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Every quantum computer is fundamentally a sampler that starts with a simple probability distribution over all possible measurement outcomes, computes a more complicated distribution, and samples an outcome via a measurement. Quantum Machine Learning 1.0 | Maria Schuld - Xanadu - Medium

Quantum computing is a type of non-classical computing based on the quantum state of subatomic particles. It differs fundamentally from classic computers, which operate using binary bits. Basic properties of quantum world:

  • Superposition - quantum computing uses quantum bits, or Qubits. One Qubit can represent a range of values, which is known as ‘superpositioning’; in addition the two states, 0 and 1, a quantum system can be in the two states at a time; the power to be a wave and a particle, at the same time.

Waterbear.jpg Wikipedia - Tardigrade


What Will We Do With Quantum Computing?

A large-scale quantum computer would be able to solve problems that existing classical computers would take much longer than the age of the universe to solve. This would have dramatic implications for cryptography, chemistry, material science, nuclear physics and probably other areas that are still un- known. But what about quantum computers that will be available in the next few years? What Will We Do With Quantum Computing?

The CIO’s guide to quantum computing | TechRadar Pro - Quantum computing could enable breakthroughs in:

  • Machine learning: Improved ML through faster structured prediction. Examples include Boltzmann machines, quantum Boltzmann machines, semi-supervised learning, unsupervised learning and deep learning;
  • Artificial intelligence: Faster calculations could improve perception, comprehension, and circuit fault diagnosis/binary classifiers;
  • Chemistry: New fertilizers, catalysts, battery chemistry will all drive improvements in resource utilization;
  • Biochemistry: New drugs, tailored drugs, and maybe even hair restorer;
  • Finance: Quantum computing could enable faster, more complex Monte Carlo simulations; for example, trading, trajectory optimization, market instability, price optimization and hedging strategies;
  • Healthcare: DNA gene sequencing, such as radiotherapy treatment optimization/brain tumor detection, could be performed in seconds instead of hours or weeks;
  • Materials: super strong materials; corrosion proof paints; lubricants and semiconductors;
  • Computer science: Faster multidimensional search functions; for example, query optimization, mathematics and simulations.

Source: McKinsey & Company Quantum computers are coming. Get ready for them to change everything | Daphne Leprince-Ringuet - ZDNet



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If quantum computers became practical, they will destroy the security of our currently deployed public-key cryptographic solutions (such as RSA or ECDSA). The majority of our telecommunications channels make use of these encryption schemes to provide confidentiality, authentication and integrity.

Quantum cryptography is the science of exploiting quantum mechanical properties to perform cryptographic tasks. The best known example of quantum cryptography is quantum key distribution which offers an information-theoretically secure solution to the key exchange problem. The advantage of quantum cryptography lies in the fact that it allows the completion of various cryptographic tasks that are proven or conjectured to be impossible using only classical (i.e. non-quantum) communication. For example, it is impossible to copy data encoded in a quantum state. If one attempts to read the encoded data, the quantum state will be changed due to wave function collapse (no-cloning theorem). This could be used to detect eavesdropping in quantum key distribution (QKD). - Quantum Cryptography | Wikipedia

Quantum Communications

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Quantum entanglement, and what it mean for the future of the Internet where information is created, stored and moved around in ways that mirror the bizarre behavior of the quantum world - sending quantum information instead of classical information.

Detection/Sensor Technology

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A quantum sensor is a quantum device that responds to a stimulus. Usually this refers to a sensor which has quantized energy levels, uses quantum coherence to measure physical quantity, or uses entanglement to improve measurements beyond what can be done with classical sensors. ... You can coherently manipulate the sensor. Quantum sensor | Wikipedia

Quantum Radar

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Quantum radar is a speculative remote-sensing technology based on quantum-mechanical effects, such as the uncertainty principle or quantum entanglement. Broadly speaking, a quantum radar can be seen as a device working in the microwave range, which exploits quantum features, from the point of view of the radiation source and/or the output detection, and is able to outperform a classical counterpart. One approach is based on the use of input quantum correlations (in particular, quantum entanglement) combined with a suitable interferometric quantum detection at the receiver (strongly related to the protocol of quantum illumination). ... One way to defeat conventional radar systems is to broadcast signals on the same frequencies used by the radar, making it impossible for the receiver to distinguish between their own broadcasts and the spoofing signal (or "jamming"). However, such systems cannot know, even in theory, what the original quantum state of the radar's internal signal was. Lacking such information, their broadcasts will not match the original signal and will be filtered out in the correlator. Environmental sources, like ground clutter and aurora, will similarly be filtered out. Wikipedia


At the core of the proposed method is a means of interconverting microwave and optical signals using a so-called electro-optomechanical converter. This device would consist of optical and microwave cavities for storing each kind of radiation, with a nanoscale vibrating object (such as a piezoelectric crystal or a metallic membrane) serving as the connection between the two. The oscillator can couple electromagnetic vibrations in the two cavities, despite their different frequencies.


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Quantum Natural Language Processing

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One particularly interesting aspect of this graphical framework for linguistics was that the networks were inherited from previous work that provided quantum theory with an entirely network-like language.

