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Preface – This post is part of the Quantum Computing series.

Introduction

What comes to your mind when someone asks you “what do you find mysterious?” Well, there are a million answers that one can think of. For example, you may find someone’s behavior or actions strange, you may think of the Bermuda Triangle, or those paranormal activities may catch your attention, or what about the time? Does it really exist? How does it work? Truly, there are so many things that puzzle us most of the time. Most of us believe that nothing is more peculiar than the universe itself! This ginormous place of galaxies, black holes, and planets,… (you name it, we include it, and this list will just go on like that) has so much to offer us, but at the same time, it leaves us with some mind-boggling questions that are really fascinating. However, it throws us into a never-ending, exhausting journey. If we look at the bigger picture, then we can say that nature is weird, and it comprises every damn living or non-living thing, an eco-system, etc., and the universe is just a part of it! What if someone tells you that science has all the answers? Then it will be a eureka moment for us. It may seem easy at first but we have to go through a lot of conundrums to solve the mystery. One of the most bizarre topics that we are going to talk about is quantum physics, the mind-blowing theory of modern physics which may help us find some interesting facts about nature.

Imagine you are driving a car. While driving, there are several parameters that you consider. For example, what should be the speed? What about the velocity? When you are going to press the accelerator, should the speed be slow or fast? How far you have come, when you are going to stop, etc. All these parameters can be solved using classical mechanics. This is all about the things that are visible and sensed by our brain. By using the sophisticated system of math and physics, we can measure certain quantities. Among all these, the most fundamental entity we can think of is an atom. Does it have any connection with quantum mechanics? Is it even possible to talk about an atom that cannot be seen with the naked eye? If we get into the nitty-gritty, it will leave us awestruck with some really difficult ideas about different subatomic particles. Wait a minute! Why is it so herculean to comprehend this subtle idea of quantum mechanics? Well, the spectrum of human perception is not big enough to perceive the spooky actions that happen at such a minute level.

What is Quantum Mechanics exactly?

Quantum means discrete quantity i.e. the smallest bit of something. For instance, a single water molecule from the Pacific Ocean or a speck of dust. Mechanics means the behavior and motion of something. So this term together stands for the interaction and motion of individual objects of a bigger quantity. But what if we want to study the constituent particles that are smaller than an atom? To predict the activities that are going on at the subatomic scale, we need a wave function which is typically a mathematical function. For example, using a wave function it is possible to find the location of a particle. This is just a small part of this tiny realm about what it truly means. To understand the underpinnings of quantum strangeness let’s talk about the double-slit experiment. Which was performed by one of the greatest scientists of the 18^th century Thomas young.

Consider two scenarios: 1. Classical case. 2. Quantum Case

Case 1: Classical case

Think of an experimental setup shown in the given figure.

• Imagine you have a source from which marbles are coming out.
• There are two slits on the slide and a screen is kept at a distance d from the slide to observe the pattern.
• These marbles pass through slits and hit the screen. You can imagine the bands much like slits appeared on the screen.

Marble can be thought as a particle. Now replace the source of marble with a source of light. Light behaves like a wave. The pattern of dark and light fringes can be shown as following. There will be an interference: Constructive (Bright) and Destructive (light).

Image source: Google

 

This is the real picture of interference. Here Diffraction also plays a role. As of now, we won’t get into the nitty gritty of it.

Case 2: Quantum Case

Think of a beam consists of electrons. As we all are aware, electrons are particles. Block one slit and let electrons pass through another slit. The pattern will be as similar to the pattern made with marbles (a single band). However, when they pass through double slits, they must behave like marbles as per our understanding (figure 1, two bands). However, they exactly behave like a wave and create the interference pattern.

