Programming with Quantum Mechanics Evandro C. R. da Rosa QuantuloopClaudio Lima

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Programming with Quantum Mechanics

Evandro C. R. da Rosa

Quantuloop

Claudio Lima

Quantuloop

Abstract—Quantum computing is an emerging paradigm that

opens a new era for exponential computational speedup. Still,

quantum computers have yet to be ready for commercial use.

However, it is essential to train and qualify today the workforce

that will develop quantum acceleration solutions to get the

quantum advantage in the future. This tutorial gives a broad view

of quantum computing, abstracting most of the mathematical

formalism and proposing a hands-on with the quantum pro-

gramming language Ket. The target audience is undergraduate

and graduate students starting in quantum computing – no

prerequisites for following this tutorial.

Index Terms—quantum computation, quantum programming,

quantum simulation

CONTENTS

I Introduction 1

II Setup 2

III Quantum Bits 2

III-A Bloch Sphere . . . . . . . . . . . . . . 2

III-B Qubits in Ket . . . . . . . . . . . . . . 3

IV Quantum Gates 3

IV-A Pauli Gates . . . . . . . . . . . . . . . . 4

IV-B Rotation Gates . . . . . . . . . . . . . . 4

IV-C Hadamard Gate . . . . . . . . . . . . . 4

IV-D Phase Gate . . . . . . . . . . . . . . . . 4

IV-E Entanglement and Controlled Gates . . 4

IV-F Inverse Quantum Gate . . . . . . . . . . 5

V Quantum Measurement 6

V-A Ket Runtime Architecture . . . . . . . . 6

V-B Measuring Entangled Qubits . . . . . . 7

VI Quantum Computer vs. Simulator 7

VII Final Considerations 8

References 8

I. INTRODUCTION

Quantum computing opens up new ways for processing

information, solving in a few minutes problems that would

take decades for a classical computer to ﬁnd the solution. This

claim was known in theory for a long time. Now we begin to

see the ﬁrst quantum computers that outperform classical ones.

This tutorial walks through some quantum mechanics concepts

with the eyes of quantum computing, demonstrating how to

use them to program a quantum application.

The idea of quantum computers dates back to the 80s when

Richard Feynman proposed a quantum computer as a universal

quantum simulator. His motivation was the difﬁculty of simu-

lating quantum systems in classical computers, a problem with

a time complexity that grows exponentially with the number of

variables. The proof that quantum computers can signiﬁcantly

increase processing power came in the late 90s with the work

of Peter Shor. His integer factorization algorithm, known as

Shor’s algorithm, reveals how to solve a problem that tokens

exponential time in a classical computer in polynomial time

with the help of a quantum computer.

Shor’s algorithm boosted the development of quantum com-

puters and uplifted the creation of post-quantum cryptography.

A research ﬁeld to ﬁnd quantum-resistant alternatives for

today’s standard public-key cryptography algorithms since

Shor’s algorithm breaks all of them. While this may sound

alarming, the industry still lacks powerful quantum computers

to break a standard cryptography scheme. Moreover, NIST

is working to standardize new quantum-resistant asymmetric

encryption algorithms.

Quantum computers can speed up several processes, in-

cluding but not limited to optimization, logistics, machine

learning, and quantum chemistry simulation. However, we

are in the Noisy Intermediate-Scale Quantum (NISQ) era,

with quantum computers featuring few qubits very susceptible

to noise that limits the complexity of quantum executions.

Although, we went further than was expected 20 years ago,

even reaching the quantum advantage milestone with quantum

computers outperforming classical ones for some tasks. The

task, in this case, is not solving any real-world problem. It

is just a trial projected speciﬁcally for the quantum advantage

demonstration. However, we are not far from large-scale fault-

tolerant quantum computers, being on the roadmap of many

companies to deliver them by the end of the decade (until

2030).

Quantum engineers are a highly in-demand workforce today,

even though we have not yet unlocked the full potential of

quantum computing. We expect this demand to grow in the

foreseeable future. A new domain that comes with the growth

of quantum technologies is the quantum developer, profes-

sionals who adapt solutions to take advantage of quantum

computers and programming quantum applications.

