The Bloch sphere is a representation of a qubit, the fundamental building block of quantum computers. In Quantum mechanics, the Bloch sphere is a geometrical representation of the Pure state space of a two-level quantum mechanical system named after A qubit is not to be confused with a Cubit, which is an ancient measure of length

A quantum computer is a hypothetical device for computation that makes direct use of distinctively quantum mechanical phenomena, such as superposition and entanglement, to perform operations on data. Computation is a general term for any type of Information processing. Quantum mechanics is the study of mechanical systems whose dimensions are close to the Atomic scale such as Molecules Atoms Electrons A phenomenon (from Greek φαινόμενoν, pl φαινόμενα - phenomena) is any observable occurrence Quantum superposition is the fundamental law of Quantum mechanics. Quantum entanglement is a quantum mechanical Phenomenon in which the Quantum states of two or more objects are linked together so that one object In a classical (or conventional) computer, information is stored as bits; in a quantum computer, it is stored as qubits (quantum binary digits). A bit is a binary digit, taking a value of either 0 or 1 Binary digits are a basic unit of Information storage and communication A qubit is not to be confused with a Cubit, which is an ancient measure of length The basic principle of quantum computation is that the quantum properties can be used to represent and structure data, and that quantum mechanisms can be devised and built to perform operations with this data. In Computer science, an instruction is a single operation of a processor defined by an Instruction set architecture. ^[1]

Although quantum computing is still in its infancy, experiments have been carried out in which quantum computational operations were executed on a very small number of qubits. A qubit is not to be confused with a Cubit, which is an ancient measure of length Research in both theoretical and practical areas continues at a frantic pace, and many national government and military funding agencies support quantum computing research to develop quantum computers for both civilian and national security purposes, such as cryptanalysis. Cryptanalysis (from the Greek kryptós, "hidden" and analýein, "to loosen" or "to untie" is the study of methods for ^[2]

If large-scale quantum computers can be built, they will be able to solve certain problems much faster than any of our current classical computers (for example Shor's algorithm). Shor's algorithm, first introduced by mathematician Peter Shor, is a quantum Algorithm for Integer factorization. Quantum computers are different from other computers such as DNA computers and traditional computers based on transistors. A computer is a Machine that manipulates data according to a list of instructions. DNA computing is a form of Computing which uses DNA, Biochemistry and Molecular biology, instead of the traditional silicon-based Computer In Electronics, a transistor is a Semiconductor device commonly used to amplify or switch electronic signals Some computing architectures such as optical computers ^[3] may use classical superposition of electromagnetic waves. An optical computer is a computer that uses light instead of electricity (i Without some specifically quantum mechanical resources such as entanglement, they have been shown to obtain no advantage over classical computers, entanglement is necessary for the exponential (or quadratic) advantage theoretically obtainable by quantum computer. Quantum entanglement is a quantum mechanical Phenomenon in which the Quantum states of two or more objects are linked together so that one object

The basis of quantum computing

A classical computer has a memory made up of bits, where each bit holds either a one or a zero. A bit is a binary digit, taking a value of either 0 or 1 Binary digits are a basic unit of Information storage and communication A quantum computer maintains a sequence of qubits. A qubit is not to be confused with a Cubit, which is an ancient measure of length A single qubit can hold a one, a zero, or, crucially, a quantum superposition of these; moreover, a pair of qubits can be in a quantum superposition of 4 states, and three qubits in a superposition of 8. Quantum superposition is the fundamental law of Quantum mechanics. In general a quantum computer with n qubits can be in up to 2ⁿ different states simultaneously (this compares to a normal computer that can only be in one of these 2ⁿ states at any one time). A quantum computer operates by manipulating those qubits with a fixed sequence of quantum logic gates. A quantum gate or quantum logic gate is a basic Quantum circuit operating on a small number of Qubits They are the analogues for Quantum computers The sequence of gates to be applied is called a quantum algorithm.

