[Paper Review] Hyperspherical Parameterization of Unitary Matrices

Samuel R. Hedemann|arXiv (Cornell University)|Mar 24, 2013

Matrix Theory and Algorithms3 references4 citations

TL;DR

This paper presents a hyperspherical parameterization of unitary matrices using Cayley-Klein parameters to provide a compact, canonical form for SU(n) matrices that embeds constraints directly into the parameterization. By expressing general unitary matrices through a product of qubit-like rotation matrices with essential parameters constrained via hyperspherical coordinates, the method enables efficient generation of unitary operators for quantum mechanics, quantum computation, and multiport interferometry without matrix exponentiation.

ABSTRACT

Unitary operators are essential to quantum mechanics, however for discrete systems larger than a qubit, it is difficult to express them in a self-contained way. This report presents just such a description, providing a compact, useful parameterization, with examples of physical applications.

Motivation & Objective

• To develop a self-contained, canonical parameterization of unitary matrices in arbitrary dimensions that embeds unitarity and special unitarity constraints intrinsically.
• To simplify the representation of general unitary operators for applications in quantum mechanics, quantum information, and quantum control by minimizing reliance on matrix exponentiation.
• To provide a practical, computationally efficient framework for generating orthonormal bases, designing multiport interferometers, and constructing entanglement-preserving transformations.
• To demonstrate the utility of the parameterization in physical systems such as coherent state processing and multi-qubit/multi-level quantum gates.
• To offer a systematic, predictive method for unitary matrix construction using hyperspherical coordinates with minimal redundancy and maximal interpretability.

Proposed method

• Derives a general factorization of SU(n) matrices as a product of n(n−1)/2 qubit-like rotation matrices, each parameterized via Cayley-Klein (CK) parameters.
• Introduces a canonical form where the first row and column of the matrix contain only single terms, simplifying interpretation and application.
• Applies hyperspherical parameterization to the essential parameters of each rotation matrix, ensuring all constraints (unitarity, determinant=1) are automatically satisfied.
• Uses a recursive structure with Kronecker delta-dependent sign conventions to maintain consistency across dimensions, particularly distinguishing n=2 from n≥3.
• Expresses the full general unitary matrix as G^{[n]} = e^{iγ}U^{[n]}, where U^{[n]} is the SU(n) matrix parameterized via the essential parameters and hyperspherical coordinates.
• Employs generalized Gell-Mann matrices and tensor product decompositions to express entanglement-preserving transformations without matrix exponentiation.

Experimental results

Research questions

• RQ1 How can a canonical, constraint-aware parameterization of SU(n) matrices be constructed to avoid redundancy and simplify quantum circuit design?
• RQ2 Can hyperspherical coordinates be effectively used to parameterize the essential degrees of freedom in unitary matrices while preserving orthonormality and determinant constraints?
• RQ3 To what extent can this parameterization replace matrix exponentiation in generating entanglement-preserving transformations in multi-party quantum systems?
• RQ4 How does the proposed method enable calibration and error isolation in multiport interferometers using classical coherent state inputs?
• RQ5 What is the minimal, physically meaningful parameter set required to fully describe the most general n×n unitary matrix without redundancy?

Key findings

• The parameterization yields a general SU(n) matrix with exactly n² real parameters, matching the dimension of the Lie group U(n), ensuring completeness and minimality.
• The method constructs the most general unitary matrix without matrix exponentiation, using only a product of n(n−1)/2 qubit-like rotation matrices with hyperspherical parameters.
• For a three-mode coherent state processor, the output state parameters transform linearly under the unitary, with the sum of squared magnitudes preserved, confirming unitarity.
• The parameterization enables direct calibration of multiport interferometers by isolating individual qubit rotations through classical input states, reducing experimental troubleshooting complexity.
• Entanglement-preserving transformations in multi-party systems can be explicitly written as tensor products of SU(n) matrices parameterized via the essential hyperspherical form, avoiding exponentiation of generators.
• The method provides a systematic way to generate orthonormal bases and unitary operations in quantum information with predictable, interpretable parameters derived from hyperspherical coordinates.