By Corey Sandler

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The state space of a composite system is the tensor product of the states of the component physical systems. For instance, for two components A and B, the total Hilbert space of the composite system becomes HAB = HA ⊗ HB . A state vector on the composite space is written as |ψ AB = |ψ A ⊗ |ψ B . The tensor product is often abbreviated as |ψ A |ψ B , or, equivalently, the labels are written in the same ket |ψA ψB . As an example, assume that the component spaces are two-dimensional, and choose a basis in each.

If we understand these states, solving the time-dependent Schrödinger equation becomes easier for any other state. 8 and Chapter 14). An excited state is any state with energy greater than the ground state. Consider an eigenstate ψα of the Hamiltonian Hψα = Eα ψα . Taking the Taylor expansion of the exponential, we observe how the time evolution operator acts on this eigenstate: e−iHt/ |ψα (0) = n = n 1 n! H i 1 n! 39) n tn |ψα (0) = e−iEα t/ |ψα (0) . 41) We define U(H, t) = e−iHt/ . This is the time evolution operator of a closed quantum system.

4). 5). Unsupervised learning is a vast field, and this chapter barely offers a glimpse. The algorithms in this chapter are the most relevant to quantum methods that have already been published. 1 Principal Component Analysis Let us assume that the data matrix X, consisting of the data instances {x1 , . . , xN }, has a zero mean. Principal component analysis looks at the eigenstructure of X X. This d × d square matrix, where d is the dimensionality of the feature space, is known as the empirical sample covariance matrix in the statistical literature.