Put on your favorite space hat and get ready to explore the nascent frontier of * quantum computation*! While

*still smacks of being futuristic science fiction magic, it's nevertheless beginning to enter into mainstream programming in the here and now.*

**quantum computation****IBM** has unveiled its cloud quantum computer for public use, **Microsoft** has revealed the **Q#** quantum programming language, and **D-Wave Systems** partnered with **NASA** to release the **D-Wave-2**.

Being the good futurologists that we are, we'll spend today's article exploring **Quantum Computation** libraries in Python and JavaScript in order to gain some increased familiarity with this powerful new technology.

## Classical Computation

**Classical bits** are represented as singular strings of

`0`

and `1`

(representing the Boolean values `false`

and `true`

). We often think of classical *bits*"one dimensionally" and they are treated as such in

**Lattice Theory**.

To summarize - for any **classical bit** B:

```
1. B = 1 or B = 0
2. B : X → {0,1}
```

We see that a **classical bit** can be thought of as the output of multiple operations on other preceding

*bits*(thinking

*algorithmically*).

Brief refresher: *Algorithms*, are finite, potentially recursive, **Boolean** operations on *bits* using familiar **Classical Logic** gates. **Classical computers** are really **Universal Turing Machines** - finite systems that implement and execute such *algorithms*!

While simple, these concepts are of course very powerful and underpin all present readily available computation.

Now, with all that in mind. Let's turn to **Quantum Computation**!

## Crash Course on Qubits and Quantum Computation

... And then throw it all out the window! Rather than think of **Quantum Computation** as directly analogous to **Classical Computation** in any way, it's better to think of them as two partly interconnected but simultaneously partly incommensurate approaches. How so?

Well, for starters, **Turing Machines** can implement virtual **Quantum Computers** (but not very well) and most Quantum programming langauges also support **Classical Computation** (such as **Q#**).

But, **Quantum Computing** mostly involves operations on the * superposition* of

*(*

**qubits***quantum bits*). A

*is expressed as a*

**qubit**`0`

or `1`

but here they represent respective *rather than*

**column vectors****Boolean**truth-values (and as such, they can have

**both**a value of

`0`

and `1`

simultaneously, while *entangled*in

*superposition*)!

We call the two * states* above (

`|0>`

and `|1>`

) the *basis states*from which all other

*are measured.*

**states****Quantum states**and

*, in general, are therefore defined by a string of*

**states***basis states*specifying the possible evolutions of those composed

*.*

**states*** Qubits* can be reasoned about in at least three ways:

### 1. The Bloch Sphere

Each * qubit* can be represented as a specific position on the surface of a sphere having a radius of

`1`

(thinking in *radians*). Technically, the surface of that sphere is a

**Complex Hilbert Space**! Indeed, the representation of a

*in this way follows as a result of the underlying*

**qubit****complex numbers**at play.

Here's a fantastic graphical representation that illuminates this a bit better:

Source: WikiMedia Commons.

See how `|0>`

and `|1>`

are located at the poles?

A helpful rule of thumb: **No** other * state* (e.g.

`|1111>`

or `|1001>`

) will be located at the poles of a **Bloch sphere**- only our

*basis states*!

### 2. Ket or Dirac Notation

As we have seen, * qubits* unlike

**classical**can be easily represented as positions on the surface of a sphere having a radius of

*bits*`1`

(*radians*-wise).

So we have seen how a * qubit* can, therefore, be represented by a sequence of

`1`

's and `0`

's specifying a location on the surface of that sphere.We thus arrive at *ket* (vector) notation which captures this intuitive notion:

Here, `|x>`

is called the * quantum state* and represents a

*linear combination*of

**probability amplitudes**and their respective

*in*

**states****superposition**.

So, given `|ψ> = a|0> + b|1>`

it follows that `|a|^2 + |b|^2 = 1`

where, `a`

and `b`

represent the *probability amplitudes* of their respective * states*.

*Ket* or *Dirac Notation* is the most common way of representing both * quantum states* and

*.*

**qubits**### 3. Via (column) Vectors

We can happily translate our *ket* notation into (column) *vector* notion like so:

Whereby each element in the *column vector* represents a **probability amplitude** of a respective * state*. Such notation is useful when performing linear operations via

**Quantum Gates**.

We easily see that operations on *qubits* are performed using a mix of *Matrix Multiplication*, *Complex Numbers*, and *Quantum Logic Operators* rather than the simple *truth-functional* operators of **Boolean Algebra**:

(*Matrix Multiplication*)

The best quantum computers that are presently publically available **can handle about 20 qubits at a time, and the most powerful quantum computers top off at around 50** simultaneously.