Quantum Machine Learning (QML)

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Quantum machine learning is an emerging interdisciplinary research area at the intersection of quantum physics and machine learning. The most common use of the term refers to machine learning algorithms for the analysis of classical data executed on a quantum computer, i.e. quantum-enhanced machine learning. While machine learning algorithms are used to compute immense quantities of data, quantum machine learning increases such capabilities intelligently, by creating opportunities to conduct analysis on quantum states and systems. This includes hybrid methods that involve both classical and quantum processing, where computationally difficult subroutines are outsourced to a quantum device. These routines can be more complex in nature and executed faster with the assistance of quantum devices. Furthermore, quantum algorithms can be used to analyze quantum states instead of classical data. Quantum machine learning | Wikipedia


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PennyLane is a cross-platform Python library for quantum machine learning, automatic differentiation, and optimization of hybrid quantum-classical computations. The first dedicated machine learning platform for quantum computers


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Noisy Intermediate-Scale Quantum Artificial Intelligence; implementing neural networks on near-term quantum computers

Quantum Neural Network (QNN)

Quantum neural networks (QNNs) are neural network models which are based on the principles of quantum mechanics. There are two different approaches to QNN research, one exploiting quantum information processing to improve existing neural network models (sometimes also vice versa), and the other one searching for potential quantum effects in the brain. Quantum Neural Network (QNN) | Wikipedia

Quantum Convolutional Neural Network (QCNN)

Machine learning techniques have so far proved to be very promising for the analysis of data in several fields, with many potential applications. However, researchers have found that applying these methods to quantum physics problems is far more challenging due to the exponential complexity of many-body systems.... "One of the objectives of the present work was to generalize a specific, well-known machine learning architecture called convolutional neural network (CNN) for a compact quantum circuit, and demonstrate its capabilities with simplistic but meaningful examples." In their study, Choi and his colleagues assumed that CNNs owe their great success to two important features. Firstly, the fact that they are made out of smaller local units (i.e., multiple layers of quasi-local quantum gates). Secondly, their ability to process input data in a hierarchical fashion. The researchers found a connection between these two characteristics and two renowned physics concepts known as Locality and Renormalization. The researchers observed that Renormalization processes share some similarities with pattern recognition applications, particularly those in which machine learning is used to identify objects in pictures. For instance, when a CNN trained for pattern recognition tasks analyzes pictures of animals, it focuses on a universal feature (i.e., trying to identify what animal is portrayed in the image), regardless of whether individual animals of the same type (e.g., cats) look slightly different. This process is somewhat similar to Renormalization techniques in theoretical physics, which can also help to distill universal information....quantum convolutional neural network (QCNN), on a quantum physics-specific problem that involved recognizing quantum states associated with a 1-D symmetry protected topological phase. Remarkably, their technique was able to recognize these quantum states, outperforming existing approaches. As it is fairly compact, the QCNN could also potentially be implemented in small quantum computers. Introducing Quantum Convolutional Neural Networks (QCNN) | Ingrid Fadelli

Machine Learning Powered By Light


A quantum photonic company - An industry-first machine learning toolbox for quantum computing. Powered by TensorFlow. Instead of using electrons to carry information and perform calculations, Xanadu uses photons. Unlike electrons, photons are very stable and are almost unaffected by random noise from heat. Xanadu uses photonic chips to generate, control, and measure photons in ways that enable extremely fast computation.

Getting Started with Quantum Programming

Quantum Logic Gate

A basic quantum circuit operating on a small number of Qubits. They are the building blocks of quantum circuits, like classical logic gates are for conventional digital circuits. It is different to electric circuits in that the Qubits exist, and instructions are applied to the Qubit(s) ... Quantum logic gates | Wikipedia

Quantum NOT gate (X, Y, Z)



  • Pauli-X gate - Pauli-x gate is the same as the not gate described above. It is the quantum equivalent of the NOT gate for classical computers (with respect to the standard basis, which distinguishes the Z-direction; It equates to a rotation around the X-axis of the Bloch sphere by pi radians. Due to this nature, it is sometimes called bit-flip.
  • Pauli-Y gate - It equates to a rotation around the Y-axis of the Bloch sphere by pi radians

  • Pauli-Z gate - It equates to a rotation around the Z-axis of the Bloch sphere by pi radians; representing a rotation around the z-axis by 180 degrees.

Phase Gate ... S & T gates

S gate is related to the Pauli-z gate; representing a rotation around the z-axis by 90 degrees.

T gate is related to the Pauli-z gate; representing a rotation around the z-axis by 45 degrees.

Both S and T gates are not reversible. Which means adding two of them in a raw will not result in the input status. Rather following diagram shows the combination of gates to achieve the inverse.

Hadamard gate

is the one-Qubit version of the quantum Fourier transform. A measurement will have equal probabilities to become 1 or 0 (i.e. creates a Superposition). The Hadamard gate can also be expressed as a 90º rotation around the Y-axis, followed by a 180º rotation around the X-axis. So H = X Y^{1/2}H=XY. Getting Qubits into Superposition at the start of an algorithm is call 'Hadamard initialization'



CNOT gate - Controlled (cX cY cZ) gates

Controlled gates act on 2 or more Qubits, where one or more Qubits act as a control for some operation. For example, the controlled NOT gate (or CNOT or cX) acts on 2 Qubits, and performs the NOT operation on the second Qubit only when the first Qubit is {\displaystyle |1\rangle }|1\rangle , and otherwise leaves it unchanged.


Toffoli gate (CCNot gate)

As an extended version of CNot gate, Toffoli gate does the Not operation depending on two control Qubits. Result of the And operation between two control bits are used as the final control bit for the Not gate.

Toffoli gate => Quantum equivalent of a universal logic gate.

Toffoli gate can be used to achieve several other logic operations as follows.

In classical computing when we need to transfer some logical state to several other logic gates we just connect all the lines as in the image. Physically also we can transfer the low, high states to several other logic gates via wires. But in quantum mechanics, we cannot send a single Qubit to two destinations. To do that we need to clone the quantum state of the Qubits. Which is not possible according to No-Cloning theorem. Instead, we must use some special gates called fanout gates. Note: Fanout gates does not copy superposed and entangled states. Creating a fanout gate using Toffoli gate:

SWAP gate

built from a circuit containing only CNOT gates. The first CNOT is the controlled by the first Qubit (CNOTa), and second one is controlled by the second Qubit (CNOTb).