Adapting and developing quantum algorithms is not a

straightforward process. Although, quantum programming is

not as hard as one may think. It is much like classical pro-

gramming. This tutorial will discuss the main characteristics

of quantum computing, demonstrating how to express it in the

arXiv:2210.15506v1 [quant-ph] 27 Oct 2022

quantum programming language Ket. We will present a broad

view of the topic, which may help one take the ﬁrst steps in

quantum development.

Ket is an embedded programming language that introduces

the ease of Python to quantum programming, letting anyone

quickly prototype and test quantum applications. We expect

the reader following this tutorial to be familiar with Python

programming and Jupyter notebook. We will introduce quan-

tum programming focusing on digital qubit-based quantum

computing, a general-purpose quantum computation model

used by several companies, including IBM, Google, and

Microsoft.

II. SETUP

We will present some examples that one can try on their

computer. There are several ways one can execute a Ket

program. Regardless, using Google Colab is the most straight-

forward approach, allowing one to create and run Ket applica-

tions on any computer quickly. However, feel free to use any

method. Get started by going to the Colab website, creating a

new notebook, and running the command presented in Listing

1. See Ket’s project website1for more ways to get Ket. Try

running the example from Listing 2 to test if it works. It should

output similarly as Fig. 1.

!pip install ket-lang qutip -q

from ket import *

Listing 1: Installing and importing Ket on a Jupyter notebook.

Figure 1: Running Ket on Google Colab.

Listing 2 allocates two qubits, uses a Hadamard gate to

create superposition, a controlled gate to entangle both qubits,

and prints the quantum state on the screen. Do not worry if

some of those terms seem unfamiliar; this tutorial will explain

all those concepts, presenting how to use them in quantum

programming.

1https://quantumket.org

1>>> from ket import *

2>>> a,b=quant(2)

3>>> H(a)

4>>> cnot(a, b)

5>>> print(dump(a+b).show())

6|00〉(50.00%)

70.707107 ∼

=1/√2

8|11〉(50.00%)

90.707107 ∼

=1/√2

Listing 2: Preparing and printing a Bell state.

III. QUANTUM BITS

Quantum computers store information in quantum bits,

aka qubits, the quantum analog for the classical bit. As its

classical counterpart, a qubit has two possible values, 0 and

1, or using the Dirac notation, |0iand |1i, known as the

computational basis. A qubit can store classical or quantum

information since it can simultaneously be in one or both

states, a phenomenon known as quantum superposition. As

for classical bits, the number of states in a set of bits grows

exponentially with the number of bits, e.g., with 32-bits, one

can express a single unsigned integer between 0 and 232 −1,

which is also true for qubits. However, with 32-qubits, one

can simultaneously represent all unsigned integers between 0

and 232 −1. The amount of memory a quantum computer

have grows exponentially with the number of qubits, which

is equivalent to saying that the number of bits needed to

represent a qubit set grows exponentially with the set length.

Fig. 2 geometrically illustrate this exponential memory growth,

displaying how much data the same number of qubits and bits

can store.

One can represent a qubit in superposition as a linear combi-

nation of the computational basis states, with an arbitrary qubit

|ψideﬁned as α|0i+β|1i. The numbers αand β, known as

the probability amplitude, weigh the probability of measuring a

basis state. We will talk more about measurement later. Those

are complex numbers with the restriction of |α|2+|β|2= 1.

Generalizing for more qubits, one can represent n-qubits as

Pkαk|ki, where Pk|α|2= 1.

Quantum mechanics limits how one can modify the state

of qubits. For instance, every quantum operation must be

reversible except for measurement, meaning there is no loss

of information during quantum computation. Reversibility is

not necessary for classical computers. We will discuss this in

the next section.

Another limitation of quantum computers is the impos-

sibility of copying a quantum state. Nonetheless, quantum

information explores this characteristic to offer communication

protocols physically secured against eavesdropping.

A. Bloch Sphere

We can picture every possible single-qubit state in the

Bloch sphere. This geometric representation helps visualize

the quantum superposition and single-qubit operation. In the

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ProgrammingwithQuantumMechanicsEvandroC.R.daRosaQuantuloopClaudioLimaQuantuloopAbstractQuantumcomputingisanemergingparadigmthatopensaneweraforexponentialcomputationalspeedup.Still,quantumcomputershaveyettobereadyforcommercialuse.However,itisessentialtotrainandqualifytodaytheworkforcethatwilldevelop...

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