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arXiv:quant-ph/0001082 · Journal of Physics A Mathematical and General

Quantum diagonalization of Hermitean matrices

Stefan Weigert

Published Jun 27, 2001• 4 citations
arXivHTMLPDF

Abstract

To measure an observable of a quantum mechanical system leaves it in one of its eigenstates and the result of the measurement is one of its eigenvalues. This process is shown to be a computational resource : Hermitean ( N × N ) matrices can be diagonalized, in principle, by performing appropriate quantum mechanical measurements. To do so, one considers the given matrix as an observable of a single spin with appropriate length s which can be measured using a generalized Stern-Gerlach apparatus. Then, each run provides one eigenvalue of the observable. As the underlying working principle is the `collapse of the wavefunction' associated with a measurement, the procedure is neither a digital nor an analogue calculation - it defines thus a new example of a quantum mechanical method of computation.

DOI
10.1088/0305-4470/34/27/312

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To measure an observable of a quantum mechanical system leaves it in one of its eigenstates and the result of the measurement is one of its eigenvalues. This process is shown to be a computational resource : Hermitean ( N × N ) matrices can be diagonalized, in principle, by performing appropriate quantum mechanical measurements. To do so, one considers the given matrix as an observable of a single spin with appropriate length s which can be measured using a generalized Stern-Gerlach apparatus. Then, each run provides one eigenvalue of the observable. As the underlying working principle is the `collapse of the wavefunction' associated with a measurement, the procedure is neither a digital nor an analogue calculation - it defines thus a new example of a quantum mechanical method of computation.
To measure an observable of a quantum mechanical system leaves it in one of its eigenstates and the result of the measurement is one of its eigenvalues. This process is shown to be a computational resource: Hermitean (N ×N) matrices can be diagonalized, in principle, by performing appropriate quantum mechanical measurements. To do so, one considers the given matrix as an observable of a single spin with appropriate length s which can be measured using a generalized Stern-Gerlach apparatus. Then, each run provides one eigenvalue of the observable. As the underlying working principle is the `collapse of the wavefunction' associated with a measurement, the procedure is neither a digital nor an analogue calculation - it defines thus a new example of a quantum mechanical method of computation.
Abstract An entirely quantum mechanical approach to diagonalize hermitean matrices has been presented recently. The method is based on the measurement of quantum mechanical observables which provides the computational resource. In brief, quantum mechanics is able to directly address and output eigenvalues of hermitean matrices. Here, the example of (2×2) matrices is considered in detail. This simple case allows one to illustrate the conceptual features of the general method which is applicable to (N×N) hermitean matrices.
The measurement problem is to explain why a system which is in a linear combination of states appears, upon measurement, to be in just one of those states. The solution given here is to first show that if one assumes linear, unitary, no collapse, no particle quantum mechanics, the different versions of reality are completely isolated from each other. This then implies only the eigenstates appropriate to the measurement setup will be perceived. In a Stern-Gerlach experiment, for example, only spin up or spin down will be perceived, but never a combination of the two.
A dynamical model for the collapse of the wave function in a quantum measurement process is proposed by considering the interaction of a quantum system (spin-1/2) with a macroscopic quantum apparatus interacting with an environment in a dissipative manner. The dissipative interaction leads to decoherence in the superposition states of the apparatus, making its behaviour classical in the sense that the density matrix becomes diagonal with time. Since the apparatus is also interacting with the system, the probabilities of the diagonal density matrix are determined by the state vector of the system. We consider a Stern-Gerlach type model, where a spin- 1/2 particle is in an inhomogeneous magnetic field, the whole set up being in contact with a large environment. Here we find that the density matrix of the combined system and apparatus becomes diagonal and the momentum of the particle becomes correlated with a spin operator, selected by the choice of the system-apparatus interaction. This allows for a measurement of spin via a momentum measurement on the particle with associated probabilities in accordance with quantum principles.
The procedure for simulating the nuclear magnetic resonance spectrum linked to the spin system of a molecule for a certain nucleus entails diagonalizing the associated Hamiltonian matrix. As the dimensions of said matrix grow exponentially with respect to the spin system's atom count, the calculation of the eigenvalues and eigenvectors marks the performance of the overall process. The aim of this paper is to provide a formalization of the complete procedure of the simulation of a spin system's NMR spectrum while also explaining how to diagonalize the Hamiltonian matrix with a quantum computer, thus enhancing the overall process's performance. Two well-known quantum algorithms for calculating the eigenvalues of a matrix are analyzed and put to the test in this context: quantum phase estimation and the variational quantum eigensolver. Additionally, we present simulated results for the later approach while also addressing the hypothetical noise found in a physical quantum computer.
While the Stern–Gerlach experiment is an old and familiar problem, no analysis of this experiment is presented in the pedagogical literature using modern quantum mechanical techniques. Expositions of the Stern–Gerlach experiment are usually based on ‘‘old quantum theory,’’ i.e., semiclassical Bohr–Sommerfeld quantum mechanics. The experiment is also a popular one in discussions of the postulate of quantum measurement, which asserts that the process of measuring an observable forces the state vector of the system into an eigenvector of that observable, and the value measured will be the eigenvalue of that eigenvector. The most enduring of the philosophical arguments associated with quantum mechanics ultimately revolves around the mechanism of quantum measurement. Since the postulate of quantum measurement is expressed in terms well outside the realm of the old quantum mechanics, the traditional analysis of the Stern–Gerlach experiment cannot reveal any information about the relationship between dynamics and measurement. Therefore, a modern analysis of the Stern–Gerlach experiment must be made if this experiment is to reveal any information about the relationship between dynamics and the mechanism of quantum measurement. This paper presents such an analysis, and develops some of the implications for the theory of quantum measurement.