Fredkin gate (Controlled Swap gate)

Fredking gate has one control Qubit and two input Qubits. When control bit is 1 it swaps the two input Qubits.

Measure operation

a special type of operations done on Qubits at the end of a circuit to get the final values of the Qubits after the operations have been conducted. Compared to the gates above, Measurement is irreversible and hence, is not a Quantum Gate.

Quantum Instruction Sets


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cQASM, also known as common QASM, is a hardware-agnostic QASM which guarantees the interoperability between all the quantum compilation and simulation tools.

  • QI Editor Guide
  • Quantum Inspire (QI) a quantum computing platform designed and built by QuTech. Quantum Inspire (QI) is initiated by QuTech the advanced research center for quantum computing and quantum internet founded by TU Delft and TNO.
  • QuTech Academy "QuTech gives students a large perspective of what is happening in the field of quantum!" Discover quantum computers and the quantum internet. Learn the principles and promises behind these developments and how they will impact our future in 2 new online courses. The course Quantum Computers and [[Quantum Internet: How can they change the world? will dive into the potential impact of a quantum computer and a quantum internet. The course Building Blocks of a Quantum Computer will give you insight in the several layers of a quantum computer, ranging from Qubits to software. Both courses are available on EdX.org.


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OpenQASM is the intermediate representation introduced by IBM for use with Qiskit and the IBM Q Experience. Wikipedia


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Quil is an instruction set architecture for quantum computing that first introduced a shared quantum/classical memory model. Many quantum algorithms (including quantum teleportation, quantum error correction, simulation, and optimization algorithms) require a shared memory architecture.

Quil is being developed for the superconducting quantum processors developed by Rigetti Computing through the Forest quantum programming API. A PPython library called pyQuil was introduced to develop Quil programs with higher level constructs. A Quil backend is also supported by other quantum programming environments. Computing developed a Quantum Virtual Machine in Common Lisp that simulates the defined Quantum Abstract Machine on a classical computer and is capable of the parsing and execution of Quil programs with possibly remote execution via HTTP

PyQuil serves three main functions:

  • Easily generating Quil programs from quantum gates and classical operations
  • Compiling and simulating Quil programs using the Quil Compiler (quilc) and the Quantum Virtual Machine (QVM)
  • Executing Quil programs on real quantum processors (QPUs) using Quantum Cloud Services (QCS)


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Blackbird is a quantum assembly language for continuous-variable quantum computation, that can be used to program Xanadu's quantum photonics hardware and Strawberry Fields simulator.

Quantum Functions & Algorithms

Schnorr's Algorithm

It is estimated that a quantum computer using 372 qubits running their algorithm could crack any of the cryptosystems in use today. Not mentioned in their work is one caveat that remains—today's quantum computers have error rates so high that it would be impossible to use them to crack cryptosystems. If systems with much lower error rates do arise in the near future, cryptosystem makers could increase the size of the prime numbers used to generate their keys—but only for so long. A more likely prospect for securing computer systems in the future will use quantum-secure communications, specifically quantum key distribution. - Modification to Shor's algorithm may mean less powerful quantum computers could crack cryptosystems | Bob Yika - Tech Xplore


Shor's Period Finding Algorithm

Performing prime factorization of integers; Shor's algorithm is difficult to understand because it mixes together ideas from quantum physics, signal processing, number theory, and computer science.

  • Quirk
  • Quantum Cryptographic Technology
  • Shor's algorithm - why doesn't the final collapse of the auxiliary qubits cripple the computation? In the factoring algorithm, there are three kinds of qubits. In the OP's notation, there are "input qubits", which start in a superposition of all possible values, and which you eventually take the Fourier transform of. There are "value qubits", in which you compute the function ya(modN), where a is the value in the input qubits. And there are "auxiliary qubits", which you use as workspace to help do this computation. In order to make the factoring algorithm work properly, you need to reset all the auxiliary qubits, which started as |0⟩ at the beginning of the computation, to |0⟩ at the end of the computation. This is called "uncomputing" these qubits. (Actually, you can set them to anything you please as long as it is a constant independent of the workings of the algorithm.) Theorems about reversible classical computation ensure that it is possible to do this. If you reset the auxiliary qubits to |0⟩, then if the environment, or somebody, measures them, nothing is revealed about the computation, and the computation is not "crippled". If you forget to reset them to |0⟩, you probably won't get the right answer, whether or not anybody measures them.
  • Shor's Quantum Factoring Algorithm | Algorithmic Assertions

Grover's Search

Searches for a specified entry in an unordered database, employing an important technique in quantum algorithm design known as amplitude amplification

Deutsch-Jozsa Algorithm

One of the first quantum algorithms with nice speedup over its classical counterpart. Consider This is an “hello world” in quantum computing. Consider a function f that takes 0 or 1 as input and outputs either 0 or 1. Our functions in mind are either balanced or constant. A function f is called balanced if it outputs 0 half the time and 1 the other half. It is a constant function if its output is a constant (1 or 0) regardless of input.