Here is what is more interesting, in 2^nd case if we choose to shoot electrons one at a time, there will be a single spot on the screen and no chance of interference. However, after some time the entire pattern is similar to the pattern of a wave. You may think, how it can be possible! Does the electron divide itself into two particles initially and later interfere with each other after passing through slits to hit the screen? Which slit an electron goes through? To observe this carefully, one idea is to place a detector by slits and watch over the electrons! It may sound funny but when we do that they behave like marbles and create the two bands’ pattern. Do electrons really know which slit they are going to pass through? This arises the question does an electron behave like a particle or wave? which leads to a wonderful phenomenon wave-particle duality.

Stern Gerlach Experiment: PDF

Postulates of Quantum mechanics: Link

Applications of Quantum Mechanics

(Kindly click the link to learn more)

• MRI scanners for medical imaging
• Lasers
• Solar cells
• Electron microscopes
• Atomic clocks used for GPS

Other areas: where Quantum Mechanics dominates – Machine learning, communication, finance, optimization, chemistry, astrophysics, etc.

These are not the actual cases if we choose to ignore the main part. For example, degenerate and non-degenerate time-independent perturbation theory, the semi-classical WKB approximation, time-dependent perturbation theory, the adiabatic approximation, and scattering theory.

Preface – This post is part of the Quantum Computing series.

Introduction

In the previous module, we discussed some of the most important concepts in Quantum Mechanics and how they are being utilized to design a functional Quantum Computer. In this module, we will discuss the basic architecture of a Quantum Computer.

The architecture of a Quantum Computer

Image Source: Fu, Xiang et al. “A heterogeneous quantum computer architecture.” Proceedings of the ACM International Conference on Computing Frontiers (2016): n. pag.

Layers of Quantum Computer

As shown in the figure above, there are total of seven layers that can be divided into five main categories as follows:

1. Qubits

The two bottom-most layers implement the main hardware of the Quantum Computer. The bottom-most layer is the Quantum Chip which contains the actual arrangement of qubits. Right above the quantum chip, there exists a Quantum-Classical Interface. The primary function of the Quantum Classical Interface is to make the measurement. Classical bits are required to make measurements on a qubit.

2. Compilers

A compiler is mainly a translator. It translates a high-level language program into the corresponding machine code. Quantum Computers also need Compilers to translate high-level instructions into corresponding low-level instructions that can be executed on the hardware level.

The Compiler block shown in the figure consists of two main parts: a Conventional Host Compiler and a Quantum Accelerator Compiler.

The Conventional Host Compiler compiles the classical logic and the Quantum Accelerator Compiler produces the quantum circuits. The quantum compiler will perform quantum gate decomposition, reversible circuit design, and circuit mapping and translates the logical quantum operations into a series of physical operations.

The third dimension in the figure represents the error correction schemes that drive the logical to the physical transformation of the quantum instructions.

3. Quantum Instruction Set Architecture (QIST)

QIST is the main divide between Quantum Hardware and Software. The Quantum Instruction set provides the programmer with the basic logical instruction set to facilitate the designing of quantum algorithms. The Instruction set also allows the programmer to go with various encoding schemes and provides error correction functionality.

The block right below QIST is divided into two parts:

1. QEC

QEC refers to Quantum Error Correction. Quantum Computers are error-prone, so some error correction schemes must be implemented to make sure the programs on the computers give accurate results. The First Error Correction Codes for Quantum Computers was given by Peter Shor called Shor codes. Since then, various error-correcting codes have been developed making QED an active area of research in the Quantum Computing Community.

2. QEX

QEX refers to Quantum Execution. QEX executes the instructions generated by the Compiler.

4. Software

The Programming Paradigm and languages block represent the high-level interface used by the programmer or algorithm designer to program a Quantum Computer. Some famous programming frameworks to program Quantum Computers are Qiskit, Cirq, Q#, and many more.

5. Quantum Algorithm

Quantum Algorithm is the topmost layer in the Quantum Architecture Stack. Quantum Algorithms are designed by exploiting the Quantum-Mechanical properties of the hardware layers. The high-level programming languages and compilers must be developed such that the algorithms can easily exploit the underlying quantum hardware.

