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Naceur, Wang and Colleagues Certify Quantum Optimization Bounds
A paper in Quantum presents a method for converting approximate semidefinite-program results into exact certificates for bounds in quantum theory. The authors test it on Bell inequalities and quantum many-body systems, including a Heisenber
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“To counter this issue, we introduce a framework for extracting rigorous certified bounds on non-commutative optimization problems from inexact numerical data, and apply it to several paradigmatic problems in quantum information theory.”
The authors write: Younes Naceur, Jie Wang, Victor Magron and Antonio Acín, authors of the paper.
Floating-point answers can miss the bound
Published on October 28, 2026, the paper addresses a gap in a standard way of calculating limits in quantum theory. Semidefinite programs can estimate bounds on problems involving non-commuting operators, including maximum violations of Bell inequalities and the energies of quantum many-body systems. But solvers work with finite-precision numbers, and their outputs can fail to satisfy the mathematical constraints exactly.
That failure can turn an apparent upper or lower bound into an invalid one. The authors note that an inaccurate energy bound can also contaminate later calculations when a result feeds into another optimization. Their method takes the solver’s approximate output and converts it into a certificate whose conditions can be checked using exact rational arithmetic.
Rational arithmetic repairs the certificate
The procedure starts with a numerical Gram matrix, the matrix used to express the optimization result as a sum of squares. First, the authors round its entries to nearby rational numbers. They then project the matrix onto the affine subspace that satisfies the certificate’s equality constraints, before adjusting it to restore positive semidefiniteness.
After rounding, those operations use exact rational arithmetic rather than another floating-point calculation. The result is an algebraic certificate of a valid bound, not merely a solver’s decimal estimate. The authors also describe an additional tightening step for cases where the projected matrix lies strictly inside the positive-semidefinite cone.
Bell tests reveal incorrect numerical bounds
A worked example shows why the distinction matters. For the CHSH Bell inequality, a solver’s level-two estimate is approximately 2.828427122879, slightly below the known maximum, 2√2. That makes it an invalid upper bound. The paper’s rationalized result is approximately 2.828427125402, above the maximum and therefore valid.
The authors also tested 88 two-party, two-outcome Bell inequalities at several relaxation orders. For the B2 and B3 cases they compared against, 6 of 20 numerical bounds in B2 and 26 of 66 in B3 were invalid; all of the rationalized bounds remained valid upper bounds against the reference values. Sparse versions of the method often reduced the correction needed to certify a bound, though the outcome depended on the relaxation order and the problem’s structure.
Spin-chain tests expose the computational trade-off
The team applied the method to ground-state energy bounds for a J1–J2 Heisenberg chain, using symmetry-adapted calculations for chains as long as 50 sites. The certified bounds agreed with upper bounds obtained through density-matrix renormalization group calculations. In this test, the authors found that the correction from numerical to rational bounds decreased as the chain grew, but exploiting symmetry also increased the certification’s computational cost.
The approach does not make every bound tighter: certification can weaken a numerical result, and larger Gram matrices can increase that loss. The paper also warns that the hierarchy’s matrix sizes grow quickly with relaxation order, which can cause certified bounds to worsen even when higher orders are attempted. For calculations of observables beyond ground-state energy, the authors say the method first needs rational, certified energy bounds; establishing certified variational upper bounds falls outside this paper’s scope.