We are given an oracle whose input is n bits and whose output is one bit. We are promised that out of the 2n possible inputs, either all of them, none of them, or half of them yield output 1. The task is to distinguish the balanced case (half of all inputs yield output 1) from the constant case (all or none of the inputs yield output 1). It was shown by Deutsch that for n=1, this can be solved on a quantum computer using one query, whereas any deterministic classical algorithm requires two. This was historically the first well-defined quantum algorithm achieving a speedup over classical computation. A single-query quantum algorithm for arbitrary n was developed by Deutsch and Jozsa. Although probabilistically easy to solve with O(1) queries, the Deutsch-Jozsa problem has exponential worst case deterministic query complexity classically. Algebraic and Number Theoretic Algorithms | Stephen Jordan

Simon’s Algorithm

Much of the functionality is the same as the Bernstien-Vazirani algorithm. Circuit | DaftWullie - Stack Exchange

Bernstein-Vazirani Algorithm

Correctly guesses a string of binary numbers in one shot. It's a restricted version of the Deutsch–Jozsa algorithm where instead of distinguishing between two different classes of functions, it tries to learn a string encoded in a function. Bernstein–Vazirani algorithm | Wikipedia, which solves the Bernstein–Vazirani problem

We are given an oracle whose input is n bits and whose output is one bit. Given input x∈{0,1}n, the output is x⊙h, where h is the "hidden" string of n bits, and ⊙ denotes the bitwise inner product modulo 2. The task is to find h. On a classical computer this requires n queries. As shown by Bernstein and Vazirani, this can be achieved on a quantum computer using a single query. Furthermore, one can construct recursive versions of this problem, called recursive Fourier sampling, such that quantum computers require exponentially fewer queries than classical computers. Algebraic and Number Theoretic Algorithms | Stephen Jordan

Bell Inequality Test (CHSH)

Determining the exclusive-or (XOR) mask over which a given black-box function is invariant

Quantum Development Algorithms & Kits

SDKs with access to quantum processors

The following software development kits can be used to run quantum circuits on prototype quantum devices, as well as on simulators.


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TensorFlow Quantum

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ProjectQ is an open-source compilation framework capable of targeting various types of hardware and a high-performance quantum computer simulator with emulation capabilities, and various compiler plug-ins.


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a new cloud computing service that gives developers and researchers a way to tinker with quantum circuits. According to an Amazon news release, users will be able to build out their own quantum circuits and applications and test them on Amazon’s machine. Amazon didn’t actually build a quantum computer — instead, it partnered with other organizations that did — Instead, it partnered with three companies — D-Wave, IonQ, and Rigetti — that did build their own. Amazon Is Now Letting Anyone Run Programs On Its Quantum Computer | Dan Robitzski


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  • Qilimanjaro build a unique first-to-market full-stack coherent quantum annealing computer with an easy-to-use advanced algorithmic toolset to effectively address complex optimization problems in multiple real-world industry use cases.
  • Address existing quantum hardware platforms
  • Development of HPC quantum simulators
  • Cloud access to quantum computing resources
  • Long Qubit coherence, low-system noise
  • High connectivity Qubit architecture
  • Cost-effective solutions
  • Quantum annealing programming: mapping mathematical models to device hardware
  • For hard combinatorial optimization problems


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The quantum bits—also known as Qubits—are the lowest energy states of the superconducting loops that make up the D-Wave QPU. These states have a circulating current and a corresponding magnetic field. As with classical bits, a Qubit can be in state of 0 or 1. But because the Qubit is a quantum object, it can also be in a Superposition of the 0 state and the 1 state at the same time. At the end of the quantum annealing process, each Qubit collapses from a Superposition state into either 0 or 1 (a classical state). The physics of this process can be shown (visualized) with an energy diagram. This diagram changes over time, as we can see in (a), (b), and (c). To begin, there is just one valley (a), with a single minimum. The quantum annealing process runs, the barrier is raised, and this turns the energy diagram into what is known as a double-well potential (b). Here, the low point of the left valley corresponds to the 0 state, and the low point of the right valley corresponds to the 1 state. The Qubit ends up in one of these valleys at the end of the anneal. Introduction to Quantum Annealing | D-Wave

Quantum Brilliance

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An Australian/German company is developing powerful quantum accelerators the size of graphics cards. They work at room temperature, undercutting and outperforming today's huge, cryo-cooled quantum supercomputers, and soon they'll be small enough for mobile devices. Properties of the nitrogen-vacancy (NV) centre in diamonds. NV centres in diamond have the longest coherence time of any room temperature quantum state. This means that as qubits they can operate anywhere a classical computer can.

The nitrogen-vacancy center (N-V center or NV center) is one of numerous point defects in diamond. Its most explored and useful property is its photoluminescence, which allows observers to read out its spin-state. The NV center's electron spin, localized at atomic scales, can be manipulated at room temperature by external factors such as magnetic, or electric fields, microwave radiation, or light, resulting in sharp resonances in the intensity of the photoluminescence. These resonances can be explained in terms of electron spin related phenomena such as quantum entanglement, spin–orbit interaction and Rabi oscillations, and analysed using advanced quantum optics theory. An individual NV center can be used as a basic unit for a quantum computer, a qubit, used f.e. for quantum cryptography. Further potential applications in novel fields of electronics and sensing include spintronics, masers, and quantum sensors. If the charge is not specified the term "NV center" refers to the negatively charged NV− center.Wikipedia


Quantum Software Development Kit (SDK)

SDKs based on simulators

Public access to quantum devices is currently planned for the following SDKs, but not yet implemented.

= Intel Quantum Software Development Kit

| Intel]

The Intel Quantum SDK includes an intuitive user interface based on C++, a low-level virtual machine (LLVM)-based compiler toolchain with a quantum runtime environment optimized for executing hybrid quantum-classical algorithms and a high-performance Intel® Quantum Simulator (IQS) qubit target backend. Future releases will feature different qubit target backends, including a quantum dot qubit simulator and, eventually, an Intel quantum dot qubit device as the target backend will also be offered.