## Linear Operators (Operations on Qubits)

**Quantum Logic Gates** are fundamentally distinct from their **Classical Logic Gate** brethren, and we can see just how much so by exploring the most fundamental logical operation and its equivalent in both kinds of computational systems.

A **Boolean NOT** operation is a simple *truth-function* that *negates* or "reverses" the supplied inputted *truth* value (a `0`

returns a `1`

and a `1`

returns a `0`

):

**Quantum NOT** (the *Pauli-X linear operator* below) transforms one *basis state* into its corresponding, "opposite", pair (`|0>`

returns a `|1>`

and `|1>`

and `|0>`

) using *matrix multiplication* tables:

Thus, while the **Quantum NOT** operator (the *Pauli-X linear operator*) bears a superficial resemblance to the **Boolean NOT** operator, those superficial distinctions are immediately dispelled upon close inspection - we readily see how much more is going on with **Quantum Computation** for something as simple as even the most fundamental logical operation.

## Unitary Evolution

It's common knowledge that *observation* of the *wave-function* results in a single observed outcome for a * quantum state* in

**superposition**. Whether that's the product of

*observation*(measurement) itself,

*wave-function collapse*, or

*branching world-lines*(à la

**Many Worlds Interpretation**or

**Bohmian Mechanics**) is still hotly debated in the

**Foundations of Physics**and the Philosophy of Science.

For our intents and purposes, it suffices to note that *measurement* of a **quantum state** in **superposition** results in a resultant outcome (setting aside the fun and interesting philosophical quibbles).

**Unitary Evolution**, on the other hand, represents manipulation of the underlying * quantum system* prior to

*measurement*rather than the

*measurement*or collapse of the

*wave-function*to a specific observed

*. Essentially,*

**state****Unitary Evolution**allows us to transform an entangled

*in superposition prior to generating a final value.*

**quantum system**## Quantum Programming

Now that we have an idea of what's going on under-the-hood, let's take a look at two great **Quantum Programming** implementations: one in Python - the **qiskit-sdk** used in **IBM's** quantum cloud and one in JavaScript - **jsqubits** so we cover two of the most popular programming languages!

## QISKit-SDK

The new **qiskit-sdk** provided by **IBM** in Python allows for local and true cloud **Quantum Computing** (using IBM's shiny new Quantum Computer available to all)!We'll also use Python 3.6 to get us going!

Per the official docs, there are four parts to running a **QISKit** program:

1. Declaring registers and configuration

In this stage, we declare how many *qubits* and classical *bits* we'll be handling:

```
q = QuantumRegister(2)
c = ClassicalRegister(2)
qc = QuantumCircuit(q, c)
```

2. Putting the system into superposition

Then we can apply *linear operations* on our selected *qubits*:

```
qc.y(q[0])
```

3. Measuring the system

We can perform measurements on the *quantum* system returning and allowing access to a resultant * state* throughout:

```
qc.measure(q, c)
```

4. Running the program

A choice of backends is available to execute our program. We can run the program in the cloud or locally:

```
job_sim = execute(qc, "local_qasm_simulator")
sim_result = job_sim.result()
```

### Basic Operations

Let's apply our quantum knowledge to **Python**!

A simple *Pauli-X* operation can be implemented like so:

```
from qiskit import QISKitError, execute
from qiskit import QuantumCircuit, ClassicalRegister, QuantumRegister
if __name__ == '__main__':
try:
# Prepare program
q = QuantumRegister(2)
c = ClassicalRegister(2)
qc = QuantumCircuit(q, c)
# Put qubit in superposition
qc.x(q[0])
# Measure quantum state
qc.measure(q, c)
# Execute program
job_sim = execute(qc, "local_qasm_simulator")
sim_result = job_sim.result()
print("simulation: ", sim_result)
print(sim_result.get_counts(qc))
except QISKitError as ex:
print('Quantum fluctuations! Eh Gad!'.format(ex))
```

Let's do the same for *Hadamard*:

```
from qiskit import QISKitError, execute
from qiskit import QuantumCircuit, ClassicalRegister, QuantumRegister
if __name__ == '__main__':
try:
# Prepare program
q = QuantumRegister(2)
c = ClassicalRegister(2)
qc = QuantumCircuit(q, c)
# Put qubit in superposition
qc.h(q[0])
# Measure quantum state
qc.measure(q, c)
# Execute program
job_sim = execute(qc, "local_qasm_simulator")
sim_result = job_sim.result()
print("simulation: ", sim_result)
print(sim_result.get_counts(qc))
except QISKitError as ex:
print('Quantum fluctuations! Eh Gad!'.format(ex)
```

Now let's collapse a *wave-function* or two, shall we:

Oh yeah!