An example of an implementation of qubits for a quantum computer could start with the use of particles with two spin states: "up" and "down" (typically written and ). In Quantum mechanics, spin is a fundamental property of atomic nuclei, Hadrons and Elementary particles For particles with non-zero spin But in fact any system possessing an observable quantity A which is conserved under time evolution and such that A has at least two discrete and sufficiently spaced consecutive eigenvalues, is a suitable candidate for implementing a qubit. In Physics, particularly in Quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical In Mathematics, given a Linear transformation, an of that linear transformation is a nonzero vector which when that transformation is applied to it changes This is true because any such system can be mapped onto an effective spin-1/2 system. In Quantum mechanics, spin is an intrinsic property of all elementary particles.

Bits vs. Qubits

Consider first a classical computer that operates on a 3-bit register. In Computer architecture, a processor register is a small amount of storage available on the CPU whose contents can be accessed more quickly than storage At any given time, the bits in the register are in a definite state, such as 101. In a quantum computer, however, the qubits can be in a superposition of all the classically allowed states. In fact, the register is described by a wavefunction:

Qubits are made up of controlled particles and the means of control (e. A wave function or wavefunction is a mathematical tool used in Quantum mechanics to describe any physical system g. devices that trap particles and switch them from one state to another). ^[4]

where the coefficients a, b, c,. . . , h are complex numbers whose amplitudes squared are the probabilities to measure the qubits in each state- for example, is the probability to measure the register in the state 010. Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted It is important that these numbers are complex, because the phases of the numbers can constructively and destructively interfere with one another; this is an important feature for quantum algorithms. The phase of an oscillation or wave is the fraction of a complete cycle corresponding to an offset in the displacement from a specified reference point at time t = 0 ^[5] The basis made up of 0 and 1 (true and false) is called the computational basis, but other bases can be used. Another common basis, used for example in measurement based quantum computation is the Hadamard basis of and . Any two orthogonal vectors can be used as a basis. In Mathematics, two Vectors are orthogonal if they are Perpendicular, i

Recording the state of a quantum register requires an exponential number of complex numbers (the 3-qubit register above requires 2³ = 8 complex numbers). A quantum register (also known as a qregister is the quantum mechanical analogue of a classical Processor register. The number of classical bits required even to estimate the complex numbers of some quantum state grows exponentially with the number of qubits. For a 300-qubit quantum register, somewhere on the order of classical registers are required, more than there are atoms in the observable universe. In Big Bang Cosmology, the observable universe is the region of space bounded by a Sphere, centered on the observer that is small enough that ^[6]

Initialization, execution and termination

In our example, the contents of the qubit registers can be thought of as an 8-dimensional complex vector. In Linear algebra, a coordinate vector is an explicit representation of a vector in an abstract vector space as an ordered list of numbers or equivalently as an An algorithm for a quantum computer must initialize this vector in some specified form (dependent on the design of the quantum computer). In each step of the algorithm, that vector is modified by multiplying it by a unitary matrix. In Mathematics, a unitary matrix is an n by n complex matrix U satisfying the condition U^* U = UU^* The matrix is determined by the physics of the device. The unitary character of the matrix ensures the matrix is invertible (so each step is reversible). Reversible computing includes any Computational process that is (at least to some close approximation Reversible, i

Upon termination of the algorithm, the 8-dimensional complex vector stored in the register must be somehow read off from the qubit register by a quantum measurement. The framework of Quantum mechanics requires a careful definition of measurement, and a thorough discussion of its practical and philosophical implications However, by the laws of quantum mechanics, that measurement will yield a random 3-bit string (and it will destroy the stored state as well). Randomness is a lack of order Purpose, cause, or predictability This random string can be used in computing the value of a function because (by design) the probability distribution of the measured output bitstring is skewed in favor of one particular value: the correct value of the function. In Probability theory and Statistics, a probability distribution identifies either the probability of each value of an unidentified Random variable By repeated runs of the quantum computer and measurement of the output, the correct value can be determined, to a high probability, by majority polling of the outputs. In brief, quantum computations are probabilistic; see quantum circuit for a more precise formulation. In Quantum information theory, a quantum circuit is a model for Quantum computation in which a computation is a sequence of reversible transformations