Information Source: Fu, Xiang et al. “A heterogeneous quantum computer architecture.” Proceedings of the ACM International Conference on Computing Frontiers (2016): n. pag.

 

Preface – This post is part of the Quantum Computing series.

Introduction

To recap, in the last module, we explored the inner workings of a Quantum Computer by mathematically defining the notion of a Quantum State. We introduced the mathematical notation used to define quantum states and, in the end, discussed the design and operating principles of a D-Wave Quantum Computer.

In this module, we will provide rigorous definitions and corresponding intuition of some technical terms we have encountered in the previous modules.

Qubit

Qubit is short for Quantum bit. A qubit is a fundamental unit of Quantum Computation. The book Quantum Computation and Quantum Information by Neilson and Chung -which is as close to a standardized textbook on Quantum Computing as we’ve got so far- defines a qubit as an abstract, mathematical object. Although – as some of you may recall from previous modules – a qubit is a physical entity, defining it as an abstract, mathematical object will yield to us the benefit of describing it as a generalized object, independent of its specific, physical method of implementation.

A qubit is the fundamental unit of quantum information. A qubit, just like a classical bit has a state. The state of a qubit is described using Dirac’s notation (“|  ⟩”) which is the standardized notation used to describe quantum states. A qubit can be in states |0⟩, |1⟩, or a linear combination of states |0⟩ and |1⟩ and as shown below:

|ψ⟩=α|0⟩+β|1⟩

Here,α and β are complex numbers describing probability amplitudes of the qubit being in the |0⟩ or |1⟩ respectively. |0⟩ and |1⟩ are the orthonormal basis vectors in a two-dimensional vector space. Thus, a qubit can be described as a unit vector in a two-dimensional vector space.

Using vector notation, the above equation can be written as:

Superposition

Superposition is one of the fundamental principles of Quantum Mechanics. The wave-particle duality describes that an electron behaves as both a wave and a particle, but what does it actually mean? How can something behave as both a wave and a particle? Classically, we know that an object can either be discrete or continuous. But how can it be both? Let’s find out!

The Double Slit Experiment

Thomas Young in 1802 designed an experiment to demonstrate the wave nature of light. The Double Slit experiment was designed such that a light source randomly launches photons at a barrier containing two slits as shown in figure.

————–

Image Source: 老陳, CC BY-SA 4.0, via Wikimedia Commons

The screen behind the barrier should record the pattern which if the photons are assumed to be particles should show something like two vertical lines of light passing through the barrier.

Instead, the screen records an interference pattern as shown below.

Interference patterns are generally generated when waves constructively and destructively interfere with each other. So, although Mr. Young succeeded in demonstrating the wave nature of light no one really understood what was really going on until the theory of Quantum Mechanics was formalized.

So, what on Earth is really going on? Well, the thing is no one really understands what is going on. The path of the photons, the slit they go through and the path they take is not exactly traceable. As we know, measuring a quantum system collapses the superposition, so we cannot detect exactly how photons bring about an interference pattern. But we do know that there must be something fundamentally wave-like in the nature of photons to bring about the interference pattern.

Superposition

Later, it was discovered that after going into the slit, the photon exists in superposition. Think of it like a dense cloud, the photon can exist anywhere within the cloud with each position corresponding to a probability amplitude. This is Superposition.

Superposition is defined as the ability of a quantum system to be in a combination of multiple states at the same instance.

Measurement

Measurement collapses the superposition. When a quantum system is measured, the superposition collapses and the system comes down to one of its basis states.

Two-level superposition

Four level superposition

Entanglement

Entanglement is the phenomenon associated with quantum systems where two systems are correlated to each other such that the measurement or manipulation of one system will affect the other immediately even when separated by vast distances.

To understand Entanglement better, let’s take an example. Let’s assume we have two balls that are entangled to each other. The balls are in a superposition of black and white before observation, once observed a ball can either be black or white. Let’s say we have two observers Alice and Bob each observing one ball. The balls are entangled so when Alice observes her ball and sees a black ball Bob will immediately see his ball turn white or vice a versa. (Alice and bob can both see the ball turn the same color as well, it depends upon how the objects are entangled).

