# quantum computing maths

Log in here. Below are summarized some of the major quantum algorithms in terms of historical significance and practical importance. Explore. Quantum computers thus have the ability to perform some computations using exponentially fewer states than classical computers. [3] Montanaro, Ashley. Quantum computing may be just around the corner or it may be, for all practical purposes, permanently out of reach: The physics needed for a useful quantum computer has not yet been discovered, and may in fact not exist.A quantum computer, real or potential, is essentially different from an adding machine. □_\square□​. The NAND (not-AND) gate takes two bits xxx and yyy, acts on them with AND, and acts on the result with NOT. Checking this directly, (1−2N)2+(2N−1N)2=1+4N2−4N+4(N−1)N2=1+4N2−4N+4N−4N2=1.\begin{aligned} On reading this Reddit thread I realized that even after a couple months of learning about quantum computing I've absolutely no clue about how a quantum computer actually works.. To make the question more precise, let's say we have a superconducting qubit based 5-qubit quantum computer (like the 5-qubit IBM Quantum Computer). About how many classical bits does this correspond to? This is an improvement on the O(N)\mathcal{O}(N)O(N) steps that a random classical algorithm would take in the best-case scenario to find xxx. the problem of computing TTT is thus reduced to period-finding for this function fff. Part of a quantum computer designed by D-Wave. Then one must only show that sin⁡θ=2N−1N\sin \theta= \frac{2\sqrt{N-1}}{N}sinθ=N2N−1​​ for the same θ\thetaθ. If kkk is a factor of N,N,N, the problem is solved; otherwise, gcd(k,N)\text{gcd}(k,N)gcd(k,N) must equal one. Because superposition is a quantum property, these two-state systems are called quantum bits or more commonly qubits. The process for computing NAND for the other choices of input bits is similar. to invert that function. The NOT gate acts on one bit to map "0" to "1" and vice versa. Accessed January 19, 2016. http://www.dwavesys.com/press-releases/d-wave-systems-breaks-1000-qubit-quantum-computing-barrier. 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. New user? 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​​). Since: f(x+T)=kx+T mod N=kTkx mod N=kx mod N=f(x),f(x+T) = k^{x+T}\:\text{mod}\:N = k^T k^{x}\:\text{mod}\:N = k^{x}\:\text{mod}\:N = f(x),f(x+T)=kx+TmodN=kTkxmodN=kxmodN=f(x). As Taught In. npj Quantum Information 2, 15023 (2016). The first experimental implementations of these algorithms began around 1998 and have continued into the twenty-first century. Most quantum computers invented so far have only been able to manipulate a small number of quantum states, on the order of tens at once. This makes the Hadamard gate an essential component of quantum algorithms such as those described below. Major quantum computational breakthrough is shaking up physics and maths by Ittay Weiss, The Conversation Quantum computers may be more trustworthy. In Grover's algorithm, a gate RRR is constructed that rotates the uniform superposition of states towards ∣x⟩|x\rangle∣x⟩, where. Show that RRR is in fact a rotation matrix. This prevents use of classical error correction techniques for quantum algorithms, since such techniques rely on making backup copies of states as templates to fix errors. After O(n)\mathcal{O}(n)O(n) iterations of this process, one obtains n−1n-1n−1 vectors orthogonal to T⃗\vec{T}T. Since T⃗\vec{T}T lives in an nnn-dimensional vector space, this is sufficient to identify the period TTT. Surveys and Presentations in Information Systems Engineering, Volume 4, 1997. However, the action of the NAND gate is not reversible, because it takes two inputs to one output and the outputs are not unique. 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.