For more details on the sequences of operations used for various algorithms, see universal quantum computer, Shor's algorithm, Grover's algorithm, Deutsch-Jozsa algorithm, quantum Fourier transform, quantum gate, quantum adiabatic algorithm and quantum error correction. In Quantum mechanics, the universal quantum computer or universal quantum Turing machine (UQTM is a theoretical machine that combines both Church-Turing Shor's algorithm, first introduced by mathematician Peter Shor, is a quantum Algorithm for Integer factorization. Grover's algorithm is a Quantum algorithm for searching an unsorted Database with N entries in O(N1/2 time and using The Deutsch-Jozsa algorithm is a quantum algorithm, proposed by David Deutsch and Richard Jozsa in 1992 with improvements by R The quantum Fourier transform is the Discrete Fourier transform with a particular decomposition into a product of simpler unitary matrices. A quantum gate or quantum logic gate is a basic Quantum circuit operating on a small number of Qubits They are the analogues for Quantum computers Adiabatic Quantum Computation relies on the Adiabatic theorem to do calculations Quantum error correction is used in quantum computing to protect Quantum information from errors due to Decoherence and other quantum Noise. Also refer to the growing field of quantum programming. Quantum programming is a set of computer programming languages that allow the expression of quantum algorithms using high-level constructs

The power of quantum computers

Integer factorization is believed to be computationally infeasible with an ordinary computer for large integers that are the product of only a few prime numbers (e. In Mathematics, a prime number (or a prime) is a Natural number which has exactly two distinct natural number Divisors 1 g. , products of two 300-digit primes). ^[7] By comparison, a quantum computer could efficiently solve this problem using Shor's algorithm to find its factors. Shor's algorithm, first introduced by mathematician Peter Shor, is a quantum Algorithm for Integer factorization. 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 bits of the integer) algorithm for solving the problem. Cryptography (or cryptology; from Greek grc κρυπτός kryptos, "hidden secret" and grc γράφω gráphō, "I write" In particular, most of the popular public key ciphers are based on the difficulty of factoring integers (or the related discrete logarithm problem which can also be solved by Shor's algorithm), including forms of RSA. Public-key cryptography, also known as asymmetric cryptography, is a form of Cryptography in which the key used to encrypt a message differs from the key In Mathematics, specifically in Abstract algebra and its applications discrete logarithms are group-theoretic analogues of ordinary Logarithms In Cryptography, RSA is an Algorithm for Public-key cryptography. 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. The only way to increase the security of an algorithm like RSA would be to increase the key size and hope that an adversary does not have the resources to build and use a powerful enough quantum computer. In Cryptography, RSA is an Algorithm for Public-key cryptography.

A way out of this dilemma would be to use some kind of quantum cryptography. Quantum cryptography, or quantum key distribution (QKD uses Quantum mechanics to guarantee secure communication There are also some digital signature schemes that are believed to be secure against quantum computers. A digital signature or digital signature scheme is a type of asymmetric cryptography used to simulate the security properties of a handwritten Signature See for instance Lamport signatures. In Cryptography, a Lamport signature or Lamport one-time signature scheme is a method for constructing a Digital signature.

This dramatic advantage of quantum computers has only been discovered for factorization and discrete logarithms so far. 2008 ( MMVIII) is the current year in accordance with the Gregorian calendar, a Leap year that started on Tuesday of the Common However, there is no proof that the advantage is real: an equally fast classical algorithm may still be discovered. There is one other problem where quantum computers have a smaller, though significant (quadratic) advantage. It is quantum database search, and can be solved by Grover's algorithm. Grover's algorithm is a Quantum algorithm for searching an unsorted Database with N entries in O(N1/2 time and using In this case the advantage is provable. This establishes beyond doubt that (ideal) quantum computers are superior to classical computers for at least one problem.

