| Summary: | We find the free-energy in the thermodynamic limit of a one-dimensional XY model associated to a system of <i>N</i> qubits. The coupling among the <inline-formula> <math display="inline"> <semantics> <msubsup> <mi>σ</mi> <mi>i</mi> <mi>z</mi> </msubsup> </semantics> </math> </inline-formula> is a long range two-body random interaction. The randomness in the couplings is the typical interaction of the Hopfield model with <i>p</i> patterns (<inline-formula> <math display="inline"> <semantics> <mrow> <mi>p</mi> <mo><</mo> <mi>N</mi> </mrow> </semantics> </math> </inline-formula>), where the patterns are <i>p</i> sequences of <i>N</i> independent identically distributed random variables (i.i.d.r.v.), assuming values <inline-formula> <math display="inline"> <semantics> <mrow> <mo>±</mo> <mn>1</mn> </mrow> </semantics> </math> </inline-formula> with probability <inline-formula> <math display="inline"> <semantics> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </semantics> </math> </inline-formula>. We show also that in the case <inline-formula> <math display="inline"> <semantics> <mrow> <mi>p</mi> <mo>≤</mo> <mi>α</mi> <mi>N</mi> </mrow> </semantics> </math> </inline-formula>, <inline-formula> <math display="inline"> <semantics> <mrow> <mi>α</mi> <mo>≠</mo> <mn>0</mn> </mrow> </semantics> </math> </inline-formula>, the free-energy is asymptotically independent from the choice of the patterns, i.e., it is self-averaging.
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