Additional complexity classes captured by BEQC

Determine which complexity classes, beyond those established in the paper, can be captured by completeness results for Bosonic Energy-Preserving Quantum Computation (BEQC) under suitable choices of time, energy, space, precision, locality, and Hamiltonian degree parameters.

Background

The paper establishes several complexity-theoretic characterizations for BEQC, including equality with BQP in the polynomial-time and polynomial-energy regime and PSPACE-completeness for non-local polynomial-degree Hamiltonians. It leaves open the broader classification of complexity classes obtainable by varying the model’s resource parameters.

References

A natural open question is which other complexity classes can be captured via completeness results in this manner.

— 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 “Computational properties”

Is the computational power of the model unchanged if, instead of a gate-based computation, we consider unit-time evolution under a many-body, energy-preserving Hamiltonian supplied with energy resource states, sometimes referred to as a ballistic model?

— 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 “Variants of the model,” subitem “Ballistic versus gate-based computation”

Does allowing homodyne measurements change the computational power of the model? In particular, when phase-sensitive measurements are available, can pure coherent resource states enable computations that are inaccessible in the thermal model, whose resource states contain no phase information?

— 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 “Variants of the model,” subitem “Homodyne measurements and phase sensitivity”

More generally, it would be interesting to fully characterize the tradeoffs between the fundamental resources of bosonic computation---time (or circuit depth), space (number of modes and ancillas), and energy. In particular, can additional energy or ancillary modes compensate for limited circuit depth, and conversely, what lower bounds relate these resources? A complementary question concerns the effect of noise. How does the computational power of these restricted models change in the presence of physically motivated noise, such as photon loss or dephasing, and how do the required energy, depth, and ancillary resources scale with the noise strength?

— 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 “Restricted models of bosonic computation”

It is interesting to ask whether we can generalize this to other energy-preserving Hamiltonians over multiple modes, which we leave as an open question.

— 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 “Fast-forwarding of energy”

A similar no-go can be shown for exponential energy states, if we can upper bound unitaries generated by simulating local constant-degree Hamiltonians on exponential-energy states to a class smaller than $\PSPACE$. Such a simulation remains open as of yet (\cref{sec:open-questions}).

— A physical and universal model of bosonic computations with Solovay-Kitaev theorem  (2609.40226 - Rudolph et al., 30 Sep 2026) in Remark following Corollary “no-compile,” Section on non-local energy-preserving gates