### EITC/QI/QIF Quantum Information Fundamentals is the European IT Certification programme on theoretical and practical aspects of quantum information and quantum computation, based on the laws of quantum physics rather than of classical physics and offering qualitative advantages over their classical counterparts.

The curriculum of the EITC/QI/QIF Quantum Information Fundamentals covers introduction to quantum mechanics (including consideration of the double slit experiment and matter wave interference), introduction to quantum information (qubits and their geometric representation), light polarization, uncertainty principle, quantum entanglement, EPR paradox, Bell inequality violation, abandonment of local realism, quantum information processing (including unitary transformation, single-qubit and two-qubit gates), no-cloning theorem, quantum teleportation, quantum measurement, quantum computation (including introduction to multi-qubit systems, universal family of gates, reversibility of computation), quantum algorithms (including Quantum Fourier Transform, Simon’s algorithm, extended Churh-Turing thesis, Shor’q quantum factoring algorithm, Grover’s quantum search algorithm), quantum observables, Shrodinger’s equation, qubits implementations, quantum complexity theory, adiabatic quantum computation, BQP, introduction to spin, within the following structure, encompassing comprehensive video didactic content as a reference for this EITC Certification.

Quantum information is the information of the state of a quantum system. It is the basic entity of study in quantum information theory, and can be manipulated using quantum information processing techniques. Quantum information refers to both the technical definition in terms of Von Neumann entropy and the general computational term.

Quantum information and computation is an interdisciplinary field that involves quantum mechanics, computer science, information theory, philosophy and cryptography among other fields. Its study is also relevant to disciplines such as cognitive science, psychology and neuroscience. Its main focus is in extracting information from matter at the microscopic scale. Observation in science is a fundamental distinctive criturium of reality and one of the most important ways of acquiring information. Hence measurement is required in order to quantify the observation, making it crucial to the scientific method. In quantum mechanics, due to the uncertainty principle, non-commuting observables cannot be precisely measured simultaneously, as an eigenstate in one basis is not an eigenstate in the other basis. As both variables are not simultaneously well defined, a quantum state can never contain definitive information about both variables. Due to this fundamental property of the measurement in quantum mechanics, this theory can be generally characterized as being nondeterministic in contrast in contrast to classical mechanics, which is fully deterministic. The indeterminism of quantum states characterizes information defined as states of quantum systems. In mathematical terms these states are in superpositions (linear combinations) of classical systems’ states.

As information is always encoded in the state of a physical system, it is physical in itself. While quantum mechanics deals with examining properties of matter at the microscopic level, quantum information science focuses on extracting information from those properties, and quantum computation manipulates and processes quantum information – performs logical operations – using quantum information processing techniques.

Quantum information, like classical information, can be processed using computers, transmitted from one location to another, manipulated with algorithms, and analyzed with computer science and mathematics. Just like the basic unit of classical information is the bit, quantum information deals with qubits, which can exists in superposition of 0 and 1 (simultaneously being somewhat true and false). Quantum information can also exist in so called entangled states, which manifest purely non-classical non-local correlations in their measurements, enabling applications such as the quantum teleportation. The level of entanglement can be measured using Von Neumann entropy, which is also a measure of quantum information. Recently, the field of quantum computing has become a very active research area because of the possibility to disrupt modern computation, communication, and cryptography.

The history of quantum information began at the turn of the 20th century when classical physics was revolutionized into quantum physics. The theories of classical physics were predicting absurdities such as the ultraviolet catastrophe, or electrons spiraling into the nucleus. At first these problems were brushed aside by adding ad hoc hypothesis to classical physics. Soon, it became apparent that a new theory must be created in order to make sense of these absurdities, and the theory of quantum mechanics was born.

Quantum mechanics was formulated by Schrödinger using wave mechanics and Heisenberg using matrix mechanics. The equivalence of these methods was proven later. Their formulations described the dynamics of microscopic systems but had several unsatisfactory aspects in describing measurement processes. Von Neumann formulated quantum theory using operator algebra in a way that it described measurement as well as dynamics. These studies emphasized the philosophical aspects of measurement rather than a quantitative approach to extracting information via measurements.