Consider a problem that has these four properties:

1. The only way to solve it is to guess answers repeatedly and check them,
2. There are n possible answers to check,
3. Every possible answer takes the same amount of time to check, and
4. There are no clues about which answers might be better: generating possibilities randomly is just as good as checking them in some special order.

An example of this is a password cracker that attempts to guess the password for an encrypted file (assuming that the password has a maximum possible length). Password cracking is the process of recovering Passwords from data that has been stored in or transmitted by a Computer system.

For problems with all four properties, the time for a quantum computer to solve this will be proportional to the square root of n (it would take an average of (n + 1)/2 guesses to find the answer using a classical computer. ) That can be a very large speedup, reducing some problems from years to seconds. It can be used to attack symmetric ciphers such as Triple DES and AES by attempting to guess the secret key. Symmetric-key algorithms are a class of Algorithms for Cryptography that use trivially related often identical Cryptographic keys for both decryption In Cryptography, Triple DES is a Block cipher formed from the Data Encryption Standard (DES Cipher by using it three times In Cryptography, the Advanced Encryption Standard ( AES) also known as Rijndael, is a Block cipher adopted as an Encryption Regardless of whether any of these problems can be shown to have an advantage on a quantum computer, they nonetheless will always have the advantage of being an excellent tool for studying quantum mechanical interactions, which of itself is an enormous value to the scientific community.

Grover's algorithm can also be used to obtain a quadratic speed-up [over a brute-force search] for a class of problems known as NP-complete. Grover's algorithm is a Quantum algorithm for searching an unsorted Database with N entries in O(N1/2 time and using In Computational complexity theory, the Complexity class NP-complete (abbreviated NP-C or NPC) is a class of problems having two properties

Problems and practicality issues

There are a number of practical difficulties in building a quantum computer, and thus far quantum computers have only solved trivial problems. David DiVincenzo, of IBM, listed the following requirements for a practical quantum computer:^[8]

• scalable physically to increase the number of qubits
• qubits can be initialized to arbitrary values
• quantum gates faster than decoherence time
• universal gate set
• qubits can be read easily

To summarize the problems from the perspective of an engineer, one needs to solve the challenge of building a system which is isolated from everything except the measurement and manipulation mechanism. In Quantum mechanics, quantum decoherence is the mechanism by which quantum systems interact with their environments to exhibit probabilistically additive behavior Furthermore, one needs to be able to turn off the coupling of the qubits to the measurement so as to not decohere the qubits while performing operations on them.

Quantum decoherence

One major problem is keeping the components of the computer in a coherent state, as the slightest interaction with the external world would cause the system to decohere. In Quantum mechanics, quantum decoherence is the mechanism by which quantum systems interact with their environments to exhibit probabilistically additive behavior This effect causes the unitary character (and more specifically, the invertibility) of quantum computational steps to be violated. Decoherence times for candidate systems, in particular the transverse relaxation time T₂ (terminology used in NMR and MRI technology, also called the dephasing time), typically range between nanoseconds and seconds at low temperature. ^[5] The issue for optical approaches are more difficult as these timescales are orders of magnitude lower and an often cited approach to overcome it uses an optical pulse shaping approach. 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.

If the error rate is small enough, it is thought to be possible to use quantum error correction, which corrects errors due to decoherence, thereby allowing the total calculation time to be longer than the decoherence time. An often cited (but rather arbitrary) figure for required error rate in each gate is 10⁻⁴. This implies that each gate must be able to perform its task 10,000 times faster than the decoherence time of the system.