## jsqubits

**jsqubits** is another great library for learning how to program quantum-ly!

In this example, we'll combine some of the most basic operations along with **Angular 5** to build a lightweight quantum application!

### Basic Operations

We'll divide our **NG5** app into two views, one for each set of basic operations.

Our *injectable* `QuantumService`

will start with the three Pauli linear operators:

```
import {Injectable} from '@angular/core'
import {jsqubits} from 'jsqubits'
@Injectable()
export class QuantumService {
private QUBIT_ZERO = '|00>';
private QUBIT_ONE = '|01>';
private QUBIT_COMPLEX_ONE = '(i)|01>';
private QUBIT_MINUS_ONE = '-|01>';
constructor() {}
pauliX() {
let x = jsqubits(this.QUBIT_ONE).x(0);
return {
initial: this.QUBIT_ONE,
compute: x.toString(),
expected: this.QUBIT_ZERO
}
}
pauliY() {
let y = jsqubits(this.QUBIT_ZERO).y(0);
return {
initial: this.QUBIT_ONE,
compute: y.toString(),
expected: this.QUBIT_COMPLEX_ONE
}
}
pauliZ() {
let z = jsqubits(this.QUBIT_ONE).z(0);
return {
initial: this.QUBIT_ONE,
compute: z.toString(),
expected: this.QUBIT_MINUS_ONE
}
}
}
```

We can then inject those helpers directly into our `PauliComponent`

:

```
import {Component, OnInit} from "@angular/core";
import {QuantumService} from '../../services/quantum.service'
@Component({
//...
})
export class PauliComponent implements OnInit {
public pauliX;
public pauliY;
public pauliZ;
constructor(private _quantumService: QuantumService) { }
ngOnInit() { this.quantumGo(); }
quantumGo() {
this.pauliX = this._quantumService.pauliX();
this.pauliY = this._quantumService.pauliY();
this.pauliZ = this._quantumService.pauliZ();
console.log('Pauli linear operator tests done!');
}
}
```

Let's enrich our `QuantumService`

with two other basic operations - the *Hadamard* operator and a simple *Fourier Transformation* method:

```
import {Injectable} from '@angular/core'
import {jsqubits} from 'jsqubits'
@Injectable()
export class QuantumService {
private QUBIT_ZERO = '|00>';
private QUBIT_ONE_ZERO_ZERO = '|100>';
private HADAMARD_RESULT = '(0.7071)|00> + (0.7071)|01>';
private QFT_RESULT = '(0.5)|000> + (-0.5)|010> + (0.5)|100> + (-0.5)|110>';
constructor() { }
hadamard() {
let h = jsqubits(this.QUBIT_ZERO).hadamard(0);
return {
initial: this.QUBIT_ZERO,
compute: h.toString(),
expected: this.HADAMARD_RESULT
}
}
fourier() {
let f = jsqubits(this.QUBIT_ONE_ZERO_ZERO).qft([1,2]);
return {
initial: this.QUBIT_ONE_ZERO_ZERO,
compute: f.toString(),
expected: this.QFT_RESULT
}
}
}
```

Like before, we can easily inject our new `QuantumService`

helpers right into our `HadamardComponent`

:

```
import {Component, OnInit} from "@angular/core"
import {QuantumService} from "../../services/quantum.service";
@Component({
//...
})
export class HadamardComponent implements OnInit {
public hadamard;
public fourier;
constructor(private _quantumService: QuantumService) { }
ngOnInit() { this.quantumGo(); }
quantumGo() {
this.hadamard = this._quantumService.hadamard();
console.log('Hadamard test done!');
this.fourier = this._quantumService.fourier();
console.log('Fourier transform test done!');
}
}
```

## Conclusion

That concludes our quick look at **Quantum Computation** and the great libraries **qiskit-sdk** used in **IBM's** quantum cloud and the great JavaScript library **jsqubits**!

All the qubits and code used in this article can be clone from GitHub here!

Until next time!

## Shout Outs!

Thanks to these great photographers for their outstanding work:

For more quantum math check out: