A Reproducible Software Workflow for Unanchored Approximate MUB Optimization: A Case Study in Dimension Six

Jul 16, 2026

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A reproducible, parameter-driven software workflow optimizes approximate mutually unbiased basis (AMUB) configurations in arbitrary dimensions d using Lie-algebra unitary parameterization. Execution is portable across CPU, Apple MPS, CUDA GPUs, and HPC backends via a Taylor-series matrix-exponential compatibility layer. For the dimension-six case, unanchored configurations are optimized across 100 random seeds for n=3–6, yielding exact recovery for three bases, a recurrent four-basis partial-exact hub-and-triangle structure, and no near-exact pairs under a primary tolerance for n=5–6. A hardware validation embeds representative d=6, n=4 transitions into 8×8 unitaries and benchmarks on an ibm_marrakesh 156-qubit processor using subspace postselection.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages26

Reading Time1 h 6 min

Introduction

Mutually unbiased bases and open problems in dimension six

Motivation for unanchored numerical approaches

Workflow overview and parameter-driven optimization

Unitary parameterization via Lie-algebra exponentiation

Implementation, backends, and accelerator compatibility

Case study in dimension six

A Reproducible Software Workflow for Unanchored Approximate MUB Optimization: A Case Study in Dimension Six arXiv :2607 . 10615v1 [ cs .MS] 12 Jul 2026 Abdul Fatah , Ian McLoughlin, and Saim Ghafoor Atlantic Technological University, Ireland We present a reproducible, parameterdriven software workflow for optimizing approximate mutually unbiased basis (AMUB) configurations in arbitrary dimensions d using a Lie-algebra unitary parameterization. The workflow is designed for portable execution across CPU, Apple MPS, CUDA-capable GPU, and HPC backends, using a Taylor-series matrixexponential layer as an accelerator compatibility pathway. As a dimension-six case study, we optimize unanchored configurations across 100 random seeds for basis counts n = 3 , 4 , 5 , 6 in complex128 and complex64 arithmetic. The workflow recovers exact three-basis configurations, identifies a recurrent four-basis partial-exact hub-and-triangle structure, and finds no near-exact pairs for n = 5 or n = 6 in the reported campaigns under the primary tolerance. As a hardware-execution check, we embed the representative d = 6, n = 4 transition unitaries into three-qubit 8 × 8 unitaries and execute the resulting circuits on the 156-qubit Heron processor ibm_marrakesh using subspace postselection. The measured QPU pairwise losses are dominated by a hardware and compilation noise floor of approximately 0.02–0.08, associated with compiled circuits averaging 37 native CZ gates, which obscures the distinction between classically near-exact and defective pairs. The results provide a reproducible computational framework for exploring AMUB Abdul Fatah: [abdul.fatah@research.atu.ie](abdul.fatah@research.atu.ie) Ian McLoughlin: [ian.mcloughlin@atu.ie](ian.mcloughlin@atu.ie) Saim Ghafoor: [saim.ghafoor@atu.ie](saim.ghafoor@atu.ie) landscapes, together with an initial assessment of the challenges involved in executing optimized dimension-six unitaries on current quantum hardware. 1 Introduction Mutually unbiased bases (MUBs) are central objects in quantum information theory, combinatorial design, and finite-dimensional Hilbert space geometry. Two orthonormal bases B and C in Cd are mutually unbiased [1] if |⟨b, c⟩| 2 = 1d for all b ∈ B, c ∈ C. A collection of bases is mutually unbiased if every pair of bases in the collection satisfies this condition. Such structures play a fundamental role in quantum state tomography, quantum cryptography [2], and operator theory. In dimensions that are prime or powers of primes, complete sets of d + 1 mutually unbiased bases are known to exist [3] . In composite dimensions [4], and in particular in d = 6 , the existence of a complete set of seven MUBs remains a longstanding open problem [5 , 6] . While numerous analytic and computational approaches have been explored [7], no definitive resolution is known. Computational searches in dimension six often reduce the search space by fixing one or more bases to canonical representatives, such asthe identity, Fourier-type matrices, or prescribed families of complex Hadamard matrices. Fixing a single basis can be interpreted as a gauge choice under the common left-unitary action and is not, by itself, necessarily restrictive. However, searches that impose additional canonical forms or restrict subsequent bases to particular Hadamard families can bias the explored landscape toward selected representatives. This mo- tivates complementary unanchored numerical approaches in which all candidate bases are optimized simultaneously and the resulting pairwise defect geometry is analysed after optimization. This paper adopts a mathematical-software and computational-physics perspective. We present a generalized, parameter-driven software workflow for optimizing and analysing approximate mutually unbiased basis (AMUB) configurations across dimensions, candidate basis counts, random seeds, numerical precisions, and hardware backends. The aim is not to provide a proof of ex

FAQs

How does the document validate results on quantum hardware?

It embeds representative d=6, n=4 transition unitaries into three-qubit 8×8 unitaries, runs the resulting circuits on IBM’s ibm_marrakesh processor using subspace postselection, and compares measured pairwise losses against noise and compilation effects.

Jul 16, 2026

Upload by Riley