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).
Simon's problem asks the following: determine if a function f:{0,1}n→{0,1}n is one-to-one or if there exists a unique s∈{0,1}n such that f(x)=f(x⊕s) for all x∈{0,1}n, given the promise that exactly one of the two holds. A classical algorithm that can solve this problem for every f requires 2Ω(n) queries to f. Simon showed that there is a quantum algorithm that can solve this promise problem for every f using only O(n) quantum queries to f. A matching lower bound on the number of quantum queries was given by Koiran et al., even for functions f:Fpn→Fpn. We give a short proof that O(n) quantum queries is optimal even when we are additionally promised that f is linear. This is somewhat surprising because for linear functions there even exists a classical n-query algorithm.
The query model offers a concrete setting where quantum algorithms are provably superior to randomized algorithms. Beautiful results by Bernstein-Vazirani, Simon, Aaronson, and others presented partial Boolean functions that can be computed by quantum algorithms making much fewer queries compared to their randomized analogs. To date, separations of O(1) vs. √N between quantum and randomized query complexities remain the state-of-the-art (where N is the input length), leaving open the question of whether O(1) vs. N1/2+Ω(1)separations are possible? We answer this question in the affirmative. Our separating problem is a variant of the Aaronson-Ambainis k-fold Forrelation problem. We show that our variant: 1)Can be solved by a quantum algorithm making 2O(k)queries to the inputs. 2)Requires at least ~Ω(N2(k-1)/(3k-1)) queries for any randomized algorithm. For any constant , this gives a O(1) vs. N1/2-εseparation between the quantum and randomized query complexities of partial Boolean functions. Our proof is Fourier analytical and uses new bounds on the Fourier spectrum of classical decision trees, which could be of independent interest. Looking forward, we conjecture that the Fourier bounds could be further improved in a precise manner, and show that such conjectured bounds imply optimal O(1) vs. N1-εseparations between the quantum and randomized query complexities of partial Boolean functions.