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Register a new userPartial Boolean functions with exact quantum 1-query complexity
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Daowen Qiu
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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We provide two sufficient and necessary conditions to characterize any $n$-bit partial Boolean function with exact quantum 1-query complexity. Using the first characterization, we present all $n$-bit partial Boolean functions that depend on $n$ bits and have exact quantum 1-query complexity. Due to the second characterization, we construct a function $F$ that maps any $n$-bit partial Boolean function to some integer, and if an $n$-bit partial Boolean function $f$ depends on $k$ bits and has exact quantum 1-query complexity, then $F(f)$ is non-positive. In addition, we show that the number of all $n$-bit partial Boolean functions that depend on $k$ bits and have exact quantum 1-query complexity is not bigger than $n^{2}2^{2^{n-1}(1+2^{2-k})+2n^{2}}$ for all $ngeq 3$ and $kgeq 2$.
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We prove two new results about the randomized query complexity of composed functions. First, we show that the randomized composition conjecture is false: there are families of partial Boolean functions $f$ and $g$ such that $R(fcirc g)ll R(f) R(g)$. In fact, we show that the left hand side can be polynomially smaller than the right hand side (though in our construction, both sides are polylogarithmic in the input size of $f$). Second, we show that for all $f$ and $g$, $R(fcirc g)=Omega(mathop{noisyR}(f)cdot R(g))$, where $mathop{noisyR}(f)$ is a measure describing the cost of computing $f$ on noisy oracle inputs. We show that this composition theorem is the strongest possible of its type: for any measure $M(cdot)$ satisfying $R(fcirc g)=Omega(M(f)R(g))$ for all $f$ and $g$, it must hold that $mathop{noisyR}(f)=Omega(M(f))$ for all $f$. We also give a clean characterization of the measure $mathop{noisyR}(f)$: it satisfies $mathop{noisyR}(f)=Theta(R(fcirc gapmaj_n)/R(gapmaj_n))$, where $n$ is the input size of $f$ and $gapmaj_n$ is the $sqrt{n}$-gap majority function on $n$ bits.
It has been proved that almost all $n$-bit Boolean functions have exact classical query complexity $n$. However, the situation seemed to be very different when we deal with exact quantum query complexity. In this paper, we prove that almost all $n$-bit Boolean functions can be computed by an exact quantum algorithm with less than $n$ queries. More exactly, we prove that ${AND}_n$ is the only $n$-bit Boolean function, up to isomorphism, that requires $n$ queries.
State complexity of quantum finite automata is one of the interesting topics in studying the power of quantum finite automata. It is therefore of importance to develop general methods how to show state succinctness results for quantum finite automata. One such method is presented and demonstrated in this paper. In particular, we show that state succinctness results can be derived out of query complexity results.
In 1986, Saks and Wigderson conjectured that the largest separation between deterministic and zero-error randomized query complexity for a total boolean function is given by the function $f$ on $n=2^k$ bits defined by a complete binary tree of NAND gates of depth $k$, which achieves $R_0(f) = O(D(f)^{0.7537ldots})$. We show this is false by giving an example of a total boolean function $f$ on $n$ bits whose deterministic query complexity is $Omega(n/log(n))$ while its zero-error randomized query complexity is $tilde O(sqrt{n})$. We further show that the quantum query complexity of the same function is $tilde O(n^{1/4})$, giving the first example of a total function with a super-quadratic gap between its quantum and deterministic query complexities. We also construct a total boolean function $g$ on $n$ variables that has zero-error randomized query complexity $Omega(n/log(n))$ and bounded-error randomized query complexity $R(g) = tilde O(sqrt{n})$. This is the first super-linear separation between these two complexity measures. The exact quantum query complexity of the same function is $Q_E(g) = tilde O(sqrt{n})$. These two functions show that the relations $D(f) = O(R_1(f)^2)$ and $R_0(f) = tilde O(R(f)^2)$ are optimal, up to poly-logarithmic factors. Further variations of these functions give additional separations between other query complexity measures: a cubic separation between $Q$ and $R_0$, a $3/2$-power separation between $Q_E$ and $R$, and a 4th power separation between approximate degree and bounded-error randomized query complexity. All of these examples are variants of a function recently introduced by goos, Pitassi, and Watson which they used to separate the unambiguous 1-certificate complexity from deterministic query complexity and to resolve the famous Clique versus Independent Set problem in communication complexity.
We study the composition question for bounded-error randomized query complexity: Is R(f o g) = Omega(R(f) R(g)) for all Boolean functions f and g? We show that inserting a simple Boolean function h, whose query complexity is only Theta(log R(g)), in between f and g allows us to prove R(f o h o g) = Omega(R(f) R(h) R(g)). We prove this using a new lower bound measure for randomized query complexity we call randomized sabotage complexity, RS(f). Randomized sabotage complexity has several desirable properties, such as a perfect composition theorem, RS(f o g) >= RS(f) RS(g), and a composition theorem with randomized query complexity, R(f o g) = Omega(R(f) RS(g)). It is also a quadratically tight lower bound for total functions and can be quadratically superior to the partition bound, the best known general lower bound for randomized query complexity. Using this technique we also show implications for lifting theorems in communication complexity. We show that a general lifting theorem for zero-error randomized protocols implies a general lifting theorem for bounded-error protocols.
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