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 bits 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 about 10⁴ qubits without error correction. ^[9] With error correction, the figure would rise to about 10⁷ qubits. Note that computation time is about L² or about 10⁷ steps and on 1 MHz, about 10 seconds. The hertz (symbol Hz) is a measure of Frequency, informally defined as the number of events occurring per Second. The second ( SI symbol s) sometimes abbreviated sec, is the name of a unit of Time, and is the International System of Units

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. A topological quantum computer is a theoretical Quantum computer that employs two-dimensional Quasiparticles called Anyons whose World lines cross In Mathematics and Physics, an anyon is a type of particle that only occurs in two-dimensional systems In Topology, braid theory is an abstract geometric Theory studying the everyday Braid concept and some generalisations ^[10]

Candidates

There are a number of quantum computing candidates, among those:

1. Superconductor-based quantum computers (including SQUID-based quantum computers)
2. Trapped ion quantum computer
3. Electrons on helium quantum computers
4. Nuclear magnetic resonance on molecules in solution"-based
5. Quantum dot on surface (e. Superconductivity is a phenomenon occurring in certain Materials generally at very low Temperatures characterized by exactly zero electrical resistance Squid are marine Cephalopods of the order Teuthida, which comprises around 300 species A Trapped ion quantum computer is a type of Quantum computer. In Chemistry, a molecule is defined as a sufficiently stable electrically neutral group of at least two Atoms in a definite arrangement held together by In Chemistry, a solution is a Homogeneous Mixture composed of two or more substances A quantum dot is a Semiconductor whose Excitons are confined in all three Spatial dimensions. g. the Loss-DiVincenzo quantum computer)
6. Cavity quantum electrodynamics (CQED)
7. Molecular magnet
8. Fullerene-based ESR quantum computer
9. Solid state NMR Kane quantum computers
10. Optic-based quantum computers (Quantum optics)
11. Topological quantum computer
12. Spin-based quantum computer
13. Adiabatic quantum computation^[11]
14. Diamond-based quantum computer^[12][13]
15. Bose–Einstein condensate-based quantum computer^[14]
16. Transistor-based quantum computer - string quantum computers with entrainment of positive holes using a electrostatic trap

The large number of candidates shows explicitly that the topic, in spite of rapid progress, is still in its infancy. The Loss-DiVincenzo quantum computer (or spin-qubit quantum computer is a scalable semiconductor-based quantum computer that was proposed by Daniel Loss and A single-molecule magnet or SMM is an object that is composed of Molecules each of which behaves as an individual superparamagnet. "C60" and "C-60" redirect here For other uses see C60 (disambiguation. Electron paramagnetic resonance (EPR or electron spin resonance (ESR Spectroscopy is a technique for studying Chemical species that have one or more unpaired The Kane quantum computer is a proposal for a scalable Quantum computer proposed by Bruce Kane in 1998, then at the University of New South Wales Quantum optics is a field of research in Physics, dealing with the application of Quantum mechanics to phenomena involving Light and its interactions A topological quantum computer is a theoretical Quantum computer that employs two-dimensional Quasiparticles called Anyons whose World lines cross Spintronics (a Neologism meaning "spin transport electronics" also known as magnetoelectronics is an Emerging technology which exploits the intrinsic Adiabatic Quantum Computation relies on the Adiabatic theorem to do calculations The Nitrogen-vacancy center (N-V center is a crystallographic defect in the structure of a Diamond that can be exploited to capture the electron spin and A Bose–Einstein condensate (BEC is a State of matter of Bosons confined in an external Potential and cooled to Temperatures very near to But at the same time there is also a vast amount of flexibility.

In 2005, researchers at the University of Michigan built a semiconductor chip which functioned as an ion trap. The University of Michigan Ann Arbor ( U of M, U-M, UM or simply Michigan) is a top-ranked Coeducational public research Microchipsjpg|right|thumb|200px|Microchips ( EPROM memory with a transparent window showing the integrated circuit inside An ion trap is a combination of electric or magnetic fields that captures Ions in a region of a vacuum system or tube Such devices, produced by standard lithography techniques, may point the way to scalable quantum computing tools. Lithography is a method for Printing using a plate or stone with a completely smooth surface ^[15] An improved version was made in 2006.

