Super-polynomial complex quantum states

The Berry phase is a fundamental geometric quantity for characterizing the geometry and topology of quantum many-body systems. Previous work established a super-polynomial quantum advantage in Berry phase estimation at inverse-polynomial precision, given an ansatz state approximating the initial ground state. However, whether this hardness persists for Berry phases quantized by symmetry, which can be distinguished at constant precision, remained open. In this work, we characterize the computational complexity of quantized Berry phase estimation. First, we prove that deciding whether the Berry phase is exactly $0$ or $π$ is $\mathsf{BQP}$ -complete when the spectral gap is inverse-polynomially small. This result implies that distinguishing the quantized Berry phase still has a super-polynomial quantum advantage, assuming $\mathsf{BPP}\neq \mathsf{BQP}$ . To prove this result, we introduce an encoding that maps the output of any quantum computation to a Berry phase that is exactly $0$ or $π$ . Second, we extend this hardness to physically motivated Hamiltonians on a 2D square lattice, including Heisenberg and XY interactions. Third, we provide a classical polynomial-time algorithm for geometrically local Hamiltonians on fixed-dimensional lattices with constant spectral gaps. Therefore we identify that the spectral gap is a key resource for the classical tractability, rather than precision. These results establish a super-polynomial quantum advantage for computing a quantized topological invariant that is unchanged under symmetry- and gap-preserving deformations, and provide new steps towards practical quantum advantage in quantum physics.The Berry phase is a fundamental geometric quantity for characterizing the geometry and topology of quantum many-body systems. Previous work established a super-polynomial quantum advantage in Berry phase estimation at inverse-polynomial precision, given an ansatz state approximating the initial ground state. However, whether this hardness persists for Berry phases quantized by symmetry, which can be distinguished at constant precision, remained open. In this work, we characterize the computational complexity of quantized Berry phase estimation. First, we prove that deciding whether the Berry phase is exactly $0$ or $π$ is $\mathsf{BQP}$-complete when the spectral gap is inverse-polynomially small. This result implies that distinguishing the quantized Berry phase still has a super-polynomial quantum advantage, assuming $\mathsf{BPP}\neq \mathsf{BQP}$. To prove this result, we introduce an encoding that maps the output of any quantum computation to a Berry phase that is exactly $0$ or $π$. Second, we extend this hardness to physically motivated Hamiltonians on a 2D square lattice, including Heisenberg and XY interactions. Third, we provide a classical polynomial-time algorithm for geometrically local Hamiltonians on fixed-dimensional lattices with constant spectral gaps. Therefore we identify that the spectral gap is a key resource for the classical tractability, rather than precision. These results establish a super-polynomial quantum advantage for computing a quantized topological invariant that is unchanged under symmetry- and gap-preserving deformations, and provide new steps towards practical quantum advantage in quantum physics.

quant-pharXiv:2609.37199Sep 30, 20260 citations

Composite Complex Quantum Systems

Springer eBooksDec 12, 20050 citations

The Structure of Promises in Quantum Speedups

Shalev Ben-David

In 1998, Beals, Buhrman, Cleve, Mosca, and de Wolf showed that no super-polynomial quantum speedup is possible in the query complexity setting unless there is a promise on the input. We examine several types of "unstructured" promises, and show that they also are not compatible with super-polynomial quantum speedups. We conclude that such speedups are only possible when the input is known to have some structure. Specifically, we show that there is a polynomial relationship of degree 18 between D(f) and Q(f) for any Boolean function f defined on permutations (elements of [n]^n in which each alphabet element occurs exactly once). More generally, this holds for all f defined on orbits of the symmetric group action (which acts on an element of [M]^n by permuting its entries). We also show that any Boolean function f defined on a "symmetric" subset of the Boolean hypercube has a polynomial relationship between R(f) and Q(f) - although in that setting, D(f) may be exponentially larger.In 1998, Beals, Buhrman, Cleve, Mosca, and de Wolf showed that no super-polynomial quantum speedup is possible in the query complexity setting unless there is a promise on the input. We examine several types of "unstructured" promises, and show that they also are not compatible with super-polynomial quantum speedups. We conclude that such speedups are only possible when the input is known to have some structure. Specifically, we show that there is a polynomial relationship of degree 18 between D(f) and Q(f) for any Boolean function f defined on permutations (elements of [n]^n in which each alphabet element occurs exactly once). More generally, this holds for all f defined on orbits of the symmetric group action (which acts on an element of [M]^n by permuting its entries). We also show that any Boolean function f defined on a "symmetric" subset of the Boolean hypercube has a polynomial relationship between R(f) and Q(f) - although in that setting, D(f) may be exponentially larger.

Leibniz international proceedings in informaticsJan 1, 20162 citations

Entangled Quantum States

Willi-Hans Steeb, Yorick Hardy

International Journal of Theoretical PhysicsDec 1, 20002 citations

Average/Worst-Case Gap of Quantum Query Complexities by On-Set Size

Andris Ambainis, Kazuo Iwama, Masaki Nakanishi +4 more

This paper considers the query complexity of the functions in the family F_{N,M} of N-variable Boolean functions with onset size M, i.e., the number of inputs for which the function value is 1, where 1<= M <= 2^{N}/2 is assumed without loss of generality because of the symmetry of function values, 0 and 1. Our main results are as follows: (1) There is a super-linear gap between the average-case and worst-case quantum query complexities over F_{N,M} for a certain range of M. (2) There is no super-linear gap between the average-case and worst-case randomized query complexities over F_{N,M} for every M. (3) For every M bounded by a polynomial in N, any function in F_{N,M} has quantum query complexity Theta (sqrt{N}). (4) For every M=O(2^{cN}) with an arbitrary large constant c<1, any function in F_{N,M} has randomized query complexity Omega (N).This paper considers the query complexity of the functions in the family F_{N,M} of N-variable Boolean functions with onset size M, i.e., the number of inputs for which the function value is 1, where 1<= M <= 2^{N}/2 is assumed without loss of generality because of the symmetry of function values, 0 and 1. Our main results are as follows: (1) There is a super-linear gap between the average-case and worst-case quantum query complexities over F_{N,M} for a certain range of M. (2) There is no super-linear gap between the average-case and worst-case randomized query complexities over F_{N,M} for every M. (3) For every M bounded by a polynomial in N, any function in F_{N,M} has quantum query complexity Theta (sqrt{N}). (4) For every M=O(2^{cN}) with an arbitrary large constant c<1, any function in F_{N,M} has randomized query complexity Omega (N).

arXiv:0908.2468Aug 18, 20097 citations

Complex Quantum Systems

Heinz K. H. Siedentop

Lecture notes series, Institute For Mathematical SciencesJan 24, 20132 citations

0

π

\mathsf{BQP}

-complete when the spectral gap is inverse-polynomially small. This result implies that distinguishing the quantized Berry phase still has a super-polynomial quantum advantage, assuming

\mathsf{BPP}\neq \mathsf{BQP}

. To prove this result, we introduce an encoding that maps the output of any quantum computation to a Berry phase that is exactly

0

π

. Second, we extend this hardness to physically motivated Hamiltonians on a 2D square lattice, including Heisenberg and XY interactions. Third, we provide a classical polynomial-time algorithm for geometrically local Hamiltonians on fixed-dimensional lattices with constant spectral gaps. Therefore we identify that the spectral gap is a key resource for the classical tractability, rather than precision. These results establish a super-polynomial quantum advantage for computing a quantized topological invariant that is unchanged under symmetry- and gap-preserving deformations, and provide new steps towards practical quantum advantage in quantum physics.