In 1960s, Stratonovich, Helstrom and Gordon proposed a formulation of optical communications using quantum mechanics. This was the first historical appearance of quantum information theory. They mainly studied error probabilities and channel capacities for communication. Later, Holevo obtained an upper bound of communication speed in the transmission of a classical message via a quantum channel.

In the 1970s, techniques for manipulating single-atom quantum states, such as the atom trap and the scanning tunneling microscope, began to be developed, making it possible to isolate single atoms and arrange them in arrays. Prior to these developments, precise control over single quantum systems was not possible, and experiments utilized coarser, simultaneous control over a large number of quantum systems. The development of viable single-state manipulation techniques led to increased interest in the field of quantum information and computation.

In the 1980s, interest arose in whether it might be possible to use quantum effects to disprove Einstein’s theory of relativity. If it were possible to clone an unknown quantum state, it would be possible to use entangled quantum states to transmit information faster than the speed of light, disproving Einstein’s theory. However, the no-cloning theorem showed that such cloning is impossible. The theorem was one of the earliest results of quantum information theory.

Development from cryptography

Despite all the excitement and interest over studying isolated quantum systems and trying to find a way to circumvent the theory of relativity, research in quantum information theory became stagnant in the 1980s. However, around the same time another avenue started dabbling into quantum information and computation: Cryptography. In a general sense, cryptography is the problem of doing communication or computation involving two or more parties who may not trust one another.

Bennett and Brassard developed a communication channel on which it is impossible eavesdrop without being detected, a way of communicating secretly at long distances using the BB84 quantum cryptographic protocol. The key idea was the use of the fundamental principle of quantum mechanics that observation disturbs the observed, and the introduction of a eavesdropper in a secure communication line will immediately let the two parties trying to communicate would know of the presence of the eavesdropper.

Development from computer science and mathematics

With the advent of Alan Turing’s revolutionary ideas of a programmable computer, or Turing machine, he showed that any real-world computation can be translated into an equivalent computation involving a Turing machine. This is known as the Church–Turing thesis.

Soon enough, the first computers were made and computer hardware grew at such a fast pace that the growth, through experience in production, was codified into an empirical relationship called Moore’s law. This ‘law’ is a projective trend that states that the number of transistors in an integrated circuit doubles every two years. As transistors began to become smaller and smaller in order to pack more power per surface area, quantum effects started to show up in the electronics resulting in inadvertent interference. This led to the advent of quantum computing, which used quantum mechanics to design algorithms.

At this point, quantum computers showed promise of being much faster than classical computers for certain specific problems. One such example problem was developed by David Deutsch and Richard Jozsa, known as the Deutsch–Jozsa algorithm. This problem however held little to no practical applications. Peter Shor in 1994 came up with a very important and practical problem, one of finding the prime factors of an integer. The discrete logarithm problem as it was called, could be solved efficiently on a quantum computer but not on a classical computer hence showing that quantum computers are more powerful than Turing machines.

Development from information theory

Around the time computer science was making a revolution, so was information theory and communication, through Claude Shannon. Shannon developed two fundamental theorems of information theory: noiseless channel coding theorem and noisy channel coding theorem. He also showed that error correcting codes could be used to protect information being sent.

Quantum information theory also followed a similar trajectory, Ben Schumacher in 1995 made an analogue to Shannon’s noiseless coding theorem using the qubit. A theory of error-correction also developed, which allows quantum computers to make efficient computations regardless of noise, and make reliable communication over noisy quantum channels.

Qubits and information theory

Quantum information differs strongly from classical information, epitomized by the bit, in many striking and unfamiliar ways. While the fundamental unit of classical information is the bit, the most basic unit of quantum information is the qubit. Classical information is measured using Shannon entropy, while the quantum mechanical analogue is Von Neumann entropy. A statistical ensemble of quantum mechanical systems is characterized by the density matrix. Many entropy measures in classical information theory can also be generalized to the quantum case, such as Holevo entropy and the conditional quantum entropy.

