Amplitude amplification

Amplitude amplification

Amplitude amplification is a technique in quantum computing which generalizes the idea behindthe Grover's search algorithm, and gives rise to a family of
quantum algorithms.It was discovered by Gilles Brassard and
Peter Høyer in 1997,cite journal
author=Gilles Brassard, Peter Høyer
title=An exact quantum polynomial-time algorithm for Simon's problem
journal=Proceedings of Fifth Israeli Symposium on Theory of Computing and Systems
publisher=IEEE Computer Society Press
] and independently rediscovered by Lov Grover in 1998.cite journal
author=Grover, Lov K.
title=Quantum Computers Can Search Rapidly by Using Almost Any Transformation
journal=Phys. Rev. Lett.

In a quantum computer, amplitude amplification can be used to obtain aquadratic speedup over several classical algorithms.


The derivation presented here roughly follows the one given incite journal
author=Gilles Brassard, Peter Høyer, Michele Mosca, Alain Tapp
title=Quantum Amplitude Amplification and Estimation
] .Assume we have an N-dimensional Hilbert space mathcal{H}representing the state space of our quantum system, spanned by theorthonormal computational basis states mathbf{B} := { |x angle }_{x=0}^{N-1}.Furthermore assume we have a Boolean function:chi:mathbb{Z} o {0,1}.chi can be used to partitionmathcal{H} into a direct sum of two mutually orthogonal subspaces,:the "good" subspace mathcal{H}_1 = operatorname{span}{|x angle in mathbf{B} ;|; chi(x) = 1}, and:the "bad" subspace mathcal{H}_0 = operatorname{span}{|x angle in mathbf{B} ;|; chi(x) = 0}.

Given a (normalized) state vector |psi angle in mathcal{H}, we canuniquely decompose it as :|psi angle = a |psi_1 angle + b |psi_0 angle,where|psi_1 angle and |psi_0 angle are thenormalized projections of |psi angle into thesubspaces mathcal{H}_1 and mathcal{H}_0,respectively, and 0 < a < 1. This decomposition defines a two-dimensional subspacemathcal{H}_psi, spanned by the vectors|psi_0 angle and |psi_1 angle. The probability of finding the system in a "good" state when measuredis a^2. Furthermore we have a^2+b^2=1.

Define a unitary operatorQ(psi, chi) := -S_psi S_chi ,!,where:S_psi = I - 2|psi angle langlepsi|quad and:S_chi |x angle = (-1)^{chi(x)} |x angle.The action of this operator on mathcal{H}_psi is given by:Q |psi_0 angle = -S_psi |psi_0 angle = (1-2a^2)|psi_0 angle +2ab|psi_1 angle and:Q |psi_1 angle = S_psi |psi_1 angle = -2ab|psi_0 angle+(1-2a^2)|psi_1 angle.By defining heta := arcsin(a) ,!,0 < heta < pi/2 ,!, we can clearly see that in this subspace Qcorresponds to a rotation by the angle 2 heta,!::Q = egin{pmatrix}cos(2 heta) & sin(2 heta)\-sin(2 heta) & cos(2 heta)end{pmatrix}.

Applying Q(psi, chi),! repeatedly on the state|psi anglegives:Q^n |psi angle = cos((2n+1) heta) |psi_0 angle +sin((2n+1) heta) |psi_1 angle,rotating the state between the "good" and "bad" subspaces.After n applications the probability of finding thesystem in a "good" state is sin^2((2n+1) heta),!. The probability is maximized if we choose:n = leftlfloorfrac{pi}{4 heta} ight floor.Up until this point each application of Q increases the amplitude of the "good" states, hencethe name of the technique.


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