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note
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It is known that finding the ground state of Hamiltonians formed from simple sums of the inequality operator xi xj is NP complete on a planar graph.
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note
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It is understood that a term in a Hamiltonian such as σi σj is the operator σ acting on the i th and j th qubit with the omitted identity operator acting on the rest of the Hilbert space. The tensor product symbol (□) is omitted between operators.
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44349143137
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note
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A simplistic Hamiltonian with vectors in the ground space L corresponding to logical AND, that is L=span { | 000, | 010, | 100, | 111 } (ordered | x1 x2 | z, where z = x1 x2), has the form: H=δ (1- | 000 000 | - | 010 010 | - | 100 100 | - | 111 111 |).
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note
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For the purpose of this section one is actually only concerned with the null space of the Hamiltonian and the spectral gap δ so Hprop > Hin is sufficient.
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noe
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Assume that Hprop represents a circuit and is given as an oracle Hamiltonian. One wishes to search for an input bit string x that will make the circuit output z =1. In this case, we will force an energy penalty any time the circuit outputs 0 by acting on the output qubit, z, with the Hamiltonian Hin = |0 0|. After successful adiabatic evolution, qubits x1, x2, and x3 can be measured to determine an input causing the circuit to output 1. If the circuit never outputs 1, successful adiabatic evolution will return an input that minimizes the Hamming distance from an input that would cause the circuit to output 1.
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http://en.wikipedia.org
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note
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Where exclusive OR (XOR) is given as f (x1, x2) = def x1 x2 = x̄ 1 x2 x1 x̄ 2 = x1 + x2 -2 x1 x2, and equivalence (EQV) as f (x1, x2) = def x1 x2 = x̄ 1 x̄ 2 x1 x2 =1- x1 - x2 +2 x1 x2.
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This spectrum corresponds to the Walsh function represented by the eighth column of the matrix H/3, where H is the 2×2 Hardamard matrix. We remark that { { 0,1 }, } is the Galois field Z2.
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