
measurements.measure(1,1)
qc = qc_hardy + measurements
print('\nResults for two x measurement on qubit 0 and z measurement on qubit 1:')
counts = execute(qc,Aer.get_backend('qasm_simulator')).result().get_counts()
plot_histogram(counts)
```

Out[15]:

The probability of `11`

is zero. You'll see the same if you swap around the measurements. These qubits therefore also satisfy property 2.

Finally, let's look at an x measurement of both.

In [16]:

```
measurements = QuantumCircuit(2,2)
measurements.h(0)
measurements.measure(0,0)
measurements.h(1)
measurements.measure(1,1)
qc = qc_hardy + measurements
print('\nResults for two x measurement on both qubits:')
counts = execute(qc,Aer.get_backend('qasm_simulator')).result().get_counts()
plot_histogram(counts)
```

Out[16]:

We reasoned that, given properties 1 and 2, it would be impossible to get the output `11`

. From the results above, we see that our reasoning was not correct: one in every dozen results will have this 'impossible' result.

So where did we go wrong? Our mistake was in the following piece of reasoning.

By applying property 2 we can deduce what the result would have been if we had made z measurements instead

We used our knowledge of the x outputs to work out what the z outputs were. Once we’d done that, we assumed that we were certain about the value of both. More certain than the uncertainty principle allows us to be. And so we were wrong.

Our logic would be completely valid if we weren’t reasoning about quantum objects. If it was some non-quantum variable, that we initialized by some random process, the x and z outputs would indeed both be well defined. They would just be based on some pre-determined list of random numbers in our computer, or generated by some deterministic process. Then there would be no reason why we shouldn't use one to deduce the value of the other, and our reasoning would be perfectly valid. The restriction it predicts would apply, and it would be impossible for both x measurements to output `1`

.

But our qubits behave differently. The uncertainty of quantum mechanics allows qubits to dodge restrictions placed on classical variables. It allows them to do things that would otherwise be impossible. Indeed, this is the main thing to take away from this section:

A physical system in a definite state can still behave randomly.

This is the first of the key principles of the quantum world. It needs to become your new intuition, as it is what makes quantum systems different to classical systems. It's what makes quantum computers able to outperform classical computers. It leads to effects that allow programs made with quantum variables to solve problems in ways that those with normal variables cannot. But just because qubits don’t follow the same logic as normal computers, it doesn’t mean they defy logic entirely. They obey the definite rules laid out by quantum mechanics.

If you’d like to learn these rules, we’ll use the remainder of this chapter to guide you through them. We'll also show you how to express them using math. This will provide a foundation for later chapters, in which we'll explain various quantum algorithms and techniques.

In [ ]:

```
```