Undecidability of bounded expected energy

Prove or refute the conjecture that, for a finite value \(c\), deciding whether the expected energy of a bosonic computation is at most \(c\) is undecidable.

Background

Prior work cited by the paper establishes undecidability of whether a bosonic computation consumes finite energy. The authors conjecture a stronger result: undecidability persists even when the question asks whether the expected energy is below a specified finite threshold. Establishing this would further undermine expected energy as a Blum-style complexity measure for the standard Lloyd–Braunstein model.

References

We, however, conjecture that, given a finite value c, deciding if the value of expected energy is \leq c is also undecidable.

— A physical and universal model of bosonic computations with Solovay-Kitaev theorem  (2609.40226 - Rudolph et al., 30 Sep 2026) in Section 1. Results, subsection “Open questions,” item “Energy as a computational measure”