Complexity with constant spectral gap and growing Hamiltonian variation

Determine the complexity of Berry phase estimation when the spectral gap is constant but the Hamiltonian variation bound B_1 grows polynomially with the system size, including constant-precision estimation with an accurate guiding state, and establish whether this regime remains classically tractable or permits BQP-hardness without a small spectral gap.

Background

The paper proves BQP-hardness for quantized Berry phase estimation when the spectral gap is inverse-polynomial, even with constant B_1, and provides classical algorithms for geometrically local Hamiltonians when both the spectral gap and B_1 are constant. The unresolved intermediate regime allows B_1 to grow polynomially while retaining a constant gap and constant precision.

Resolving this problem would clarify whether large variation of the Hamiltonian path can restore quantum computational hardness despite the absence of a small spectral gap. The authors relate the question to the computational power of constant-gap adiabatic evolution, including settings where constant-gap evolution can support universal computation through degenerate ground spaces.

References

Several open problems remain. First, what is the complexity of Berry phase estimation when the spectral gap remains constant but $B_1$ is allowed to grow polynomially in $n$, even at constant precision and with an accurate guiding state? Does this regime remain classically tractable, or can sufficiently large variation revive $\mathsf{BQP}$-hardness without a small gap?

— Encoding universal quantum computation into quantized Berry phases: Hardness results and classical algorithms  (2609.37199 - Sakamoto et al., 29 Sep 2026) in Section Conclusion, first open-problem paragraph

Second, can our classical algorithms be extended to guiding states with smaller overlap? In particular, it remains open whether efficient classical estimation is possible with an arbitrary nonzero constant, or even inverse-polynomial, overlap with the initial ground state.

— Encoding universal quantum computation into quantized Berry phases: Hardness results and classical algorithms  (2609.37199 - Sakamoto et al., 29 Sep 2026) in Section Conclusion, second open-problem paragraph

Third, what is the complexity of constant-precision Berry phase estimation without a guiding state? Previous work establishes $\mathsf{UQMA}\cap\mathsf{co}\text{-}\mathsf{UQMA}$-completeness at inverse-polynomial precision when an energy threshold that separates the ground energy and first excited energy is given. It is essential to clarify how this characterization changes under exact phase quantization or a constant spectral gap.

— Encoding universal quantum computation into quantized Berry phases: Hardness results and classical algorithms  (2609.37199 - Sakamoto et al., 29 Sep 2026) in Section Conclusion, third open-problem paragraph

These results suggest analogous questions for Berry phase estimation. Can Berry phase estimation be performed faster than the complexity of $O(2{n/2})$, and can fine-grained upper and lower bounds characterize how the achievable speedup depends on the phase precision and spectral gap?

— Encoding universal quantum computation into quantized Berry phases: Hardness results and classical algorithms  (2609.37199 - Sakamoto et al., 29 Sep 2026) in Section Conclusion, fourth open-problem paragraph