Unlike classical digital states (which are discrete), a qubit is continuous-valued, describable by a direction on the Bloch sphere. Despite being continuously valued in this way, a qubit is the smallest possible unit of quantum information, and despite the qubit state being continuous-valued, it is impossible to measure the value precisely. Five famous theorems describe the limits on manipulation of quantum information:

- no-teleportation theorem, which states that a qubit cannot be (wholly) converted into classical bits; that is, it cannot be fully “read”,
- no-cloning theorem, which prevents an arbitrary qubit from being copied,
- no-deleting theorem, which prevents an arbitrary qubit from being deleted,
- no-broadcasting theorem, which prevents an arbitrary qubit from being delivered to multiple recipients, although it can be transported from place to place (e.g. via quantum teleportation),
- no-hiding theorem, which demonstrates the conservation of quantum information,These theorems prove that quantum information within the universe is conserved and they open up unique possibilities in quantum information processing.

Quantum information processing

The state of a qubit contains all of its information. This state is frequently expressed as a vector on the Bloch sphere. This state can be changed by applying linear transformations or quantum gates to them. These unitary transformations are described as rotations on the Bloch Sphere. While classical gates correspond to the familiar operations of Boolean logic, quantum gates are physical unitary operators.

Due to the volatility of quantum systems and the impossibility of copying states, the storing of quantum information is much more difficult than storing classical information. Nevertheless, with the use of quantum error correction quantum information can still be reliably stored in principle. The existence of quantum error correcting codes has also led to the possibility of fault-tolerant quantum computation.

Classical bits can be encoded into and subsequently retrieved from configurations of qubits, through the use of quantum gates. By itself, a single qubit can convey no more than one bit of accessible classical information about its preparation. This is Holevo’s theorem. However, in superdense coding a sender, by acting on one of two entangled qubits, can convey two bits of accessible information about their joint state to a receiver.

Quantum information can be moved about, in a quantum channel, analogous to the concept of a classical communications channel. Quantum messages have a finite size, measured in qubits; quantum channels have a finite channel capacity, measured in qubits per second.

Quantum information, and changes in quantum information, can be quantitatively measured by using an analogue of Shannon entropy, called the von Neumann entropy.

In some cases quantum algorithms can be used to perform computations faster than in any known classical algorithm. The most famous example of this is Shor’s algorithm that can factor numbers in polynomial time, compared to the best classical algorithms that take sub-exponential time. As factorization is an important part of the safety of RSA encryption, Shor’s algorithm sparked the new field of post-quantum cryptography that tries to find encryption schemes that remain safe even when quantum computers are in play. Other examples of algorithms that demonstrate quantum supremacy include Grover’s search algorithm, where the quantum algorithm gives a quadratic speed-up over the best possible classical algorithm. The complexity class of problems efficiently solvable by a quantum computer is known as BQP.

Quantum key distribution (QKD) allows unconditionally secure transmission of classical information, unlike classical encryption, which can always be broken in principle, if not in practice. Do note that certain subtle points regarding the safety of QKD are still hotly debated.

The study of all of the above topics and differences comprises quantum information theory.

Relation to quantum mechanics

Quantum mechanics is the study of how microscopic physical systems change dynamically in nature. In the field of quantum information theory, the quantum systems studied are abstracted away from any real world counterpart. A qubit might for instance physically be a photon in a linear optical quantum computer, an ion in a trapped ion quantum computer, or it might be a large collection of atoms as in a superconducting quantum computer. Regardless of the physical implementation, the limits and features of qubits implied by quantum information theory hold as all these systems are mathematically described by the same apparatus of density matrices over the complex numbers. Another important difference with quantum mechanics is that, while quantum mechanics often studies infinite-dimensional systems such as a harmonic oscillator, quantum information theory concerns both with continuous-variable systems and finite-dimensional systems.

Quantum computation

Quantum computing is a type of computation that harnesses the collective properties of quantum states, such as superposition, interference, and entanglement, to perform calculations. The devices that perform quantum computations are known as quantum computers.: I-5 Though current quantum computers are too small to outperform usual (classical) computers for practical applications, they are believed to be capable of solving certain computational problems, such as integer factorization (which underlies RSA encryption), substantially faster than classical computers. The study of quantum computing is a subfield of quantum information science.

