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Register a new userLower Bounds for Function Inversion with Quantum Advice
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Function inversion is the problem that given a random function $f: [M] to [N]$, we want to find pre-image of any image $f^{-1}(y)$ in time $T$. In this work, we revisit this problem under the preprocessing model where we can compute some auxiliary information or advice of size $S$ that only depends on $f$ but not on $y$. It is a well-studied problem in the classical settings, however, it is not clear how quantum algorithms can solve this task any better besides invoking Grovers algorithm, which does not leverage the power of preprocessing. Nayebi et al. proved a lower bound $ST^2 ge tildeOmega(N)$ for quantum algorithms inverting permutations, however, they only consider algorithms with classical advice. Hhan et al. subsequently extended this lower bound to fully quantum algorithms for inverting permutations. In this work, we give the same asymptotic lower bound to fully quantum algorithms for inverting functions for fully quantum algorithms under the regime where $M = O(N)$. In order to prove these bounds, we generalize the notion of quantum random access code, originally introduced by Ambainis et al., to the setting where we are given a list of (not necessarily independent) random variables, and we wish to compress them into a variable-length encoding such that we can retrieve a random element just using the encoding with high probability. As our main technical contribution, we give a nearly tight lower bound (for a wide parameter range) for this generalized notion of quantum random access codes, which may be of independent interest.
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In function inversion, we are given a function $f: [N] mapsto [N]$, and want to prepare some advice of size $S$, such that we can efficiently invert any image in time $T$. This is a well studied problem with profound connections to cryptography, data structures, communication complexity, and circuit lower bounds. Investigation of this problem in the quantum setting was initiated by Nayebi, Aaronson, Belovs, and Trevisan (2015), who proved a lower bound of $ST^2 = tildeOmega(N)$ for random permutations against classical advice, leaving open an intriguing possibility that Grovers search can be sped up to time $tilde O(sqrt{N/S})$. Recent works by Hhan, Xagawa, and Yamakawa (2019), and Chung, