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The SIC Question: History and State of Play
Christopher A. Fuchs, Michael C. Hoang, Blake C. Stacey
TL;DR
The paper addresses whether SIC measurements exist in every dimension and reviews their interdisciplinary significance. It surveys group-covariant numerical searches and related symmetry-based reductions, reporting solutions through d = 151, additional cases through d = 844, and likely completeness of the catalogue through d = 90.
Problem
Whether maximal complex equiangular sets can be constructed for every dimension remains open despite exact and high-precision numerical solutions.
Method
The paper combines a historical and interdisciplinary review with numerical searches that exploit group covariance, Weyl–Heisenberg structure, optimization, parallelization, and symmetry-based reductions.
Results
Numerical solutions are known in all dimensions through d = 151, in additional dimensions through d = 844, and Scott's catalogue is strongly expected to be complete through d = 90.
Takeaways & Limitations
The expanded numerical record supports broader study of SICs' connections with algebraic number theory and their additional symmetry beyond the defining conditions.
Abstract
from arXiv · showhide
Recent years have seen significant advances in the study of symmetric informationally complete (SIC) quantum measurements, also known as maximal sets of complex equiangular lines. Previously, the published record contained solutions up to dimension 67, and was with high confidence complete up through dimension 50. Computer calculations have now furnished solutions in all dimensions up to 151, and in several cases beyond that, as large as dimension 844. These new solutions exhibit an additional type of symmetry beyond the basic definition of a SIC, and so verify a conjecture of Zauner in many new cases. The solutions in dimensions 68 through 121 were obtained by Andrew Scott, and his catalogue of distinct solutions is, with high confidence, complete up to dimension 90. Additional results in dimensions 122 through 151 were calculated by the authors using Scott's code. We recap the history of the problem, outline how the numerical searches were done, and pose some conjectures on how the search technique could be improved. In order to facilitate communication across disciplinary boundaries, we also present a comprehensive bibliography of SIC research.
I. INTRODUCTION
SICs are maximal sets of complex equiangular lines that connect quantum measurement with several areas of mathematics and physics. Their existence remains open in general, despite extensive exact and numerical progress, including solutions through dimension 151 and beyond.
- SICs consist of d^2 unit vectors in C^d whose pairwise squared inner products equal 1/(d + 1).
- The SIC problem asks whether such maximal equiangular sets can be constructed for every dimension, and remains open despite exact and high-precision numerical solutions.
- Numerical solutions are now known in every dimension through d = 151, with additional solutions reaching d = 844.The published numerical record previously reached d = 67.
- Scott extended searches through d = 121, while the authors used his code on the Chimera supercomputer to calculate solutions in dimensions 122 through 151.
- Computational research helps reveal likely complete catalogues in many dimensions and supports investigation of SICs' connections with algebraic number theory.
- SICs also define quantum measurements whose outcome probabilities determine the probabilities of any other experiment on the same d-level system.
II. GENERATING SICS WITH GROUPS
The search for SICs is reduced by exploiting group covariance, especially Weyl–Heisenberg symmetry. All known SICs have this additional symmetry, although it is not known whether every SIC must have it.
- A group-covariant SIC is generated by applying group elements to one fiducial vector, greatly reducing the search space.
- In all but one known case, the generating group is a Weyl–Heisenberg group built from shift and phase operators in dimension d.
- The Weyl–Heisenberg generators obey a commutation relation in which exchanging their order introduces a dimension-dependent phase factor.
- Weyl–Heisenberg displacement operators are closed under multiplication up to phase, enabling them to form the relevant group after phase factors are included.
- The known exception is the Hoggar SIC in d = 8; a Weyl–Heisenberg SIC also exists in that dimension.
III. HISTORICAL OVERVIEW
SIC research developed across geometry, quantum information, algebra, and number theory. Its history includes early low-dimensional constructions, Zauner's symmetry-based formulation, and increasingly broad interdisciplinary bibliographic coverage.
- SICs emerged at the intersection of geometric equiangular-line questions and quantum-information motivations, prompting an explicitly interdisciplinary historical account.
- The Hesse SIC traces to Coxeter's 1940 Hessian polyhedron, while explicit mathematical study of dimensions 2 and 3 appeared by the 1970s.
- Zauner began studying SICs in the 1990s, proved existence through d = 5 by 1999, and proposed simplifying Weyl–Heisenberg searches using a particular unitary operator.
- Caves independently developed SICs as quantum measurements, initially motivated by attempts to prove the quantum de Finetti theorem.
- Exact SIC expressions led to connections with Galois theory and algebraic number theory, which remain under active investigation.
- The real analogue has an upper bound of d(d + 1)/2 equiangular lines, unlike the complex bound d^2, and the real bound is known to be attained only in dimensions 2, 3, 7, and 23.
- The complex problem uses continuous inner-product phases, whereas the real problem has a discrete sign choice, giving the two settings different mathematical character.
IV. HOW TO SEARCH FOR SICS NUMERICALLY
Numerical SIC searches transform fiducial-vector constraints into conditions on correlation and Fourier matrices, then minimize a lower-bounded objective with repeated parallel optimization. The resulting search found solutions through dimension 151, while search time varied substantially across dimensions and motivated a linear-complexity conjecture.
