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## Linear algebra for quantum computers

This is material for a course on linear algebra and mathematical requirements for quantum computing. It contains Jupyter notebooks that can be downloaded or opened in Binder as an online interactive notebook.

## Why linear algebra?

Linear algebra, the language of matrices and vectors, is the fundamental language of quantum computing and quantum information. The linear algebra approach to quantum computing is the approach taken in the most cited textbook on the subject: Quantum Computing and Quantum Information by Nielsen & Chuang. This approach is also the most mathematically accessible for the absolute beginner. If you have little experience in mathematics, physics and computer programming, this approach is for you!

## What Quantum Computing Is and Isn't

Quantum computing is a way to process information in a fundamentally different way than the way your laptop or smartphone processes data. The “classic” way of processing information is in the form of bits, binary values of zeros and ones. Quantum computing uses three different properties of quantum physics to process information.

## overlap

**overlap**is the fancy word used to describe how**Qubits**, the classic version of bits, can exist in a combination of zeros and ones. Qubits can also be measured in different bases, enabling things like adaptive state measurement and measurement-based quantum computing. Because qubits can be in a state that is a mix of zero and one, this gives them computational properties that classical bits don't have.

## entanglement

**entanglement**is the word used to describe how qubits can have states that correlate with other qubit states. The qubits interact and intertwine, meaning their states are now statistically correlated and information about one can sometimes reveal information about the other. Multiple qubits can become entangled and unique and complicated behavior can emerge. One area of research that scientists have shown great interest in is multipart entanglement. This is a complex form of entanglement that exhibits highly non-classical behavior. This can be used in quantum teleportation and quantum cryptography protocols. It is also the source of a famous experiment named after Einstein, Podolsky, and Rosen, as well as John Bell's famous experiments showing that local hidden variable theories are highly improbable.

## interference

**interference**, like the interference we see in light, sound and water waves is shown by quantum computers. Quantum computers can have constructive (additive) interference or they can have destructive interference, where the waves are out of sync and cancel each other out. This interesting and strange behavior is often the subject of discussion when it comes to wave-particle duality and the famous measurement problem of quantum physics. Interference can be used to increase "good information" and cancel "bad information" in calculations. This allows us to configure calculations in intelligent ways and use inference as an information processing technique to solve problems.

## Quantum computer on ASICs

ASICs bzw**Application specific integrated circuits**are specialized computer chips that are built for very specific types of computing tasks. A good example of this is the GPU.**graphics card**, or Google's TPU (Tensor Processing Unit), which have drastically improved machine learning in recent years. Other examples could be these**Neuromorph Computer Chips**that Intel designed. All of them are designed to solve specific types of problems, such as: B. improving machine learning tasks. Quantum computers can be imagined in a very similar way. There is a classic computer interface that delegates certain computing tasks to quantum hardware. The quantum hardware then processes this arithmetic task and the classical computer receives the output data. There are many types of problems that quantum computers can solve. In fact, any arithmetic problem that a classical computer can solve can also be solved by a quantum computer. However, classic computers are still more efficient at certain things, so they're not going anywhere. There are many algorithms and problems that can be solved exponentially faster on a quantum computer compared to all known classical methods.

Figuring out where this computational advantage comes from is one of the central tasks of quantum computing research. There is a zoo of quantum algorithms listing many of the known algorithms that offer speedup over classical methods. Much of the focus of The Singularity is exploring these different applications and algorithms and their implementation in different quantum computing languages. Some applications include using Shor's algorithm to break RSA public key cryptography and other algorithms that break elliptic curve cryptography. If you are interested in discovering the inner workings of quantum algorithms and understanding where this quantum superiority or advantage comes from, watch these lectures.

## Top 10 what is the advantage of interference in quantum computing, edited by 5 WS

### Quantum computing in a nutshell¶

**Author:**qiskit.org**Published:**23.09.2022**Analysis:**4.68 (478 votes)**Summary:**Interference¶. Like a classical computer, a quantum computer works with bits. However, while classic bits are only found in states 0 and 1,**Matching search results:**Building complex quantum circuits with minimal effort is at the heart of Qiskit, which supports a wide range of functions and can be passed to a variety of quantum computers or classical simulators. With just a few lines of code, it's...