Microsoft Quantum Development Kit

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  • Cirq | Google
  • Cirq - a Python library for writing, manipulating, and optimizing quantum circuits and running them against quantum computers and simulators. | Google]
  • PennyLane a cross-platform Python library for quantum machine learning

A Python framework for creating, editing, and invoking Noisy Intermediate Scale Quantum (NISQ) circuits; running them against quantum computers and simulators.

Strawberry Fields

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  • Xanadu
  • PennyLane a cross-platform Python library for quantum machine learning
  • Blackbird Quantum Instruction Set
  • TensorFlow Quantum
  • Quantum Machine Learning 1.0 | Maria Schuld - Xanadu - Medium
  • Strawberry Fields - is a full-stack Python library for designing, simulating, and optimizing continuous variable quantum optical circuits. | Xanadu
    • An open-source software architecture for photonic quantum computing
    • A full-stack quantum software platform, implemented in Python specifically targeted to the CV model
    • Quantum circuits are written using the easy-to-use and intuitive Blackbird quantum programming language
    • Powers the Strawberry Fields Interactive web app, which allows anyone to run a quantum computing simulation via drag and drop
    • Includes quantum computer simulators implemented using NumPy and Tensorflow - these built-in quantum compiler tools convert and optimize Blackbird code for classical simulation
    • Future releases will aim to target experimental backends, including photonic quantum computing chips



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A quantum programming environment developed by Cambridge Quantum Computing, that targets simulators and quantum hardware. It is for use with the hardware produced by the NQIT hub, as well as Oxford Quantum Circuits.

Silicon Devices

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TensorFlow Quantum

TensorFlow Quantum simplifies the swift development of hybrid quantum-classical machine learning models. Those engaged in quantum algorithm and application research can harness the power of Google's quantum computing frameworks seamlessly through TensorFlow. TensorFlow Quantum is dedicated to quantum data processing and constructing hybrid quantum-classical models. It seamlessly incorporates quantum computing algorithms and logical constructs from Cirq while offering quantum computing primitives that seamlessly blend with existing TensorFlow APIs. Additionally, it comes equipped with high-performance quantum circuit simulators for efficient experimentation.

Simulating and Graphing


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QuTiP is open-source software for simulating the dynamics of open quantum systems. The QuTiP library depends on the excellent Numpy, SciPy, and Cython numerical packages. In addition, graphical output is provided by Matplotlib. QuTiP aims to provide user-friendly and efficient numerical simulations of a wide variety of Hamiltonians, including those with arbitrary time-dependence, commonly found in a wide range of physics applications such as quantum optics, trapped ions, superconducting circuits, and quantum nanomechanical resonators. QuTiP is freely available for use and/or modification on all major platforms such as Linux, Mac OSX, and Windows*. Being free of any licensing fees, QuTiP is ideal for exploring quantum mechanics and dynamics in the classroom.


Concepts: Intrinsic Properties

  • Bell State - a maximally entangled quantum state of two qubits. a concept in quantum information science, are specific quantum states of two Qubits that represent the simplest (and maximal) examples of quantum entanglement. The Bell states are a form of entangled and normalized basis vectors. This normalization implies that the overall probability of the particle being in one of the mentioned states is 1. Bell states can be generalized to represent specific quantum states of multi-Qubit systems, such as the Greenberger–Horne–Zeilinger (GHZ) State for 3 subsystems. ...Wikipedia


  • Greenberger–Horne–Zeilinger (GHZ) State - a certain type of entangled quantum state that involves at least three subsystems (particles). Extremely non-classical properties of the state have been observed. There is no standard measure of multi-partite entanglement because different, not mutually convertible, types of multi-partite entanglement exist. Nonetheless, many measures define the GHZ state to be maximally entangled state. ... ...Wikipedia To create such a GHZ state, we start with a three-qubit quantum register. By default, each qubit in the register is initialized to |0⟩. To make the GHZ state, we apply the following gates:
  • Hadamard gate 𝐻 on qubit 0, which puts it into the superposition state (|0⟩+|1⟩)/2⎯⎯√.
  • controlled-Not operation (𝐶𝑋) between qubit 0 and qubit 1.
  • controlled-Not operation between qubit 0 and qubit 2.



  • Quantum tunneling - able to bypass any barriers i.e move through walls ... ...Wikipedia

  • Quantum Spin - intrinsic form of angular momentum carried by elementary particles, composite particles (hadrons), and atomic nuclei. Spin is one of two types of angular momentum in quantum mechanics, the other being orbital angular momentum. The orbital angular momentum operator is the quantum-mechanical counterpart to the classical angular momentum of orbital revolution and appears when there is periodic structure to its wavefunction as the angle varies. ... ...Wikipedia

  • Quantum Wave - a complex-valued probability amplitude, and the probabilities for the possible results of measurements made on the system can be derived from it. If you've studied light, you may already know a bit about quantum theory. You might know that a beam of light sometimes behaves as though it's made up of particles (like a steady stream of cannonballs), and sometimes as though it's waves of energy rippling through space (a bit like waves on the sea). That's called wave-particle duality and it's one of the ideas that comes to us from quantum theory. It's hard to grasp that something can be two things at once—a particle and a wave. ...Wikipedia and Quantum computing | Chris Woodford - Explain That Stuff


  • Wave Function Collapse - occurs when a wave function—initially in a superposition of several eigenstates—reduces to a single eigenstate due to interaction with the external world. This interaction is called an "observation". It is the essence of a measurement in quantum mechanics which connects the wave function with classical observables like position and momentum. Collapse is one of two processes by which quantum systems evolve in time; the other is the continuous evolution via the Schrödinger equation. ... ...Wikipedia

Quantum coherence and quantum entanglement are two landmark features of quantum physics, and now physicists have demonstrated that the two phenomena are "operationally equivalent"—that is, equivalent for all practical purposes, though still conceptually distinct.