Albert Einstein never could reconcile with the theory of entanglement. He referred to entanglement as “spooky”. And it was a perfectly valid reaction as scientists did not know how exactly entanglement worked at the time. How can an object pass information to another object separated by a distance as large as two different galaxies almost immediately? This is only possible if the objects are communicating faster than the speed of light which completely breaks Einstein’s theory of special relativity. This is a common misconception that was later proved false after conducting various experiments. Even Quantum Physics cannot achieve faster than light communication.

Teleportation

Quantum Teleportation is the process of sending quantum information from a sender to a receiver. Teleportation is achieved through entanglement, the sender and receiver’s qubits are entangled in Bell States. The Bell State is described as follows in a two-qubit system:

Here, in |00⟩ and |11⟩ the first 0/1 represents the first qubit and the second 0/1 represents the second qubit. Both the states are in equal superposition with probability amplitude :

 

Teleportation sends classical information across the channel so it cannot achieve faster than light speed as we discussed in the previous section.

Preface – This post is part of the Quantum Computing series.

Introduction

So far, we have discussed how Quantum Computing in contrast to Classical Computing. In this module and upcoming modules, we will start to discuss the inner workings of a Quantum Computer. In this module, we will discuss Quantum Processors Units or QPUs and their architectures. In the end, we will demonstrate the inner workings of the D-Wave Quantum Computer. We have much to discuss so, let’s skedaddle!

What is a Quantum Processing Unit?

A Qubit is the fundamental part of a QPU. All the operations in a QPU are performed on Qubits. Now, we would like to point out at the very beginning that many architectures exist to arrange Qubits on Quantum Processors and all of them have their advantages and drawbacks. As of now, no architecture is proven to be better in terms of efficiency for all kinds of problems. Some promising contending architectures include Superconducting, Photonic, and Trapped ion. Now, we realize these names may sound daunting but just keep a lookout for these in the literature until we eventually discuss each architecture in more detail.

What is a Quantum Computer?

In the previous sections and modules, we talked about Quantum Computers very abstractly. There were no rigorous proofs or mathematical formulas to actually give you a technical context into the inner workings of Quantum Computers. Here, we will not go all maniac on you with mathematical formulas but we will give you enough to demonstrate our point more clearly. Don’t worry if you don’t understand all of it at once. We advise you to go through this section and do some additional readings we provide at the end of this module.

We will start by defining and representing a Quantum State mathematically. A Quantum State is represented using a vector. The vector represents the probability of the qubit being in that state. A two-qubit system can be represented as shown below.

|0⟩=1 0         |1⟩=0 1

The Quantum States are described using the bra-ket notation. We will discuss it in detail, for now “| ” also known as “ket” is used to specify that 0 and 1 are quantum states.

The vectors representing the quantum states describe the probability of the qubit being in that state. This is a very powerful idea and should be digested as we go further into Quantum Mechanics. Here, the first vector (1 0) describes that the qubit is in the |0⟩ state with probability 1 and |1⟩ state with probability 0. We index the vector starting with 0 and go further along up to the total number of qubits in the system.

Similarly, the second vector (0 1) describes that the qubit is in the |0⟩ state with probability 0 and |1⟩ state with probability 1.

Representation of a Quantum State

A pure Quantum state is a superposition of classical states written as,

Where, is a complex number representing the probability amplitude of each state in |.

Mathematically, the states |0⟩, . . . , |N-1⟩ form an orthonormal basis of an N-dimensional Hilbert space.

A quantum state |∅⟩ is a vector in this space, usually written as an N-dimensional column vector of its amplitudes:

Example of a Quantum Computer: D-Wave

The picture above shows a D-Wave Quantum Computer. Most of the apparatus you see is used to keep the Quantum Processor situated at the bottom at sub-zero temperature. The Qubits get noisy if the temperature is not maintained which results in Quantum Decoherence. Quantum Decoherence results in errors, noise, and sometimes even loss of information in qubits.

