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Register a new userQuantum Approximate Counting with Nonadaptive Grover Iterations
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Added by
Ramgopal Venkateswaran
Publication date
2020
fields
Physics
and research's language is
English
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Approximate Counting refers to the problem where we are given query access to a function $f : [N] to {0,1}$, and we wish to estimate $K = #{x : f(x) = 1}$ to within a factor of $1+epsilon$ (with high probability), while minimizing the number of queries. In the quantum setting, Approximate Counting can be done with $Oleft(minleft(sqrt{N/epsilon}, sqrt{N/K}/epsilonright)right)$ queries. It has recently been shown that this can be achieved by a simple algorithm that only uses Grover iterations; however the algorithm performs these iterations adaptively. Motivated by concerns of computational simplicity, we consider algorithms that use Grover iterations with limited adaptivity. We show that algorithms using only nonadaptive Grover iterations can achieve $Oleft(sqrt{N/epsilon}right)$ query complexity, which is tight.
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In 1998, Brassard, Hoyer, Mosca, and Tapp (BHMT) gave a quantum algorithm for approximate counting. Given a list of $N$ items, $K$ of them marked, their algorithm estimates $K$ to within relative error $varepsilon$ by making only $Oleft( frac{1}{varepsilon}sqrt{frac{N}{K}}right) $ queries. Although this speedup is of Grover type, the BHMT algorithm has the curious feature of relying on the Quantum Fourier Transform (QFT), more commonly associated with Shors algorithm. Is this necessary? This paper presents a simplified algorithm, which we prove achieves the same query complexity using Grover iterations only. We also generalize this to a QFT-free algorithm for amplitude estimation. Related approaches to approximate counting were sketched previously by Grover, Abrams and Williams, Suzuki et al., and Wie (the latter two as we were writing this paper), but in all cases without rigorous analysis.
We prove tight lower bounds for the following variant of the counting problem considered by Aaronson et al. The task is to distinguish whether an input set $xsubseteq [n]$ has size either $k$ or $k=(1+epsilon)k$. We assume the algorithm has access to * the membership oracle, which, for each $iin [n]$, can answer whether $iin x$, or not; and * the uniform superposition $|psi_xrangle = sum_{iin x} |irangle/sqrt{|x|}$ over the elements of $x$. Moreover, we consider three different ways how the algorithm can access this state: ** the algorithm can have copies of the state $|psi_xrangle$; ** the algorithm can execute the reflecting oracle which reflects about the state $|psi_xrangle$; ** the algorithm can execute the state-generating oracle (or its inverse) which performs the transformation $|0ranglemapsto |psi_xrangle$. Without the second type of resources (related to $|psi_xrangle$), the problem is well-understood, see Brassard et al. The study of the problem with the second type of resources was recently initiated by Aaronson et al. We completely resolve the problem for all values of $1/k le epsilonle 1$, giving tight trade-offs between all types of resources available to the algorithm. Thus, we close the main open problems from Aaronson et al. The lower bounds are proven using variants of the adversary bound by Belovs and employing analysis closely related to the Johnson association scheme.
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 consider the Grover search algorithm implementation for a quantum register of size $N = 2^k$ using k (or k +1) microwave- and laser-driven Rydberg-blockaded atoms, following the proposal by M{o}lmer, Isenhower, and Saffman [J. Phys. B 44, 184016 (2011)]. We suggest some simplifications for the microwave and laser couplings, and analyze the performance of the algorithm for up to k = 4 multilevel atoms under realistic experimental conditions using quantum stochastic (Monte-Carlo) wavefunction simulations.
The final goal of quantum hypothesis testing is to achieve quantum advantage over all possible classical strategies. In the protocol of quantum reading this advantage is achieved for information retrieval from an optical memory, whose generic cell stores a bit of information in two possible lossy channels. For this protocol, we show, theoretically and experimentally, that quantum advantage is obtained by practical photon-counting measurements combined with a simple maximum-likelihood decision. In particular, we show that this receiver combined with an entangled two-mode squeezed vacuum source is able to outperform any strategy based on statistical mixtures of coherent states for the same mean number of input photons. Our experimental findings demonstrate that quantum entanglement and simple optics are able to enhance the readout of digital data, paving the way to real applications of quantum reading and with potential applications for any other model that is based on the binary discrimination of bosonic loss.
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