Quantum computing began in 1980 when physicist Paul Benioff proposed a quantum mechanical model of the Turing machine. Richard Feynman and Yuri Manin later suggested that a quantum computer had the potential to simulate things a classical computer could not feasibly do. In 1994, Peter Shor developed a quantum algorithm for factoring integers with the potential to decrypt RSA-encrypted communications. In 1998 Isaac Chuang, Neil Gershenfeld and Mark Kubinec created the first two-qubit quantum computer that could perform computations. Despite ongoing experimental progress since the late 1990s, most researchers believe that “fault-tolerant quantum computing [is] still a rather distant dream.” In recent years, investment in quantum computing research has increased in the public and private sectors. On 23 October 2019, Google AI, in partnership with the U.S. National Aeronautics and Space Administration (NASA), claimed to have performed a quantum computation that was infeasible on any classical computer, but whether this claim was or is still valid is a topic of active research.

There are several types of quantum computers (also known as quantum computing systems), including the quantum circuit model, quantum Turing machine, adiabatic quantum computer, one-way quantum computer, and various quantum cellular automata. The most widely used model is the quantum circuit, based on the quantum bit, or “qubit”, which is somewhat analogous to the bit in classical computation. A qubit can be in a 1 or 0 quantum state, or in a superposition of the 1 and 0 states. When it is measured, however, it is always 0 or 1; the probability of either outcome depends on the qubit’s quantum state immediately prior to measurement.

Efforts towards building a physical quantum computer focus on technologies such as transmons, ion traps and topological quantum computers, which aim to create high-quality qubits.: 2–13 These qubits may be designed differently, depending on the full quantum computer’s computing model, whether quantum logic gates, quantum annealing, or adiabatic quantum computation. There are currently a number of significant obstacles to constructing useful quantum computers. It is particularly difficult to maintain qubits’ quantum states, as they suffer from quantum decoherence and state fidelity. Quantum computers therefore require error correction.

Any computational problem that can be solved by a classical computer can also be solved by a quantum computer. Conversely, any problem that can be solved by a quantum computer can also be solved by a classical computer, at least in principle given enough time. In other words, quantum computers obey the Church–Turing thesis. This means that while quantum computers provide no additional advantages over classical computers in terms of computability, quantum algorithms for certain problems have significantly lower time complexities than corresponding known classical algorithms. Notably, quantum computers are believed to be able to quickly solve certain problems that no classical computer could solve in any feasible amount of time—a feat known as “quantum supremacy.” The study of the computational complexity of problems with respect to quantum computers is known as quantum complexity theory.

The prevailing model of quantum computation describes the computation in terms of a network of quantum logic gates. This model can be thought of as an abstract linear-algebraic generalization of a classical circuit. Since this circuit model obeys quantum mechanics, a quantum computer capable of efficiently running these circuits is believed to be physically realizable.

A memory consisting of n bits of information has 2^n possible states. A vector representing all memory states thus has 2^n entries (one for each state). This vector is viewed as a probability vector and represents the fact that the memory is to be found in a particular state.

In the classical view, one entry would have a value of 1 (i.e. a 100% probability of being in this state) and all other entries would be zero.

In quantum mechanics, probability vectors can be generalized to density operators. The quantum state vector formalism is usually introduced first because it is conceptually simpler, and because it can be used instead of the density matrix formalism for pure states, where the whole quantum system is known.

a quantum computation can be described as a network of quantum logic gates and measurements. However, any measurement can be deferred to the end of quantum computation, though this deferment may come at a computational cost, so most quantum circuits depict a network consisting only of quantum logic gates and no measurements.

Any quantum computation (which is, in the above formalism, any unitary matrix over n qubits) can be represented as a network of quantum logic gates from a fairly small family of gates. A choice of gate family that enables this construction is known as a universal gate set, since a computer that can run such circuits is a universal quantum computer. One common such set includes all single-qubit gates as well as the CNOT gate from above. This means any quantum computation can be performed by executing a sequence of single-qubit gates together with CNOT gates. Though this gate set is infinite, it can be replaced with a finite gate set by appealing to the Solovay-Kitaev theorem.

Quantum algorithms

Progress in finding quantum algorithms typically focuses on this quantum circuit model, though exceptions like the quantum adiabatic algorithm exist. Quantum algorithms can be roughly categorized by the type of speedup achieved over corresponding classical algorithms.

