Mutually Unbiased Bases for Variational Quantum Initialization: Basis-Union Optimality and Adaptive Family Search
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arXiv
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| Natura: | Preprint |
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2026
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| author | Semre, Abed Frankel, Steven |
| author_facet | Semre, Abed Frankel, Steven |
| contents | We study mutually unbiased bases (MUBs) as structured finite initialization and adaptation families for variational quantum algorithms. The main theoretical result is that, in every dimension admitting a complete set of MUBs, the complete MUB ensemble maximizes isotropic Gaussian random-Hamiltonian width among all unions of d+1 orthonormal bases in C^d. Equivalently, within this basis-union class, it gives the smallest expected best-of-set minimum for random-Hamiltonian minimization. The proof represents each orthonormal basis as a regular-simplex Gaussian block and uses a centered-convex Gaussian correlation inequality to show that the independent-block case, realized by complete MUBs, is stochastically extremal. We also record a radial extension for Hamiltonians H=RG with R nonnegative and independent, and the unrestricted qubit case, where complete qubit MUBs are globally optimal among arbitrary six-state ensembles by a Bloch-sphere/octahedron mean-width argument.
We then separate this coverage theorem from variational training dynamics. For diagonal QUBO costs, the MUB-family dependence of a fully matched construction collapses; for the canonical b=0 label it reduces to ordinary X-mixer QAOA. The empirical method is therefore adaptive MUB-XRot warm-start QAOA rather than canonical matched-mixer MUB-QAOA. In a cross-problem benchmark over MaxCut, weighted MaxCut, MIS, weighted MIS, and knapsack, adaptive MUB-XRot is non-worse than standard QAOA in 80.0% of 1500 paired cases, with win/tie/loss 829/371/300 and mean decoded-ratio improvement +0.1616. A separate QRAO MaxCut study shows that bit-flip MUB-family search reaches mean relaxed ratio 0.921 and improves over the X-variational baseline by +0.0608. The evidence is quality-oriented and incurs substantial runtime overhead; no quantum-advantage claim is made. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_16060 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Mutually Unbiased Bases for Variational Quantum Initialization: Basis-Union Optimality and Adaptive Family Search Semre, Abed Frankel, Steven Quantum Physics Emerging Technologies 81P68, 81P45, 60G15, 90C59 F.1.2; F.2.2; G.1.6; G.3 We study mutually unbiased bases (MUBs) as structured finite initialization and adaptation families for variational quantum algorithms. The main theoretical result is that, in every dimension admitting a complete set of MUBs, the complete MUB ensemble maximizes isotropic Gaussian random-Hamiltonian width among all unions of d+1 orthonormal bases in C^d. Equivalently, within this basis-union class, it gives the smallest expected best-of-set minimum for random-Hamiltonian minimization. The proof represents each orthonormal basis as a regular-simplex Gaussian block and uses a centered-convex Gaussian correlation inequality to show that the independent-block case, realized by complete MUBs, is stochastically extremal. We also record a radial extension for Hamiltonians H=RG with R nonnegative and independent, and the unrestricted qubit case, where complete qubit MUBs are globally optimal among arbitrary six-state ensembles by a Bloch-sphere/octahedron mean-width argument. We then separate this coverage theorem from variational training dynamics. For diagonal QUBO costs, the MUB-family dependence of a fully matched construction collapses; for the canonical b=0 label it reduces to ordinary X-mixer QAOA. The empirical method is therefore adaptive MUB-XRot warm-start QAOA rather than canonical matched-mixer MUB-QAOA. In a cross-problem benchmark over MaxCut, weighted MaxCut, MIS, weighted MIS, and knapsack, adaptive MUB-XRot is non-worse than standard QAOA in 80.0% of 1500 paired cases, with win/tie/loss 829/371/300 and mean decoded-ratio improvement +0.1616. A separate QRAO MaxCut study shows that bit-flip MUB-family search reaches mean relaxed ratio 0.921 and improves over the X-variational baseline by +0.0608. The evidence is quality-oriented and incurs substantial runtime overhead; no quantum-advantage claim is made. |
| title | Mutually Unbiased Bases for Variational Quantum Initialization: Basis-Union Optimality and Adaptive Family Search |
| topic | Quantum Physics Emerging Technologies 81P68, 81P45, 60G15, 90C59 F.1.2; F.2.2; G.1.6; G.3 |
| url | https://arxiv.org/abs/2605.16060 |