Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userImproved Approximate Degree Bounds For k-distinctness
81
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
An open problem that is widely regarded as one of the most important in quantum query complexity is to resolve the quantum query complexity of the k-distinctness function on inputs of size N. While the case of k=2 (also called Element Distinctness) is well-understood, there is a polynomial gap between the known upper and lower bounds for all constants k>2. Specifically, the best known upper bound is O(N^{(3/4)-1/(2^{k+2}-4)}) (Belovs, FOCS 2012), while the best known lower bound for k >= 2 is Omega(N^{2/3} + N^{(3/4)-1/(2k)}) (Aaronson and Shi, J.~ACM 2004; Bun, Kothari, and Thaler, STOC 2018). For any constant k >= 4, we improve the lower bound to Omega(N^{(3/4)-1/(4k)}). This yields, for example, the first proof that 4-distinctness is strictly harder than Element Distinctness. Our lower bound applies more generally to approximate degree. As a secondary result, we give a simple construction of an approximating polynomial of degree O(N^{3/4}) that applies whenever k <= polylog(N).
rate research
Read More
Based on the recent breakthrough of Huang (2019), we show that for any total Boolean function $f$, $bullet quad mathrm{deg}(f) = O(widetilde{mathrm{deg}}(f)^2)$: The degree of $f$ is at most quadratic in the approximate degree of $f$. This is optimal as witnessed by the OR function. $bullet quad mathrm{D}(f) = O(mathrm{Q}(f)^4)$: The deterministic query complexity of $f$ is at most quartic in the quantum query complexity of $f$. This matches the known separation (up to log factors) due to Ambainis, Balodis, Belovs, Lee, Santha, and Smotrovs (2017). We apply these results to resolve the quantum analogue of the Aanderaa--Karp--Rosenberg conjecture. We show that if $f$ is a nontrivial monotone graph property of an $n$-vertex graph specified by its adjacency matrix, then $mathrm{Q}(f)=Omega(n)$, which is also optimal. We also show that the approximate degree of any read-once formula on $n$ variables is $Theta(sqrt{n})$.
We consider the task of approximating the ground state energy of two-local quantum Hamiltonians on bounded-degree graphs. Most existing algorithms optimize the energy over the set of product states. Here we describe a family of shallow quantum circuits that can be used to improve the approximation ratio achieved by a given product state. The algorithm takes as input an $n$-qubit product state $|vrangle$ with mean energy $e_0=langle v|H|vrangle$ and variance $mathrm{Var}=langle v|(H-e_0)^2|vrangle$, and outputs a state with an energy that is lower than $e_0$ by an amount proportional to $mathrm{Var}^2/n$. In a typical case, we have $mathrm{Var}=Omega(n)$ and the energy improvement is proportional to the number of edges in the graph. When applied to an initial random product state, we recover and generalize the performance guarantees of known algorithms for bounded-occurrence classical constraint satisfaction problems. We extend our results to $k$-local Hamiltonians and entangled initial states.
We study quantum algorithms that are given access to trusted and untrusted quantum witnesses. We establish strong limitations of such algorithms, via new techniques based on Laurent polynomials (i.e., polynomials with positive and negative integer exponents). Specifically, we resolve the complexity of approximate counting, the problem of multiplicatively estimating the size of a nonempty set $S subseteq [N]$, in two natural generalizations of quantum query complexity. Our first result holds in the standard Quantum Merlin--Arthur ($mathsf{QMA}$) setting, in which a quantum algorithm receives an untrusted quantum witness. We show that, if the algorithm makes $T$ quantum queries to $S$, and also receives an (untrusted) $m$-qubit quantum witness, then either $m = Omega(|S|)$ or $T=Omega bigl(sqrt{N/left| Sright| } bigr)$. This is optimal, matching the straightforward protocols where the witness is either empty, or specifies all the elements of $S$. As a corollary, this resolves the open problem of giving an oracle separation between $mathsf{SBP}$, the complexity class that captures approximate counting, and $mathsf{QMA}$. In our second result, we ask what if, in addition to a membership oracle for $S$, a quantum algorithm is also given QSamples -- i.e., copies of the state $left| Srightrangle = frac{1}{sqrt{left| Sright| }} sum_{iin S}|irangle$ -- or even access to a unitary transformation that enables QSampling? We show that, even then, the algorithm needs either $Theta bigl(sqrt{N/left| Sright| }bigr)$ queries or else $Theta bigl(min bigl{left| Sright| ^{1/3}, sqrt{N/left| Sright| }bigr}bigr)$ QSamples or accesses to the unitary. Our lower bounds in both settings make essential use of Laurent polynomials, but in different ways.
We study quantum algorithms for testing bipartiteness and expansion of bounded-degree graphs. We give quantum algorithms that solve these problems in time O(N^(1/3)), beating the Omega(sqrt(N)) classical lower bound. For testing expansion, we also prove an Omega(N^(1/4)) quantum query lower bound, thus ruling out the possibility of an exponential quantum speedup. Our quantum algorithms follow from a combination of classical property testing techniques due to Goldreich and Ron, derandomization, and the quantum algorithm for element distinctness. The quantum lower bound is obtained by the polynomial method, using novel algebraic techniques and combinatorial analysis to accommodate the graph structure.
We prove lower bounds on the error probability of a quantum algorithm for searching through an unordered list of N items, as a function of the number T of queries it makes. In particular, if T=O(sqrt{N}) then the error is lower bounded by a constant. If we want error <1/2^N then we need T=Omega(N) queries. We apply this to show that a quantum computer cannot do much better than a classical computer when amplifying the success probability of an RP-machine. A classical computer can achieve error <=1/2^k using k applications of the RP-machine, a quantum computer still needs at least ck applications for this (when treating the machine as a black-box), where c>0 is a constant independent of k. Furthermore, we prove a lower bound of Omega(sqrt{log N}/loglog N) queries for quantum bounded-error search of an ordered list of N items.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
