A long-standing open question in query complexity asks whether there is a total Boolean function f with R(f) << C(f), where R(f) and C(f) denote its bounded-error randomized query complexity and certificate complexity, respectively. We construct a function with R(f) = O~(sqrt{C(f)}), which is optimal up to log factors. The same function also has $Q(f) = O~(C(f)^{1/4}), where Q(f) is the bounded-error quantum query complexity of f, which is also nearly optimal.
A long-standing open question in query complexity asks whether there is a total Boolean function f with R(f) << C(f), where R(f) and C(f) denote its bounded-error randomized query complexity and certificate complexity, respectively. We construct a function with R(f) = O~(sqrt{C(f)}), which is optimal up to log factors. The same function also has $Q(f) = O~(C(f)^{1/4}), where Q(f) is the bounded-error quantum query complexity of f, which is also nearly optimal.
We study how large the certificate complexity C(f) of a total Boolean function can be relative to its randomized and quantum query complexities. We construct a total Boolean function f whose randomized query complexity with one-sided error satisfies R1(f)=Θ(C(f)). This separation is optimal even when two-sided error is allowed.
From this construction, we obtain another total Boolean function F with bounded-error quantum query complexity Q(F)=O(C(F)1/4), attaining the general quartic bound up to polylogarithmic factors. For the same function, both exact and zero-error quantum query complexities are O(C(F)). We also prove matching lower bounds for these two measures on F.
We construct a total Boolean function f satisfying R(f)=Ω~(Q(f)5/2), refuting the long-standing conjecture that R(f)=O(Q(f)2) for all total Boolean functions. Assuming a conjecture of Aaronson and Ambainis about optimal quantum speedups for partial functions, we improve this to R(f)=Ω~(Q(f)3). Our construction is motivated by the Göös-Pitassi-Watson function but does not use it.
Certificate complexity (C(f)) is a fundamental measure of complexity of Boolean functions f which counts the number of bits of an input that need to be known in order for the value of the function to be determined. A certificate can be viewed as a partial assignment, or a boolean subcube where the function is constant. Certificate complexity is well understood for deterministic query (or decision tree) complexity (D) and other query models such as bounded-error randomized and quantum complexity (R,Q), but not as well for quantum zero-error (Q0) and exact query complexity (QE), where there is no agreed-upon certificate 'object' (even for Q). Instead, we study an operational notion of certification and apply it to various query-based models, with a focus on zero-error and exact quantum query complexity, but also on polynomial degree measures. We give new characterizations of C,RC (randomized certificate complexity) and QC (quantum certificate complexity), in terms of various measures such as classical and quantum sabotage complexity, unambiguous certificate complexity, and variants of polynomial degree. Certification complexity also gives rise to new lower bounds on QE and Q0, the quantum analogues of D and R0, complexity measures for which few lower bound techniques are known which are not already lower bounds for two-sided error quantum query complexity. We exhibit a total Boolean function for which our certification complexity measure gives a tight lower bound for Q0, but rational degree and Q are asymptotically smaller.
Buhrman, Cleve and Wigderson (STOC'98) observed that for every Boolean function f:{-1,1}ⁿ → {-1,1} and •:{-1,1}² → {-1,1} the two-party bounded-error quantum communication complexity of (f ∘ •) is O(Q(f) log n), where Q(f) is the bounded-error quantum query complexity of f. Note that the bounded-error randomized communication complexity of (f ∘ •) is bounded by O(R(f)), where R(f) denotes the bounded-error randomized query complexity of f. Thus, the BCW simulation has an extra O(log n) factor appearing that is absent in classical simulation. A natural question is if this factor can be avoided. Razborov (IZV MATH'03) showed that the bounded-error quantum communication complexity of Set-Disjointness is Ω(√n). The BCW simulation yields an upper bound of O(√n log n). Høyer and de Wolf (STACS'02) showed that this can be reduced to c^(log^* n) for some constant c, and subsequently Aaronson and Ambainis (FOCS'03) showed that this factor can be made a constant. That is, the quantum communication complexity of the Set-Disjointness function (which is NOR_n ∘ ∧) is O(Q(NOR_n)). Perhaps somewhat surprisingly, we show that when • = ⊕, then the extra log n factor in the BCW simulation is unavoidable. In other words, we exhibit a total function F:{-1,1}ⁿ → {-1,1} such that Q^{cc}(F ∘ ⊕) = Θ(Q(F) log n). To the best of our knowledge, it was not even known prior to this work whether there existed a total function F and 2-bit function •, such that Q^{cc}(F ∘ •) = ω(Q(F)).