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Journal of Mathematical Physics

Householder factorizations of unitary matrices

Jesús Urı́as

Published Jul 1, 2010• 15 citations
Publisher

Abstract

A method to construct all representations of finite dimensional unitary matrices as the product of Householder reflections is given. By arbitrarily severing the state space into orthogonal subspaces, the method may, e.g., identify the entangling and single-component quantum operations that are required in the engineering of quantum states of composite (multipartite) systems. Earlier constructions are shown to be extreme cases of the unifying scheme that is presented here.

DOI
10.1063/1.3451111

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A method to construct all representations of finite dimensional unitary matrices as the product of Householder reflections is given. By arbitrarily severing the state space into orthogonal subspaces, the method may, e.g., identify the entangling and single-component quantum operations that are required in the engineering of quantum states of composite (multipartite) systems. Earlier constructions are shown to be extreme cases of the unifying scheme that is presented here.
Abstract We propose a fault-tolerant implementation of the quantum Householder reflection, which is a key operation in various quantum algorithms, quantum-state engineering, generation of arbitrary unitaries, and entanglement characterization. We construct this operation using the modular approach of composite pulses and a relation between the Householder reflection and the quantum phase gate. The proposed implementation is highly insensitive to variations in the experimental parameters, which makes it suitable for high-fidelity quantum information processing.
Abstract A natural symmetry group of a multicomponent quantum system is a special combination of a symmetry group acting within a single component (“local group”) and a group that permutes the components (“spatial symmetry group”). This combination is called the wreath product. Unitary representations of wreath products describe quantum evolutions of multipartite systems. It is known that any unitary representation of a finite group is contained in some permutation representation. We describe an algorithm for decomposing permutation representations of wreath products into irreducible components. This decomposition makes it possible to study the quantum behavior (entanglement, non-local correlations, etc.) of multipartite systems in invariant subspaces of the permutation Hilbert space.
Algorithms to resolve multiple-qubit unitary transformations into a sequence of simple operations on one-qubit subsystems are central to the methods of quantum-circuit simulators. We adapt Householder’s theorem to the tensor-product character of multi-qubit state vectors and translate it to a combinatorial procedure to assemble cascades of quantum gates that recreate any unitary operation U acting on n-qubit systems. U may be recreated by any cascade from a set of combinatorial options that, in number, are not lesser than super-factorial of 2n, [Formula: see text]. Cascades are assembled with one-qubit controlled-gates of a single type. We complement the assembly procedure with a new algorithm to generate Gray codes that reduce the combinatorial options to cascades with the least number of CNOT gates. The combined procedure —factorization, gate assembling, and Gray ordering — is illustrated on an array of three qubits.
We consider finite-dimensional Hilbert spaces [Formula: see text] with [Formula: see text] with n ≥ 2 and unitary operators. In particular, we consider the case n = 2m, where m ≥ 2 in order to study entanglement of states in these Hilbert spaces. Two normalized states ψ and ϕ in these Hilbert spaces [Formula: see text] are connected by a unitary transformation (n×n unitary matrices), i.e. ψ = Uϕ, where U is a unitary operator UU* = I. Given the normalized states ψ and ϕ, we provide an algorithm to find this unitary operator U for finite-dimensional Hilbert spaces. The construction is based on a modified Gram–Schmidt orthonormalization technique. A number of applications important in quantum computing are given. Symbolic C++ is used to give a computer algebra implementation in C++.
Unitary transformations and density matrices are central objects in quantum physics and various tasks require to introduce them in a parameterized form. In this paper we present a parameterization of the unitary group of arbitrary dimension d which is constructed in a composite way. We show explicitly how any element of can be composed of matrix exponential functions of generalized anti-symmetric σ-matrices and one-dimensional projectors. The specific form makes it considerably easy to identify and discard redundant parameters in several cases. In this way, redundancy-free density matrices of arbitrary rank k can be formulated. Our construction can also be used to derive an orthonormal basis of any k -dimensional subspaces of with the minimal number of parameters. As an example it is shown that this feature leads to a significant reduction of parameters in the case of investigating distillability of quantum states via lower bounds of an entanglement measure (the m -concurrence).
Local unitary equivalence is an important ingredient for quantifying and classifying entanglement. Verifying whether or not two quantum states are local unitary equivalent is a crucial problem, where only the case of multipartite pure states is solved. For mixed states, however, the verification of local unitary equivalence is still a challenging problem. In this paper, based on the coefficient matrices of generalized Bloch representations of quantum states, we find a variety of local unitary invariants for arbitrary-dimensional bipartite quantum states. These invariants are operational and can be used as necessary conditions for verifying the local unitary equivalence of two quantum states. Furthermore, we extend the construction to the arbitrary-dimensional multipartite case. We finally apply these invariants to estimate concurrence, a vital entanglement measure, showing the practicability of local unitary invariants in characterizing entanglement.