A systematic account of projection operators (projectors) and orthogonalization techniques together with applications to selected areas of physics is presented. This unified approach is shown to have advantages over other approaches in that the mathematical statements are more precise. The mathematical level, however, is aimed at the practicing physicist and lies between rigorous mathematics and current use in physics. Further, the techniques presented have practical applications as is demonstrated by examples in the quantum theory of measurement, in the relationship between second quantization and configuration space techniques, and in an account of generalized Wannier and Bloch functions. Attention is paid to the problem of construction of orthogonal projection operators (orthogonal projectors). The construction of orthogonal projectors even in approximate form would allow the solution of many practical problems ranging from the eigenvalue spectrum problem to the construction of states for many-body systems. One can almost say that any problem in quantum mechanics can be formulated as a problem involving the construction of projectors.
Solving the electronic structure problem on a universal-gate quantum computer within the variational quantum eigensolver (VQE) methodology requires constraining the search procedure to a subspace defined by relevant physical symmetries. Ignoring symmetries results in convergence to the lowest eigenstate of the Fock space for the second quantized electronic Hamiltonian. Moreover, this eigenstate can be symmetry broken due to limitations of the wavefunction ansatz. To address this VQE problem, we introduce and assess methods of exact and approximate projectors to irreducible eigensubspaces of available physical symmetries. Feasibility of symmetry projectors in the VQE framework is discussed, and their efficiency is compared with symmetry constraint optimization procedures. Generally, projectors introduce a higher number of terms for VQE measurement compared to the constraint approach. On the other hand, the projection formalism improves accuracy of the variational wavefunction ansatz without introducing additional unitary transformations, which is beneficial for reducing depths of quantum circuits.
Standard approaches to approximating solutions to the Schroedinger equation rely upon the assumption that there is a single dominant electronic configuration with a small number of low-lying excited states. As such, they are referred to as single reference approaches and are accurate only when the band gap is small in comparison with the inter-electronic repulsion of the valence electrons. Systems for which this is the case are weakly-correlated. If the inter-electronic repulsion larger than the band gap, then a single electronic reference is not an appropriate ansatz for the physical wavefunction. As the strength of the interaction is increased many Slater determinants become near-degenerate, each providing non-negligible contributions. Such systems are strongly-correlated. The goal of the thesis was to employ wavefunctions which are: non-trivial, tractable, size-consistent and extensive, and not given in an active space. We have outlined approaches based on wavefunction forms more suited to describe strong correlation. Built on the Lie algebra su(2), our first trial wavefunction was the Bethe ansatz solution to the reduced BCS Hamiltonian derived by Richardson. This wavefunction is expressed as a product of quasi-particles each parametrized in terms of quasi-momenta. Consistency among the quasi-momenta is ensured by a set of non-linear equations to be solved numerically. This approach is feasible in both variational and projected flavours. We next considered a wavefunction of the same form without requiring it to be an eigenvector of the reduced BCS Hamiltonian. A variational approach in this case is only feasible provided there are two equivalent descriptions in terms of particles and holes. In any case, a bivariational principle and a projected approach are both feasible. Also based on su(2), there is AP1roG (coupled cluster pair doubles). Open-shell singlets and triplets were constructed with the Lie algebras sp(N) and so(4N). The resulting formulae appear to be infeasible unless it is possible to efficiently compute generalizations of determinants to three-dimensional objects. Treating each spatial orbital individually, all four possible occupations close the Lie superalgebra gl(2|2).
The construction of operator functionals of the full Hamiltonian of a quantum-mechanical system which act as projection operators for a specified exact eigenstate is studied. In general these operator functionals are not equivalent to the formal operator |ψ〉〈ψ|, where |ψ〉 is the state of interest, and are not always idempotent. A formal method by which standard perturbation theoretic expansions for the exact ground-state vector and ground-state energy of the system can be recovered by expansion of a certain operator functional of this type is presented. Related work is briefly discussed.
In quantum information theory and statistical physics, symmetries of multiple copies, or replicas, of a system play a pivotal role. For unitary ensembles, these symmetries are encoded in the replicated commutant: the algebra of operators commuting with the ensemble across
k replicas. Determining the commutant is straightforward for the full unitary group, but remains a major obstacle for structured, computationally relevant circuit families. We solve this problem for matchgate circuits, which prepare fermionic Gaussian states on
n qubits. Using a Majorana fermion representation, we show that operators coupling different system copies generate the orthogonal Lie algebra
so(k), endowing the space of invariants with rich and tractable structure. This underlying symmetry decomposes the matchgate commutant into irreducible sectors, which we completely resolve via a Gelfand--Tsetlin construction. We provide an explicit orthonormal basis of the matchgate commutant for all
k and
n, together with a formula for its dimension that grows polynomially in
n. Furthermore, we characterize the commutant of the Clifford--matchgate subgroup, showing that restricting to signed permutations of Majorana modes yields a commutant that qualitatively diverges from the matchgate case for
k≥4 replicas. Ultimately, our orthonormal basis turns algebraic classification into a working toolbox. Using it, we derive closed-form expressions for matchgate twirling channels and a fermionic analogue of Weingarten calculus, the projector encoding all moments of the Gaussian state orbit, state and unitary frame potentials, the average nonstabilizerness of fermionic Gaussian states, a systematic hierarchy of non-Gaussianity measures, and a fermionic de Finetti theorem.
We solved the Schr{\\"o}dinger equation for a particle in a uniform magnetic field in the $ n $-dimensional torus. We obtained a complete set of solutions for a broad class of problems; the torus $ T^n = \\R^n / {\\mit\\Lambda} $ is defined as a quotient of the Euclidean space $ \\R^n $ by an arbitrary $ n $-dimensional lattice $ {\\mit\\Lambda} $. The lattice is not necessary either cubic or rectangular. The magnetic field is also arbitrary. However, we restrict ourselves within potential-free problems; the Schr{\\"o}dinger operator is assumed to be the Laplace operator defined with the covariant derivative. We defined an algebra that characterizes the symmetry of the Laplacian and named it the magnetic algebra. We proved that the space of functions on which the Laplacian acts is an irreducible representation space of the magnetic algebra. In this sense the magnetic algebra completely characterizes the quantum mechanics in the magnetic torus. We developed a new method for Fourier analysis for the magnetic torus and used it to solve the eigenvalue problem of the Laplacian. All the eigenfunctions are given in explicit forms.