We give exponential quantum query separations between the standard XOR interface and two forward-only interfaces that supply neither an adjoint nor an inverse oracle. Let $X=\F_2^n$, N=∣X∣, and fh,r(x)=(h(x),x,rx), where h:X→X is promised to be either a permutation or a Simon two-to-one function, and r is a fixed table of n-bit tags, unrestricted by the promise and reused on every query. The resulting problem is solvable with at most n+2 standard XOR queries, but has forward-erasing query complexity Θ(N). This answers affirmatively open question 11 in [Scott Aaronson. Open problems related to quantum query complexity. ACM Transactions on Quantum Computing, 2(4):14:1-14:9, 2021] . We also embed these instances into permutations. The detailed construction retains the copy of x in each prescribed output, but for these promises that copy can be replaced by one bit that distinguishes the two inputs in every Simon pair. This gives a permutation domain of size L=4N2 and a permutation problem with the same standard-query upper bound and forward-only in-place query complexity Θ(N)=Θ(L1/4). Both lower bounds remain valid with a clean coherent bypass. The common lower bound uses an analysis-only recording replacement. In the replacement computation, tracing out the fixed random tag table after T calls gives a sum of positive-semidefinite operator contributions, each depending on h at no more than T addresses. On such a set, the restrictions induced by random permutations and random Simon functions differ only if the set contains a hidden Simon pair, an event of probability O(T2/N).
We study the quantum query complexity of maximizing a non-negative submodular function, considering both the unconstrained setting and, for monotone functions, a cardinality constraint k on an n-element ground set. In the exact reversible digital value-oracle model, our unconstrained algorithm achieves an expected (1/2−ε)-approximation using only Oε(logn) queries. In contrast, any classical randomized algorithm that attains a fixed expected ratio above 1/4 requires Ω(n/logn) queries (Li, Feldman, Kazemi, and Karbasi, 2022), establishing an exponential separation in query complexity. For cardinality-constrained maximization, we give a bounded-error quantum algorithm that achieves a (1−1/e−ε)-approximation using Oε(min{n,n/k}) queries. When k=o(n), our algorithm achieves at least a quadratic speedup up to logarithmic factors over classical randomized algorithms (Mirzasoleiman, Badanidiyuru, Karbasi, Vondrák, and Krause, 2015; Peng and Rubinstein, 2025). Moreover, when k=cn for any fixed rational c<1−1/e−ε, the query complexity reduces to Oε,c(logn), yielding an exponential separation from the classical Ω(n/logn) lower bound (Li, Feldman, Kazemi, and Karbasi, 2022). We further prove quantum lower bounds of exp(Ω(ε2n)) queries for achieving a ratio beyond 1/2+ε without constraints, and exp(Ω(ε2k)) queries for exceeding 1−1/e+ε when k/n≤ε. These barriers demonstrate that quantum computation offers no exponential speedup at these approximation thresholds.
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).