  • Decoherence - happens when anything gets in the way of a Qubit's job, The interaction of Qubit with their environment in ways that cause their quantum behavior to decay and ultimately disappear. Qubit's quantum state is extremely fragile. The slightest vibration or change in temperature—disturbances known as “noise” in quantum-speak—can cause Qubit to tumble out of Superposition before their job has been properly done. That’s why researchers do their best to protect Qubits from the outside world in those supercooled fridges and vacuum chambers. But despite their efforts, noise still causes lots of errors to creep into calculations. Smart quantum algorithms can compensate for some of these, and adding more Qubits also helps. However, it will likely take thousands of standard Qubits to create a single, highly reliable one, known as a “logical” Qubit. This will sap a lot of a quantum computer’s computational capacity. - #1 enemy of quantum computing. Explainer: What is a quantum computer? | Martin Giles - MIT Technology Review ...Wikipedia

  • Quantum Negativity - is a measure of quantum entanglement. Quasi-probability allows negative probabilities, like a -100%, to explain concepts such as quantum entanglement. Quantum particles can carry an unlimited amount of information about things they have interacted with and could enable far more precise measurements and power new technologies, such as super-precise microscopes and quantum computers....We are used to dealing with probabilities that range from 0% (never happens) to 100% (always happens). To explain results from the quantum world however, the concept of probability needs to be expanded to include a so-called quasi-probability, which can be negative. This quasi-probability allows quantum concepts such as Einstein’s ‘spooky action at a distance’ and wave-particle duality to be explained in an intuitive mathematical language. For example, the probability of an atom being at a certain position and traveling with a specific speed might be a negative number, such as -5%. An experiment whose explanation requires negative probabilities is said to possess ‘quantum negativity.’ Quantum Negativity can be used to take more precise measurements of everything from molecular distances to gravitational waves. Ultra-Precise Measurements Powered by Quantum Negativity – “Highly Counterintuitive and Truly Amazing!” | University of Cambridge - SciTechDaily

  • Locality - an object is directly influenced only by its immediate surroundings. The concept is that for an action at one point to have an influence at another point, something in the space between those points such as a field must mediate the action. To exert an influence, something, such as a wave or particle, must travel through the space between the two points, carrying the influence. This is an alternative to the older concept of instantaneous "action at a distance". In 1935 Albert Einstein, Boris Podolsky and Nathan Rosen in their EPR paradox theorized that quantum mechanics might not be a local theory, because a measurement made on one of a pair of separated but entangled particles causes a simultaneous effect, the collapse of the wave function, in the remote particle (i.e. an effect exceeding the speed of light). In 1964 John Stewart Bell formulated the "Bell inequality", which, if violated in actual experiments, implies that quantum mechanics violates either locality or realism, another principle which relates to the value of unmeasured quantities. The two principles are commonly referred to as a single principle, local realism. ... Wikipedia
  • Renormalization - In physics, certain universal features of a quantum many-body system, such as the phase (e.g., liquid, gas, solid, etc.) of materials do not depend on (or are not sensitive to) microscopically detailed information of the system, but rather governed by only a few important hidden parameters. Renormalization is a theory technique to identify those important parameters starting from microscopic description of a quantum system. The theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering values of quantities to compensate for effects of their self-interactions. ... Wikipedia

  • Quantum Dot (QD) - are tiny semiconductor particles a few nanometres in size, having optical and electronic properties that differ from larger particles due to quantum mechanics. They are a central topic in nanotechnology. When the quantum dots are illuminated by UV light, an electron in the quantum dot can be excited to a state of higher energy. ... ...Wikipedia

  • Interference - a phenomenon in which two waves superpose to form a resultant wave of greater, lower, or the same amplitude. Constructive and destructive interference result from the interaction of waves that are correlated or coherent with each other, either because they come from the same source or because they have the same or nearly the same frequency. Interference effects can be observed with all types of waves, for example, light, radio, acoustic, surface water waves, gravity waves, or matter waves. The resulting images or graphs are called interferograms. ... ...Wikipedia


Quantum Double Slit Experiment

One of the deepest mysteries in quantum physics is the wave-particle duality: every quantum object has properties of both a wave and a particle. Nowhere is this effect more beautifully demonstrated than in the double-slit experiment: streams of particles, photons and electrons are directed at a barrier with two narrow openings. While each particle shows up at the detector individually, the population as a whole creates an interference pattern as though they are waves. Neither a pure wave nor a pure particle description has proven successful in explaining these experiments.

Quantum laser pointers brings you the infamous double slit experiment right in the palm of your hand. In 1801 English physicist Thomas Young performed this experiment to determine if light was a particle or a wave. A laser shines a coherent beam of light through a film disc containing two parallel slits. Light striking the wall behind the slits producers a classic interference pattern. This surprising result means light passes through the parallel slits not as particles but as waves. When the peaks of two waves overlap it creates a band of light. When the peak of one wave meets the valley of another, light is cancelled out. Variations of this experiment spurred public debates between Albert Einstein and Neils Bohr on the true nature of reality. It’s been called the granddaddy of all quantum weirdness.

