Quantum Search for Zeros of Polynomials

A quantum mechanical search procedure to determine the real zeros of a polynomial is introduced. It is based on the construction of a spin observable whose eigenvalues coincide with the zeros of the polynomial. Subsequent quantum mechanical measurements of the observable output directly the numerical values of the zeros. Performing the measurements is the only computational resource involved. PACS: 03.67.-a, 03.65Sq Introduction Quantum mechanical measurements are a computational resource. Various quantum algorithms use projective measurements at some stage or other to determine the period of a function e.g. [1]. In [2], projective measurements have been assigned a crucial role for a particular scheme of universal quantum computation which requires measurements on up to four qubits. Related schemes have been formulated based on measuring on triples and pairs [3], and finally on pairs of qubits only [4]. Measurements are also an essential part of Grover’s search algorithm in order to actually read the result of the computation [5, 6]. A conceptually different strategy has been applied to propose a special-purpose machine which is capable to diagonalize any finite-dimensional hermitean matrix by genuine quantum means, i.e. quantum measurements [7, 8]. In this approach of quantum diagonalization, a hermitean matrix is considered as a quantum mechanical observable of an appropriate one-spin system. Projective measurements with a generalized Stern-Gerlach apparatus provide directly the unknown eigenvalues of the matrix which solves the hard part of the diagonalization. In this paper, it will be shown how to find the real zeros of a prescribed polynomial in a similar way, using quantum mechanical measurements. 1 The quantum search procedure Consider a polynomial of degree N which is assumed to have N real zeros ζn, P (x) = N