[Paper Review] On the Monomiality of Nice Error Bases

Andreas Klappenecker, Martin Roetteler|arXiv (Cornell University)|Jan 16, 2003

Coding theory and cryptography8 references4 citations

TL;DR

This paper resolves an open problem in quantum information theory by proving that not all nice error bases are equivalent to shift-and-multiply bases, contrary to a conjecture by Schlingemann and Werner. Using group representation theory, the authors construct an explicit counterexample in dimension 165, demonstrating the existence of non-monomial nice error bases, while also showing that every nice error basis admits a sparse representation with at least half its entries zero.

ABSTRACT

Unitary error bases generalize the Pauli matrices to higher dimensional systems. Two basic constructions of unitary error bases are known: An algebraic construction by Knill, which yields nice error bases, and a combinatorial construction by Werner, which yields shift-and-multiply bases. An open problem posed by Schlingemann and Werner (see http://www.imaph.tu-bs.de/qi/problems/6.html) relates these two constructions and asks whether each nice error basis is equivalent to a shift-and-multiply basis. We solve this problem and show that the answer is negative. However, we also show that it is always possible to find a fairly sparse representation of a nice error basis.

Motivation & Objective

• To resolve the open problem posed by Schlingemann and Werner regarding the equivalence of nice error bases and shift-and-multiply bases.
• To determine whether all nice error bases can be represented as monomial matrices (i.e., with a single non-zero entry per row and column).
• To investigate the structural limitations and representations of nice error bases in higher dimensions.
• To construct explicit counterexamples to the conjecture that all nice error bases are equivalent to shift-and-multiply bases.
• To show that while not all nice error bases are monomial, they still admit a fairly sparse matrix representation with at least 50% zero entries.

Proposed method

• Constructs a unitary error basis using a projective representation of a group $ G = (H_5 \times H_{11}) \rtimes_\varphi H_3 $, where $ H_p $ is the Heisenberg group of order $ p^3 $.
• Employs the discrete Fourier transform $ F_p $ and diagonal matrices $ D_p $ to realize automorphisms of $ H_p $, enabling the semidirect product structure.
• Uses matrix conjugation to implement the action of $ H_3 $ on $ H_5 \times H_{11} $, with generators realized via $ R_5 $ and $ R_{11} $, ensuring the group is non-abelian and non-monomial.
• Derives a faithful irreducible representation of degree 165 by combining representations of $ H_3 $, $ H_5 $, and $ H_{11} $, resulting in a non-monomial unitary error basis.
• Applies equivalence relations via unitary conjugation and phase multiplication to compare the constructed basis with shift-and-multiply types.
• Uses group-theoretic tools, including central extensions and projective representations, to verify that the constructed basis satisfies the conditions of a nice error basis but is not equivalent to any shift-and-multiply basis.

Experimental results

Research questions

• RQ1 Is every nice error basis equivalent to a shift-and-multiply basis?
• RQ2 Can a nice error basis be represented using monomial matrices?
• RQ3 What is the structural complexity of nice error bases beyond small dimensions?
• RQ4 Are there unitary error bases that are neither nice nor shift-and-multiply?
• RQ5 What is the minimal sparsity achievable in a unitary error basis representation?

Key findings

• The paper constructs an explicit counterexample to the conjecture: a nice error basis in dimension 165 that is not equivalent to any shift-and-multiply basis.
• The constructed error basis arises from a faithful irreducible representation of the group $ (H_5 \times H_{11}) \rtimes_\varphi H_3 $, which is non-monomial and not equivalent to any shift-and-multiply basis.
• The authors prove that every nice error basis admits a representation where at least half of the entries in the basis matrices are zero, indicating a form of structural sparsity.
• The study confirms that shift-and-multiply bases are not equivalent to all nice error bases, and vice versa, showing that the two constructions are not unificationally equivalent.
• The result implies that nice error bases do not necessarily have a monomial structure, despite their algebraic simplicity in low dimensions.
• The counterexample is built using group representations and automorphisms realized via conjugation with discrete Fourier and diagonal matrices, demonstrating a non-trivial representation of central type.