  • Diffraction - refers to various phenomena that occur when a wave encounters an obstacle or a slit. It is defined as the bending of waves around the corners of an obstacle or through an aperture into the region of geometrical shadow of the obstacle/aperture. The diffracting object or aperture effectively becomes a secondary source of the propagating wave. ... ...Wikipedia

The delayed-choice quantum eraser experiment investigates a paradox. If a photon manifests itself as though it had come by a single path to the detector, then "common sense" (which Wheeler and others challenge) says that it must have entered the double-slit device as a particle. If a photon manifests itself as though it had come by two indistinguishable paths, then it must have entered the double-slit device as a wave. If the experimental apparatus is changed while the photon is in mid‑flight, then the photon should reverse its original "decision" as to whether to be a wave or a particle. ... ...Wikipedia


  • Absolute Zero - The lowest temperature possible, equivalent to -273.15°C (or 0° on the absolute Kelvin scale), at which point atoms cease to move altogether and molecular energy is minimal. The idea that it is impossible, through any physical process, to lower the temperature of a system to zero is known as the Third Law of Thermodynamics. The lowest limit of the thermodynamic temperature scale, a state at which the enthalpy and entropy of a cooled ideal gas reach their minimum value, taken as 0°. The fundamental particles of nature have minimum vibrational motion, retaining only quantum mechanical, zero-point energy-induced particle motion. ... ...Wikipedia

Concepts: Mechanisms for Understanding

  • Qubit - the fundamental object of information in quantum computing ... Wikipedia

  • Bloch sphere - a geometrical representation of the pure state space of a two-level quantum mechanical system (Qubit) Quantum mechanics is mathematically formulated in Hilbert space or projective Hilbert space. The space of pure states of a quantum system is given by the one-dimensional subspaces of the corresponding Hilbert space (or the "points" of the projective Hilbert space). For a two-dimensional Hilbert space, this is simply the complex projective line ℂℙ1. This is the Bloch sphere.... Wikipedia

Bloch Sphere Visualization (Python) | Tyler Dwyer - GitHub ... Python

  • Quantum Operator - a function over a space of physical states to another space of physical states. The simplest example of the utility of operators is the study of symmetry (which makes the concept of a group useful in this context). Because of this, they are very useful tools in classical mechanics. Operators are even more important in quantum mechanics, where they form an intrinsic part of the formulation of the theory.Wikipedia

  • Hilbert Space - an abstract vector space possessing the structure of an inner product that allows length and angle to be measured ...Wikipedia

220px-Standing_waves_on_a_string.gif 210px-Discrete_complex_vector_components.svg.png 214px-Continuous_complex_vector_components.svg.png

  • Dirac Notation .. Bra–ket - notation is a standard notation for describing quantum states, composed of angle brackets and vertical bars. It can also be used to denote abstract vectors and linear functionals in mathematics. It is so called because the inner product (or dot product) of two states is denoted by a ⟨bra|c|ket⟩ ...Wikipedia

  • Schrödinger's Equation - a linear partial differential equation that describes the wave function or state function of a quantum-mechanical system ... Wikipedia

  • Quantum Numbers, Atomic Orbitals, and Electron Configurations - By solving the Schrödinger equation (Hy = Ey), we obtain a set of mathematical equations, called wave functions (y), which describe the probability of finding electrons at certain energy levels within an atom. A wave function for an electron in an atom is called an atomic orbital; this atomic orbital describes a region of space in which there is a high probability of finding the electron. Energy changes within an atom are the result of an electron changing from a wave pattern with one energy to a wave pattern with a different energy (usually accompanied by the absorption or emission of a photon of light). Each electron in an atom is described by four different quantum numbers. The first three (n, l, ml) specify the particular orbital of interest, and the fourth (ms) specifies how many electrons can occupy that orbital. Angelo State University ...Wikipedia

  • MIP*=RE

The class MIP* of languages that can be decided by a classical verifier interacting with multiple all-powerful quantum provers sharing entanglement is equal to the class RE of recursively enumerable languages. MIP*=RE | Z. Ji, A. Natarajan, T. Vidick, J. Wright, and H. Yuen

MIP* = RE is not a typo. It is a groundbreaking discovery. Major quantum computational breakthrough is shaking up physics and maths | Ittay Weiss - Yahoo! News

Halting Problem

The problem of determining, from a description of an arbitrary computer program and an input, whether the program will finish running, or continue to run forever. Alan Turing proved in 1936 that a general algorithm to solve the halting problem for all possible program-input pairs cannot exist. Wikipedia

  • Oracle Function

An oracle O is a "black box" operation that is used as input to another algorithm. Quantum Oracles | Microsoft A super smart or a super slick quantum circuit). It uses the quantum parallelism concept to compute all values of f(x) from the superposition in polynomial time, instead of growing exponentially with the number of bases. … The key idea is the Oracle function will output values that after applying interference (a transformation), we will measure 0 for all constant functions or 1 for balanced functions.QC — Quantum Algorithm with an example | Jonathan Hui

Quantum Biology

Quantum biology is the study of applications of quantum mechanics and theoretical chemistry to biological objects and problems. Many biological processes involve the conversion of energy into forms that are usable for chemical transformations, and are quantum mechanical in nature. Such processes involve chemical reactions, light absorption, formation of excited electronic states, transfer of excitation energy, and the transfer of electrons and protons (hydrogen ions) in chemical processes, such as photosynthesis, olfaction and cellular respiration. Quantum biology | Wikipedia

Quantum Cryptographic Technology

How DLTs could withstand the advent of quantum computers? | Rodny Palomino - Medium

National Institute of Standards and Technology (NIST) Computer Security Resource Center (CSRC)

In recent years, there has been a substantial amount of research on quantum computers – machines that exploit quantum mechanical phenomena to solve mathematical problems that are difficult or intractable for conventional computers. If large-scale quantum computers are ever built, they will be able to break many of the public-key cryptosystems currently in use. This would seriously compromise the confidentiality and integrity of digital communications on the Internet and elsewhere. The goal of post-quantum cryptography (also called quantum-resistant cryptography) is to develop cryptographic systems that are secure against both quantum and classical computers, and can interoperate with existing communications protocols and networks.