Quantum algorithms that offer more than a polynomial speedup over the best known classical algorithm include Shor’s algorithm for factoring and the related quantum algorithms for computing discrete logarithms, solving Pell’s equation, and more generally solving the hidden subgroup problem for abelian finite groups. These algorithms depend on the primitive of the quantum Fourier transform. No mathematical proof has been found that shows that an equally fast classical algorithm cannot be discovered, although this is considered unlikely.[self-published source?] Certain oracle problems like Simon’s problem and the Bernstein–Vazirani problem do give provable speedups, though this is in the quantum query model, which is a restricted model where lower bounds are much easier to prove and doesn’t necessarily translate to speedups for practical problems.

Other problems, including the simulation of quantum physical processes from chemistry and solid-state physics, the approximation of certain Jones polynomials, and the quantum algorithm for linear systems of equations have quantum algorithms appearing to give super-polynomial speedups and are BQP-complete. Because these problems are BQP-complete, an equally fast classical algorithm for them would imply that no quantum algorithm gives a super-polynomial speedup, which is believed to be unlikely.

Some quantum algorithms, like Grover’s algorithm and amplitude amplification, give polynomial speedups over corresponding classical algorithms. Though these algorithms give comparably modest quadratic speedup, they are widely applicable and thus give speedups for a wide range of problems. Many examples of provable quantum speedups for query problems are related to Grover’s algorithm, including Brassard, Høyer, and Tapp’s algorithm for finding collisions in two-to-one functions, which uses Grover’s algorithm, and Farhi, Goldstone, and Gutmann’s algorithm for evaluating NAND trees, which is a variant of the search problem.

Cryptographic applications

A notable application of quantum computation is for attacks on cryptographic systems that are currently in use. Integer factorization, which underpins the security of public key cryptographic systems, is believed to be computationally infeasible with an ordinary computer for large integers if they are the product of few prime numbers (e.g., products of two 300-digit primes). By comparison, a quantum computer could efficiently solve this problem using Shor’s algorithm to find its factors. This ability would allow a quantum computer to break many of the cryptographic systems in use today, in the sense that there would be a polynomial time (in the number of digits of the integer) algorithm for solving the problem. In particular, most of the popular public key ciphers are based on the difficulty of factoring integers or the discrete logarithm problem, both of which can be solved by Shor’s algorithm. In particular, the RSA, Diffie–Hellman, and elliptic curve Diffie–Hellman algorithms could be broken. These are used to protect secure Web pages, encrypted email, and many other types of data. Breaking these would have significant ramifications for electronic privacy and security.

Identifying cryptographic systems that may be secure against quantum algorithms is an actively researched topic under the field of post-quantum cryptography. Some public-key algorithms are based on problems other than the integer factorization and discrete logarithm problems to which Shor’s algorithm applies, like the McEliece cryptosystem based on a problem in coding theory. Lattice-based cryptosystems are also not known to be broken by quantum computers, and finding a polynomial time algorithm for solving the dihedral hidden subgroup problem, which would break many lattice based cryptosystems, is a well-studied open problem. It has been proven that applying Grover’s algorithm to break a symmetric (secret key) algorithm by brute force requires time equal to roughly 2n/2 invocations of the underlying cryptographic algorithm, compared with roughly 2n in the classical case, meaning that symmetric key lengths are effectively halved: AES-256 would have the same security against an attack using Grover’s algorithm that AES-128 has against classical brute-force search (see Key size).

Quantum cryptography could potentially fulfill some of the functions of public key cryptography. Quantum-based cryptographic systems could, therefore, be more secure than traditional systems against quantum hacking.

Search problems

The most well-known example of a problem admitting a polynomial quantum speedup is unstructured search, finding a marked item out of a list of n items in a database. This can be solved by Grover’s algorithm using O(sqrt(n)) queries to the database, quadratically fewer than the Omega(n) queries required for classical algorithms. In this case, the advantage is not only provable but also optimal: it has been shown that Grover’s algorithm gives the maximal possible probability of finding the desired element for any number of oracle lookups.