### Understanding Quantum Computing - Azure Quantum | Microsoft Learn

**Author:**lernen.microsoft.com**Published:**19.11.2021**Analysis:**4.4 (213 votes)**Summary:**· Learn how quantum computing works, how it compares to classical computers... for which a quantum computer has an exponential advantage over the best**Matching search results:**Quantum computers are controllable quantum mechanical devices that use the properties of quantum physics to perform calculations. For some computing tasks, quantum computing offers exponential speedups. These accelerations are possible thanks to…

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### What is the advantage of interference in quantum computing?

**Author:**indiascreen.ir**Published:**15.05.2022**Analysis:**4.39 (316 votes)**Summary:**· Pseudo-telepathy is an extraordinary means of demonstrating the power of quantum computers. This power is due to the destructive interference used**Matching search results:**Today's common computers run on chips that use bits for calculations. These bits take on either the values zero or one, with zero representing the "off" position and one representing the "on" position. Several of these bits revealing a combination...

### What is Quantum Interference? – Definition by WhatIs.com

**Author:**techtarget.com**Published:**26.04.2022**Analysis:**4.18 (531 votes)**Summary:**Quantum interference is one of the most challenging principles in quantum theory. In essence, the concept states that elementary particles cannot just be**Matching search results:**Today's common computers run on chips that use bits for calculations. These bits take on either the values zero or one, with zero representing the "off" position and one representing the "on" position. Several of these bits revealing a combination...

### quantum computer? | dotQuantum.io – What is quantum computing?

**Author:**dotquantum.io**Published:**04.06.2022**Analysis:**3.92 (412 votes)**Summary:**Quantum interference, a byproduct of superposition, allows us to bias the measurement of a qubit toward a desired state or set of states. interference**Matching search results:**A brief explanation: the classical bit can only assume one of two values, it is 0 or 1, while the quantum bit (or qubit) can also be found in a superposition of the two states 0 and 1; essentially it can be 0 and 1 at the same time until one...

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### Faculty of Computer Science Wins DOE Scholarship to Study Benefits of Quantum Computing

**Author:**news.stonybrook.edu**Published:**05.09.2022**Analysis:**3.68 (553 votes)**Summary:**· Podder investigates quantum advantages in solving computational problems; a witness is data that confirms the answer to a calculation**Matching search results:**A brief explanation: the classical bit can only assume one of two values, it is 0 or 1, while the quantum bit (or qubit) can also be found in a superposition of the two states 0 and 1; essentially it can be 0 and 1 at the same time until one...

### What is quantum mechanics? | Institute of Quantum Computing

**Author:**uwaterloo.ca**Published:**22.01.2022**Analysis:**3.53 (544 votes)**Summary:**Since particles behave like waves in the quantum world, quantum interference comes into play. Constructive interference increases the likelihood of certain**Matching search results:**A brief explanation: the classical bit can only assume one of two values, it is 0 or 1, while the quantum bit (or qubit) can also be found in a superposition of the two states 0 and 1; essentially it can be 0 and 1 at the same time until one...

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### Advanced Quantum Computing: Interference and Entanglement

**Author:**worldscientific.com**Published:**01.05.2022**Analysis:**3.28 (419 votes)**Summary:**The special properties of quantum objects (atoms, ions and photons) are superposition, interference and entanglement. Superposition refers to particles**Matching search results:**A brief explanation: the classical bit can only assume one of two values, it is 0 or 1, while the quantum bit (or qubit) can also be found in a superposition of the two states 0 and 1; essentially it can be 0 and 1 at the same time until one...

### Quantum computing and quantum supremacy explained

**Author:**wired.co.uk**Published:**15.04.2022**Analysis:**3.02 (415 votes)**Summary:**· Here's what makes quantum computers different from regular computers... They're incredibly sensitive to interference**Matching search results:**Catch a Coin When you flip it, it could be heads or tails. But if you spin it, it has a chance to land heads and tails. Until you measure by stopping the coin, it could be anyone. Overlay is like a...

### An interactive introduction to quantum computing part 2

**Author:**davidbkemp.github.io**Published:**27.10.2021**Analysis:**2.97 (182 votes)**Summary:**This is part 2 of a two-part series. In Part 1, I explained some of the basics of qubits, including the arcane notions of phase and quantum interference.**Matching search results:**Describing quantum computing in terms of superpositions of states that when "measured" "collapse" into the observed state is a very conventional approach to explaining the physics behind quantum computing. However, there are some serious problems...