Physics and Cryptology

Computers and cryptographic hardware eventually are no Turing machines living in a Platonic world, but real, physical objects that can be touched and accessed, and which consume power, dissipate radiation, or are susceptible to faults and invasive manipulation. This simple observation has had a massive impact both for codemakers and codebreakers over the last decades. On the side of codebreakers, various attacks have been developed that specifically target the physical nature of electronic hardware; they include invasive and semi-invasive read-out techniques, fault injection, photonic emission analysis, power or other side channels, and also hardware Trojans, only naming a few. Complementary to that, physical phenomena like quantum effects or optical interference can allow faster computations than classical Turing machines for certain cryptographically relevant problems. This enables novel computational attack vectors based on physics, too. On the other hand, physics provides equally powerful tools for codemakers. Call for Papers: Special Section on Physics and Cryptology--an Inevitable Couple

Leading Experts

  • Peter Shor: Shor is a professor of mathematics at MIT and is known for developing Shor's algorithm, which is a quantum algorithm for integer factorization.
  • Amir Yacoby: Yacoby is a professor of physics at Harvard University and is known for his work on quantum computing with superconducting circuits.
  • Ludwig M. Duan: Duan is a professor of physics at the University of Chicago and is known for his work on quantum computing with trapped ions.
  • Raymond Laflamme: Laflamme is a professor of physics at the University of Waterloo and is known for his work on quantum computing with NMR.
  • John Preskill: Preskill is a professor of theoretical physics at Caltech and is known for his work on quantum computing theory.
  • Seth Lloyd: Lloyd is a professor of mechanical engineering and applied physics at MIT and is known for his work on quantum computing theory.
  • Christoph Simon: Simon is a professor of physics at the University of Vienna and is known for his work on quantum algorithms.
  • Daniel Gottesman: Gottesman is a professor of physics at the Perimeter Institute for Theoretical Physics and is known for his work on quantum error correction.
  • Michel Devoret: Devoret is a professor of physics at Yale University and is known for his work on quantum measurement and control.
  • Isaac Chuang: Chuang is a professor of electrical engineering and computer science at the University of Chicago and is known for his work on quantum information theory.
  • David Awschalom: Awschalom is a professor of physics at the University of Chicago and is known for his work on quantum computing with spin qubits.
  • David DiVincenzo: DiVincenzo is a professor of physics at the University of Maryland and is known for his work on the requirements for a scalable quantum computer.
  • Hartmut Neven: Neven is a vice president of Google AI and is known for his work on Google's quantum computing project, Sycamore.
  • Michele Mosca: Mosca is a professor of computer science at the University of Toronto and is known for his work on quantum algorithms.
  • Terry Rudolph: Rudolph is a professor of physics at the University of Oxford and is known for his work on quantum error correction.
  • Luigi Amico: Amico is a professor of physics at the University of Catania and is known for his work on quantum many-body systems.
  • Paolo Zanardi: Zanardi is a professor of physics at the University of Pavia and is known for his work on quantum algorithms and quantum error correction.
  • Liang Jiang: Jiang is a professor of physics at the University of Chicago and is known for his work on quantum many-body systems and quantum information theory.
  • Jianfeng Lu: Lu is a professor of physics at the University of Science and Technology of China and is known for his work on quantum information theory and quantum communication.
  • Xiao-Gang Wen: Wen is a professor of physics at the Massachusetts Institute of Technology and is known for his work on quantum many-body systems and topological quantum computation.


Learn Quantum Computation using Qiskit

Quantum Computing for the Determined | Michael Nielsen

University of Toronto

In this course we will introduce several quantum machine learning algorithms and implement them in Python. This massively open online online course (MOOC) on edX is offered by the University of Toronto on edX with an emphasis on what benefits current and near-future quantum technologies may bring to machine learning. These notebooks contain the lecture notes and the code for the course. The content is organized in four modules, with an additional introductory module to the course itself. Since the course is hands-on, we found it important that you can try the code on actual quantum computers if you want to. There isn't a single, unified programming framework that would allow to address all available quantum hardware. For this reason, the notebooks are available in two versions: one in Qiskit targeting the IBM Q hardware and the Forest SDK targetting the Rigetti quantum computer. The notebooks also cover quantum annealing -- for that, the D-Wave Ocean Suite is used. For more details on setting up your computational environment locally.

Quantum Computation at CMU | Ryan O'Donnell

Quantum Mechanics - Univ. of California, Berkeley | Umesh V. Vazirani

Quantum Computing Math Skills

Texts and Notes

Quantum Supremacy

Quantum supremacy is the potential ability of quantum computing devices to solve problems that classical computers practically cannot. Experts forecast that quantum supremacy will become a reality within a matter of years for a limited number of computing problems.

Grim, With Hope

Quantum Chess

What exactly is quantum chess? It's a complicated version of regular chess that incorporates the quantum concepts of superposition, entanglement, and interference. “It’s like you’re playing in a multiverse but the different boards [in different universes] are connected to each other,” said Caltech physicist Spiros Michalakis during a livestream of the tournament. “It makes 3D chess from Star Trek look silly.” Quantum chess (as played in the tournament) is the brainchild of Chris Cantwell of Quantum Realm Games. We have a winner in the world’s first quantum chess tournament | Jennifer Ouellette

Exploring Quantum History

Albert Einstein was not a fan of quantum mechanics. He was annoyed by the uncertain, random nature of the universe it implied (hence the famous quote "God does not play dice with the universe"). So, Einstein tried to develop a unified theory that would circumvent what he saw as quantum mechanics' flaws; the "unruly child" of quantum mechanics, and how the famed physicist came up with the Special Theory of Relativity.