Problems that can be addressed with Grover’s algorithm have the following properties:

- There is no searchable structure in the collection of possible answers,
- The number of possible answers to check is the same as the number of inputs to the algorithm, and
- There exists a boolean function that evaluates each input and determines whether it is the correct answer

For problems with all these properties, the running time of Grover’s algorithm on a quantum computer scales as the square root of the number of inputs (or elements in the database), as opposed to the linear scaling of classical algorithms. A general class of problems to which Grover’s algorithm can be applied is Boolean satisfiability problem, where the database through which the algorithm iterates is that of all possible answers. An example and (possible) application of this is a password cracker that attempts to guess a password. Symmetric ciphers such as Triple DES and AES are particularly vulnerable to this kind of attack.[citation needed] This application of quantum computing is a major interest of government agencies.

Simulation of quantum systems

Since chemistry and nanotechnology rely on understanding quantum systems, and such systems are impossible to simulate in an efficient manner classically, many believe quantum simulation will be one of the most important applications of quantum computing. Quantum simulation could also be used to simulate the behavior of atoms and particles at unusual conditions such as the reactions inside a collider. Quantum simulations might be used to predict future paths of particles and protons under superposition in the double-slit experiment.[citation needed] About 2% of the annual global energy output is used for nitrogen fixation to produce ammonia for the Haber process in the agricultural fertilizer industry while naturally occurring organisms also produce ammonia. Quantum simulations might be used to understand this process increasing production.

Quantum annealing and adiabatic optimization

Quantum annealing or Adiabatic quantum computation relies on the adiabatic theorem to undertake calculations. A system is placed in the ground state for a simple Hamiltonian, which is slowly evolved to a more complicated Hamiltonian whose ground state represents the solution to the problem in question. The adiabatic theorem states that if the evolution is slow enough the system will stay in its ground state at all times through the process.

Machine learning

Since quantum computers can produce outputs that classical computers cannot produce efficiently, and since quantum computation is fundamentally linear algebraic, some express hope in developing quantum algorithms that can speed up machine learning tasks. For example, the quantum algorithm for linear systems of equations, or “HHL Algorithm”, named after its discoverers Harrow, Hassidim, and Lloyd, is believed to provide speedup over classical counterparts. Some research groups have recently explored the use of quantum annealing hardware for training Boltzmann machines and deep neural networks.

Computational biology

In the field of computational biology, quantum computing has played a big role in solving many biological problems. One of the well-known examples would be in computational genomics and how computing has drastically reduced the time to sequence a human genome. Given how computational biology is using generic data modeling and storage, its applications to computational biology are expected to arise as well.

Computer-aided drug design and generative chemistry

Deep generative chemistry models emerge as powerful tools to expedite drug discovery. However, the immense size and complexity of the structural space of all possible drug-like molecules pose significant obstacles, which could be overcome in the future by quantum computers. Quantum computers are naturally good for solving complex quantum many-body problems and thus may be instrumental in applications involving quantum chemistry. Therefore, one can expect that quantum-enhanced generative models including quantum GANs may eventually be developed into ultimate generative chemistry algorithms. Hybrid architectures combining quantum computers with deep classical networks, such as Quantum Variational Autoencoders, can already be trained on commercially available annealers and used to generate novel drug-like molecular structures.

Developing physical quantum computers

Challenges

There are a number of technical challenges in building a large-scale quantum computer. Physicist David DiVincenzo has listed these requirements for a practical quantum computer:

- Physically scalable to increase the number of qubits,
- Qubits that can be initialized to arbitrary values,
- Quantum gates that are faster than decoherence time,
- Universal gate set,
- Qubits that can be read easily.

Sourcing parts for quantum computers is also very difficult. Many quantum computers, like those constructed by Google and IBM, need helium-3, a nuclear research byproduct, and special superconducting cables made only by the Japanese company Coax Co.

The control of multi-qubit systems requires the generation and coordination of a large number of electrical signals with tight and deterministic timing resolution. This has led to the development of quantum controllers which enable interfacing with the qubits. Scaling these systems to support a growing number of qubits is an additional challenge.

Quantum decoherence

One of the greatest challenges involved with constructing quantum computers is controlling or removing quantum decoherence. This usually means isolating the system from its environment as interactions with the external world cause the system to decohere. However, other sources of decoherence also exist. Examples include the quantum gates, and the lattice vibrations and background thermonuclear spin of the physical system used to implement the qubits. Decoherence is irreversible, as it is effectively non-unitary, and is usually something that should be highly controlled, if not avoided. Decoherence times for candidate systems in particular, the transverse relaxation time T2 (for NMR and MRI technology, also called the dephasing time), typically range between nanoseconds and seconds at low temperature. Currently, some quantum computers require their qubits to be cooled to 20 millikelvin (usually using a dilution refrigerator) in order to prevent significant decoherence. A 2020 study argues that ionizing radiation such as cosmic rays can nevertheless cause certain systems to decohere within milliseconds.

As a result, time-consuming tasks may render some quantum algorithms inoperable, as maintaining the state of qubits for a long enough duration will eventually corrupt the superpositions.

These issues are more difficult for optical approaches as the timescales are orders of magnitude shorter and an often-cited approach to overcoming them is optical pulse shaping. Error rates are typically proportional to the ratio of operating time to decoherence time, hence any operation must be completed much more quickly than the decoherence time.

As described in the Quantum threshold theorem, if the error rate is small enough, it is thought to be possible to use quantum error correction to suppress errors and decoherence. This allows the total calculation time to be longer than the decoherence time if the error correction scheme can correct errors faster than decoherence introduces them. An often cited figure for the required error rate in each gate for fault-tolerant computation is 10−3, assuming the noise is depolarizing.

Meeting this scalability condition is possible for a wide range of systems. However, the use of error correction brings with it the cost of a greatly increased number of required qubits. The number required to factor integers using Shor’s algorithm is still polynomial, and thought to be between L and L2, where L is the number of digits in the number to be factored; error correction algorithms would inflate this figure by an additional factor of L. For a 1000-bit number, this implies a need for about 104 bits without error correction. With error correction, the figure would rise to about 107 bits. Computation time is about L2 or about 107 steps and at 1 MHz, about 10 seconds.

A very different approach to the stability-decoherence problem is to create a topological quantum computer with anyons, quasi-particles used as threads and relying on braid theory to form stable logic gates.

Quantum supremacy

Quantum supremacy is a term coined by John Preskill referring to the engineering feat of demonstrating that a programmable quantum device can solve a problem beyond the capabilities of state-of-the-art classical computers. The problem need not be useful, so some view the quantum supremacy test only as a potential future benchmark.

In October 2019, Google AI Quantum, with the help of NASA, became the first to claim to have achieved quantum supremacy by performing calculations on the Sycamore quantum computer more than 3,000,000 times faster than they could be done on Summit, generally considered the world’s fastest computer. This claim has been subsequently challenged: IBM has stated that Summit can perform samples much faster than claimed, and researchers have since developed better algorithms for the sampling problem used to claim quantum supremacy, giving substantial reductions to or the closing of the gap between Sycamore and classical supercomputers.

In December 2020, a group at USTC implemented a type of Boson sampling on 76 photons with a photonic quantum computer Jiuzhang to demonstrate quantum supremacy. The authors claim that a classical contemporary supercomputer would require a computational time of 600 million years to generate the number of samples their quantum processor can generate in 20 seconds. On November 16, 2021 at the quantum computing summit IBM presented a 127-qubit microprocessor named IBM Eagle.

Physical implementations

For physically implementing a quantum computer, many different candidates are being pursued, among them (distinguished by the physical system used to realize the qubits):

