Download Quantum Algorithms via Linear Algebra: A Primer by Richard J. Lipton, Kenneth W. Regan PDF

By Richard J. Lipton, Kenneth W. Regan

This advent to quantum algorithms is concise yet entire, protecting many key algorithms. it's mathematically rigorous yet calls for minimum history and assumes no wisdom of quantum conception or quantum mechanics. The ebook explains quantum computation when it comes to undemanding linear algebra; it assumes the reader may have a few familiarity with vectors, matrices, and their simple homes, yet deals a assessment of the entire proper fabric from linear algebra. by means of emphasizing computation and algorithms instead of physics, this primer makes quantum algorithms available to scholars and researchers in desktop technology with out the problems of quantum mechanical notation, actual suggestions, and philosophical issues.

After explaining the improvement of quantum operations and computations in response to linear algebra, the ebook provides the key quantum algorithms, from seminal algorithms by means of Deutsch, Jozsa, and Simon via Shor's and Grover's algorithms to contemporary quantum walks. It covers quantum gates, computational complexity, and a few graph concept. Mathematical proofs are as a rule brief and simple; quantum circuits and gates are used to light up linear algebra; and the dialogue of complexity is anchored in computational difficulties instead of desktop models.

Quantum Algorithms through Linear Algebra is acceptable for lecture room use or as a reference for desktop scientists and mathematicians.

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Additional resources for Quantum Algorithms via Linear Algebra: A Primer

Example text

A first idea is to compute it using applications of OR and AND as follows: For every three-element subset S = {i, j, k} of {1, 2, 3, 4, 5}, we compute yS = OR(xi , xj , xk ). Define y to be the AND of each of the ten subsets S. Then y = 1 ⇐⇒ no more than 2 bits of x1 , . . , x5 are 0 ⇐⇒ MAJ(x1 , x2 , x3 , x4 , x5 ) is true. The complexity is counted as 11 operations and, importantly, 35 total arguments of those operations. On second thought, we can find a program of slightly lower complexity. 1. Expressed as a sequence of operations, in one of many possible orders, the circuit is equivalent to the following straight-line program: v1 = OR(x1 , x2 , x3 ), w1 = AND(x1 , x2 ), v2 = OR(x4 , x5 ), w2 = AND(x1 , x3 ), w4 = AND(w1 , w2 ), u = OR(w1 , w2 , w3 ), w3 = AND(x2 , x3 ), w5 = AND(v1 , x4 , x5 ) t = AND(u, v2 ), y = OR(w4 , t, w5 ).

We can give a partial answer. 3. 4 All classically feasible Boolean functions f have feasible quantum computations in the form of P f . The proof of this theorem stays entirely classical—that is, the quantum circuits are the same as Boolean circuits that are reversible, which in turn efficiently embed any given Boolean circuit computing f . 14. 5 The Toffoli gate is the ternary Boolean function TOF (x1 , x2 , x3 ) = (x1 , x2 , x3 ⊕ (x1 ∧ x2 )). The Toffoli gate induces the permutation in 8-dimensional Hilbert space that swaps the last two entries, which correspond to the strings 110 and 111, and leaves the rest the same.

The tensor product of two vectors a and b is the vector c = a ⊗ b defined by c(ij) = a(i)b(j). A vector denoting a pure quantum state is separable if it is the tensor product of two other vectors; otherwise it is entangled. The vectors e00 and e11 are separable, but their unit-scaled sum √1 (e00 + e11 ) is entangled. The standard 2 basis vectors of H1 ⊗ H2 are separable, but this is not true of many of their linear combinations, all of which still belong to H1 ⊗ H2 because it is a Hilbert space.

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