The D-Wave Quantum System is implemented superconducting qubits. There are two models based on which Quantum Computers can be built:

• Quantum Gate Model: In this model, problems are expressed in terms of quantum gates. Companies like Microsoft, IBM, Google, etc. use the Gate Model in their Quantum Computers.
• Quantum Annealing Model: In this model, problems are expressed in terms of optimization problems from which the computer then finds the optimal solution. D-Wave systems use Quantum Annealing to search for solutions in state space.

D-Wave has commercialized its quantum services through its cloud platform Leap. With their latest QPU (Quantum Processing Unit) containing over 5000 qubits, D-Wave is indeed a major player in the Quantum arena.

With this, we conclude this module. In the next module, we will discuss important Quantum Phenomena like Superposition, Entanglement, Interference, and teleportation in detail.

 

Preface – This post is part of the Quantum Computing series.

Introduction

In the previous module, we discussed classical computing in detail. In this module, we will discuss the key differences between classical and quantum computing. This is perhaps the last module before we dive deep into the nitty-gritty details of quantum computing. So, without further ado let’s get started, shall we?

Classical Computing

Once again, Classical or Modern Computers are made up of fundamental units called “bits”. A single bit can take a binary state i.e., 0 or 1, high or low. A bit can be in a single state at a time. It can either be 0 or 1, either at high or low voltage, and nothing in between.

Quantum Computing

Quantum computers on the other hand, as we discussed before are fundamentally different from Classical Computers. Fundamental units of Quantum Computers are called Quantum bits or “Qubits”. Unlike “bits”, “qubits” can be in multiple states at the same time. But before talking about the nature of qubits, let’s talk about what exactly they are made of.

Transistors that make up classical processors are basically silicone (NPN/PNP) semiconductors. The transistor is designed to represent the high or low voltage; the principle as we discussed above, on which modern computing is tethered. An interesting challenge presented itself when physicists began thinking about what should be used to represent a Qubit.

It was quite obvious that it should be a Quantum material, but basically, everything is a Quantum Material when it is cooled below a certain threshold temperature. Photons, electrons, superconducting materials, etc behave Quantum Mechanically. So, where to start?

Well, the good news is there are several different effective Quantum Computers that use different materials as qubits. Their effectiveness and efficiency vary according to various circumstances. We will discuss this further in the upcoming modules. Just remember that there is not a single material like the transistor which resides in different Quantum processors. Different groups and companies have different materials representing Qubits and which method is more effective than another is a much-debated topic in the Quantum Computing Community.

For now, just keep in mind that a qubit can either be in state 1, state 0, or a combination of states 1 and 0. And this is because our Qubits are “Quantum Mechanical”.

Difference between Classical Computing and Quantum Computing

Okay, enough with the jargon. Let’s summarize some points we discussed until now for some much-needed clarity.

Classical Computing	Quantum Computing
A classical computer is fundamentally made up of binary digits or “bits”.	A Quantum Computer is fundamentally made up of “Qubits” or Quantum Bits
A bit can either be in state 0 or state 1 at a given instance.	A Qubit can either be in state 0, state 1, or a combination of state 1 and state 0 at a given instance.
Bits are made up of transistors in a classical processor.	Qubits are made up of “Quantum” materials like ions, photons, or superconducting materials.
Classical computers are relatively less error-prone than quantum computers	Quantum Computers are a lot more error-prone than classical computers
Transistors do not need sub-zero temperatures to operate.	Qubits require sub-zero temperatures to preserve their Quantum nature.
Classical computers cannot run algorithms like optimization problems, machine learning models, etc. Very efficiently in most cases.	Quantum computers can theoretically run algorithms like optimization problems, machine learning models, etc. much more efficiently than Classical Computers.