- Superconducting quantum computing (qubit implemented by the state of small superconducting circuits, Josephson junctions)
- Trapped ion quantum computer (qubit implemented by the internal state of trapped ions)
- Neutral atoms in optical lattices (qubit implemented by internal states of neutral atoms trapped in an optical lattice)
- Quantum dot computer, spin-based (e.g. the Loss-DiVincenzo quantum computer) (qubit given by the spin states of trapped electrons)
- Quantum dot computer, spatial-based (qubit given by electron position in double quantum dot)
- Quantum computing using engineered quantum wells, which could in principle enable the construction of quantum computers that operate at room temperature
- Coupled quantum wire (qubit implemented by a pair of quantum wires coupled by a quantum point contact)
- Nuclear magnetic resonance quantum computer (NMRQC) implemented with the nuclear magnetic resonance of molecules in solution, where qubits are provided by nuclear spins within the dissolved molecule and probed with radio waves
- Solid-state NMR Kane quantum computers (qubit realized by the nuclear spin state of phosphorus donors in silicon)
- Electrons-on-helium quantum computers (qubit is the electron spin)
- Cavity quantum electrodynamics (CQED) (qubit provided by the internal state of trapped atoms coupled to high-finesse cavities)
- Molecular magnet (qubit given by spin states)
- Fullerene-based ESR quantum computer (qubit based on the electronic spin of atoms or molecules encased in fullerenes)
- Nonlinear optical quantum computer (qubits realized by processing states of different modes of light through both linear and nonlinear elements)
- Linear optical quantum computer (qubits realized by processing states of different modes of light through linear elements e.g. mirrors, beam splitters and phase shifters)
- Diamond-based quantum computer (qubit realized by the electronic or nuclear spin of nitrogen-vacancy centers in diamond)
- Bose-Einstein condensate-based quantum computer
- Transistor-based quantum computer – string quantum computers with entrainment of positive holes using an electrostatic trap
- Rare-earth-metal-ion-doped inorganic crystal based quantum computers (qubit realized by the internal electronic state of dopants in optical fibers)
- Metallic-like carbon nanospheres-based quantum computers
- The large number of candidates demonstrates that quantum computing, despite rapid progress, is still in its infancy.

There are a number of quantum computing models, distinguished by the basic elements in which the computation is decomposed. For practical implementations, the four relevant models of computation are:

- Quantum gate array (computation decomposed into a sequence of few-qubit quantum gates)
- One-way quantum computer (computation decomposed into a sequence of one-qubit measurements applied to a highly entangled initial state or cluster state)
- Adiabatic quantum computer, based on quantum annealing (computation decomposed into a slow continuous transformation of an initial Hamiltonian into a final Hamiltonian, whose ground states contain the solution)
- Topological quantum computer (computation decomposed into the braiding of anyons in a 2D lattice)

The quantum Turing machine is theoretically important but the physical implementation of this model is not feasible. All four models of computation have been shown to be equivalent; each can simulate the other with no more than polynomial overhead.

To acquaint yourself in-detail with the certification curriculum you can expand and analyze the table below.

The EITC/QI/QIF Quantum Information Fundamentals Certification Curriculum references open-access didactic materials in a video form. Learning process is divided into a step-by-step structure (programmes -> lessons -> topics) covering relevant curriculum parts. Unlimited consultancy with domain experts are also provided.

For details on the Certification procedure check How it Works.

### Main lecture notes

U. Vazirani lecture notes:

https://people.eecs.berkeley.edu/~vazirani/quantum.html

### Supportive lecture notes

L. Jacak et al. lecture notes (with supplementary materials):

https://drive.google.com/open?id=1cl27qPRE8FyB3TvvMGp9mwBFc-Qe-nlG

https://drive.google.com/open?id=1nX_jIheCHSRB7pYAjIdVD0ab6vUtk7tG

### Main supportive textbook

Quantum Computation & Quantum Information textbook (Nielsen, Chuang):

http://mmrc.amss.cas.cn/tlb/201702/W020170224608149940643.pdf

### Additional lecture notes

J. Preskill lecture notes:

http://theory.caltech.edu/~preskill/ph219/index.html#lecture

A. Childs lecture notes:

http://www.math.uwaterloo.ca/~amchilds/teaching/w08/co781.html

S. Aaronson lecture notes:

https://scottaaronson.blog/?p=3943

R. de Wolf lecture notes:

https://arxiv.org/abs/1907.09415

### Other recommended textbooks

Classical and Quantum Computation (Kitaev, Shen, Vyalyi)

http://www.amazon.com/exec/obidos/tg/detail/-/082182161X/qid=1064887386/sr=8-3/ref=sr_8_3/102-1370066-0776166

Quantum Computing Since Democritus (Aaronson)

http://www.amazon.com/Quantum-Computing-since-Democritus-Aaronson/dp/0521199565

The Theory of Quantum Information (Watrous)

https://www.amazon.com/Theory-Quantum-Information-John-Watrous/dp/1107180562/

Quantum Information Theory (Wilde)

http://www.amazon.com/Quantum-Information-Theory-Mark-Wilde/dp/1107034256