The AND gate maps (0,1)(0,1)(0,1) to 000. The NAND gate is computationally universal. The NAND (not-AND) gate takes two bits xxx and yyy, acts on them with AND, and acts on the result with NOT. For this basic example, the difference in computation time is commonly denoted by saying the classical computer takes O(2n)\mathcal{O}(2^n)O(2n) time while the quantum computer takes O(n)\mathcal{O}(n)O(n) time. Accessed January 19, 2016. http://www.dwavesys.com/press-releases/d-wave-systems-breaks-1000-qubit-quantum-computing-barrier. Already have an account? \left( 1 - \frac{2}{N}\right)^2 + \left(\frac{2\sqrt{N-1}}{N}\right)^2 &= 1 + \frac{4}{N^2 } - \frac{4}{N} + \frac{4(N-1)}{N^2} \\ Quantum algorithms are often (though not always) inherently probabilistic in that they are guaranteed to obtain the correct answer only a certain percentage of the time. Quantum computing is not as binary (rather literally) as classical computing. Shor's algorithm demonstrates a remarkable speedup to O((log⁡N)3])\mathcal{O}\big((\log N)^3]\big)O((logN)3]). Accessed January 19, 2016. http://www.doc.ic.ac.uk/~nd/surprise_97/journal/vol4/spb3/#1.1%20Quantum%20computer%20basics. Quantum computers, like classical computers, rely on gates acting on states in a two-state system. As a convention in quantum mechanics, these states are typically written as ∣0⟩|0\rangle∣0⟩ and ∣1⟩|1\rangle∣1⟩. This problem can be expressed formally as follows: given a function that maps NNN-digit binary values to either 000 or 111 such that one unique binary value xxx is mapped to 111, find xxx. The NAND gate can be simulated by a reversible gate that takes three bits as input called the Toffoli gate. Accessed January 19, 2016. http://www.nature.com/articles/npjqi201523. To compute TTT, consider the function f(x)=kx mod Nf(x) = k^x\:\text{mod}\:Nf(x)=kxmodN. : I-5.. This is important because it is easy to mathematically represent gates as 2×22\times 22×2 matrices acting on qubits by multiplying the corresponding vectors. Level. An early barrier to the development and viability of quantum computing was the no-cloning theorem discovered in 1982 by William Wootters, Wojciech Zurek, and Dennis Dieks, which states that it is impossible to create a guaranteed identical copy of any quantum system. The qubit is the quantum computing equivalent of a binary bit ... As you go more in-depth into quantum computing and/or quantum mechanics, math (linear algebra) becomes extremely important. □_\square□​. Next, a certain gate is constructed that rotates the uniform superposition towards the state ∣x⟩|x\rangle∣x⟩ corresponding to xxx. First, apply the Hadamard gate to a set of nnn qubits to achieve a uniform superposition of states, where N=2nN=2^nN=2n. In the early 1980s, he observed that a classical computer would require 2n2^n2n states to describe a set of nnn spin-12\frac1221​ particles such as electrons, while a quantum computer requires only nnn states. An important gate in quantum computing is the Hadamard gate. In 1995, however, Peter Shor and Andrew Steane discovered a method of quantum error correction that circumvents this problem. Once quantum computing becomes scalable, it will have enormous practical implications in a wide variety of fields including cryptography, computational chemistry, mathematics, and computer science. R=(1−2N2N−1N−2N−1N1−2N).R = \begin{pmatrix} 1 - \frac{2}{N} & \frac{2\sqrt{N-1}}{N} \\ - \frac{2\sqrt{N-1}}{N} & 1 - \frac{2}{N}\end{pmatrix} .R=(1−N2​−N2N−1​​​N2N−1​​1−N2​​). A simple physical prototype for this two-state system is the electron, which can have a spin pointing up or down. Part of a quantum computer designed by D-Wave. \end{aligned}(1−N2​)2+(N2N−1​​)2​=1+N24​−N4​+N24(N−1)​=1+N24​−N4​+N4​−N24​=1.​, Therefore, sin⁡θ=2N−1N\sin \theta= \frac{2\sqrt{N-1}}{N}sinθ=N2N−1​​ for this θ\thetaθ. When z=1z=1z=1, the Toffoli gate maps (x,y,1)(x,y,1)(x,y,1) to (x,y,x NAND y)(x,y,x\:\text{NAND}\:y)(x,y,xNANDy). Modeling quantum systems such as those useful for chemistry on a classical computer thus will require time growing exponentially with the number of states nnn that one wants to model, while modeling the same system on a quantum computer only requires time growing linearly with nnn. For example, quantum algorithms would be able to efficiently factor products of large prime numbers, thus breaking RSA encryption, which is widely used to protect online data. If the period TTT is represented as a vector T⃗\vec{T}T that is nnn bits in length, measuring the state after applying the Hadamard gate yields a vector orthogonal to T⃗\vec{T}T with high probability. Without error correction, thermal fluctuations and other sources of noise cause quantum systems to decohere and lose information. The benefit of quantum algorithms is their enormous increase in computational efficiency for certain problems.

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