Quantum computing in computational complexity theory

The suspected relationship of BQP to other problem spaces. ^[16]

This section surveys what is currently known mathematically about the power of quantum computers. It describes the known results from computational complexity theory and the theory of computation dealing with quantum computers. Computational complexity theory, as a branch of the Theory of computation in Computer science, investigates the problems related to the amounts of resources The theory of computation is the branch of Computer science that deals with whether and how efficiently problems can be solved on a Model of computation, using an

The class of problems that can be efficiently solved by quantum computers is called BQP, for "bounded error, quantum, polynomial time". In Computational complexity theory BQP stands for " B ounded error Q uantum '''P'''olynomial time " Quantum computers only run probabilistic algorithms, so BQP on quantum computers is the counterpart of BPP on classical computers. A randomized algorithm or probabilistic algorithm is an Algorithm which employs a degree of randomness as part of its logic In complexity theory, BPP is the class of Decision problems solvable by a Probabilistic Turing machine in Polynomial time, with an error It is defined as the set of problems solvable with a polynomial-time algorithm, whose probability of error is bounded away from one quarter. ^[16] A quantum computer is said to "solve" a problem if, for every instance, its answer will be right with high probability. If that solution runs in polynomial time, then that problem is in BQP.

BQP is contained in the complexity class #P (or more precisely in the associated class of decision problems P^#P) ^[17], which is a subclass of PSPACE. #P, pronounced "sharp P" or "number P" is a complexity class in Computational complexity theory. Mathematicians and computer scientists try to carefully define different types of complexity, and PSPACE is one of these types

BQP is suspected to be disjoint from NP-complete and a strict superset of P, but that is not known. In Computational complexity theory, the Complexity class NP-complete (abbreviated NP-C or NPC) is a class of problems having two properties In Computational complexity theory, P, also known as PTIME or DTIME ( n O(1 is one of the most fundamental Complexity Both integer factorization and discrete log are in BQP. In Mathematics, specifically in Abstract algebra and its applications discrete logarithms are group-theoretic analogues of ordinary Logarithms Both of these problems are NP problems suspected to be outside BPP, and hence outside P. Both are suspected to not be NP-complete. There is a common misconception that quantum computers can solve NP-complete problems in polynomial time. That is not known to be true, and is generally suspected to be false. ^[17]

Quantum gates may be viewed as linear transformations. In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that Daniel S. Abrams and Seth Lloyd have shown that if nonlinear transformations are permitted, then NP-complete problems could be solved in polynomial time. Seth Lloyd is a Professor of Mechanical engineering at Massachusetts Institute of Technology. It could even do so for #P-complete problems. #P-complete, pronounced "sharp P complete" or "number P complete" is a complexity class in complexity theory. They do not believe that such a machine is possible.

Although quantum computers may be faster than classical computers, those described above can't solve any problems that classical computers can't solve, given enough time and memory (albeit possibly an amount that could never practically be brought to bear). A Turing machine can simulate these quantum computers, so such a quantum computer could never solve an undecidable problem like the halting problem. Turing machines are basic abstract symbol-manipulating devices which despite their simplicity can be adapted to simulate the logic of any Computer Algorithm Undecidable has more than one meaning;In Mathematical logic: A Decision problem is called (recursively undecidable if no Algorithm can In computability theory, the halting problem is a Decision problem which can be stated as follows given a description of a program and a finite input The existence of "standard" quantum computers does not disprove the Church-Turing thesis. ^[16]

See also

• List of emerging technologies
• Quantum bus
• Timeline of quantum computing
• Chemical computer
• DNA computer
• Molecular computer

Notes

References