- Search representation: The search represents candidate fiducials through matrices F and G, whose Fourier and autocorrelation relations encode squared inner products.The G matrix removes phase factors and treats its indices symmetrically, making it convenient for computation.
- Optimization objective: A candidate is accepted when minimizing the inequality’s left-hand side reaches its lower bound, which occurs if and only if the input is a SIC fiducial.This converts SIC construction into a numerical optimization problem.
- Optimization procedure: Repeated optimizations from different starting points address local minima, and parallel trials enabled searches on 96 Chimera cores using L-BFGS.The authors used implementations in Mathematica, Python, and C++, with Scott’s C++ code employed on Chimera.
- Search results: Solutions through dimension 151 were obtained, but elapsed search time did not increase steadily with dimension.Dimension 146 took eleven days, dimension 148 twelve days, dimension 147 eighteen hours, and dimension 150 two hours, whereas dimension 151 took 28 days.
- Search results: The authors suspect that variation in search time reflects differing numbers of inequivalent solutions, because more solutions may make successful discovery easier.This is presented as a conjecture based on the observed variation, not as an established explanation.
- Search simplification: The 3d conjecture proposes that a subset of constraints suffices to imply all SIC equations, reducing the equation count from quadratic to linear if true.It had been verified numerically through dimension 28.
V. ZAUNER SYMMETRY
Zauner symmetry restricts SIC fiducials to eigenspaces of a particular order-3 Clifford unitary, thereby reducing the numerical search space. The symmetry also yields matrix degeneracies and proves additional cases of the 3d conjecture, including dimension 9.
- Clifford framework: The Clifford group is the normalizer of the Weyl–Heisenberg group, consisting of unitaries that map its operators to themselves under conjugation.This supplies the group-theoretic setting for studying additional fiducial symmetries.
- Zauner conjecture: Zauner’s conjecture asserts that for every dimension d > 2, a Weyl–Heisenberg SIC has a fiducial that is an eigenvector of a specific order-3 Clifford unitary.Applying the defining transformation three times returns the original operation, confirming order 3.
- Matrix consequences: Requiring a fiducial to be a Zauner eigenvector creates degeneracies in F, so specifying one column simultaneously fixes a row and a diagonal.The Zauner unitary maps the left edge of F to the top edge and then to the main diagonal.
- 3d conjecture: The Zauner condition proves additional cases of the 3d conjecture through dimension 9 by combining orbit constraints with column averages in G.The argument extends the directly established cases from dimensions 5 through 8 to dimension 9.
- Numerical search: Projecting vectors into the relevant eigenspace at each optimization iteration significantly reduces the effective numerical search space.The eigenspace projector is constructed for a unitary of order n, and known searches largely used Zauner symmetry.
VI. EXHAUSTIVE SEARCHES
The section explains how Clifford symmetry narrows SIC classification and how random numerical searches identify essentially distinct solutions. Scott’s procedure supports a catalogue strongly expected to be complete through dimension 90.
- VI. EXHAUSTIVE SEARCHES: The Hesse SIC is invariant under the entire Clifford group, whereas a related fiducial generates four distinct SICs comprising 36 Norrell-state vectors.The Norrell states are identified as significant in quantum computation.
- VI. EXHAUSTIVE SEARCHES: Clifford unitaries map Weyl–Heisenberg SICs to SICs, allowing equivalence classes to be defined through the extended Clifford group.The extended group includes complex conjugation alongside unitary operations.
- VI. EXHAUSTIVE SEARCHES: Scott’s code samples initial vectors randomly using the Haar measure, refines discovered solutions, and identifies unique extended-Clifford orbits.Orbit identification is computationally demanding.
- VI. EXHAUSTIVE SEARCHES: Scott’s exhaustive searches extend through dimension 90, and the authors strongly expect the resulting catalogue to contain all Weyl–Heisenberg SICs up to extended-Clifford equivalence.This completeness claim is explicitly described as an expectation rather than a proof.
VII. DISCUSSION
The discussion connects exact SIC phases to algebraic number theory and notes applications beyond quantum computation. Known Weyl–Heisenberg SIC phases appear related to ray class fields, while SICs have also been used in engineering-oriented signal-processing tasks.
- VII. DISCUSSION: For all known Weyl–Heisenberg SICs with d > 3, the phases are units in ray class fields and related extensions.The paper links this observation to Hilbert’s twelfth problem.
- VII. DISCUSSION: The phases can reconstruct a SIC, making their algebraic-number-theoretic structure relevant to the underlying measurement construction.The passage states that the phases determine the SIC.
- VII. DISCUSSION: SICs have applications in signal processing, including high-precision radar and speech recognition.
IX. AUTHOR CONTRIBUTIONS
The author-contributions section assigns computational work for dimensions 122–151, paper writing, research direction, bibliography contributions, and revision responsibilities among the authors.
- IX. AUTHOR CONTRIBUTIONS: MCH performed the calculations on Chimera that found SICs in dimensions 122 through 151.
- IX. AUTHOR CONTRIBUTIONS: BCS wrote the paper, while CAF directed the research and contributed to the bibliography and revisions.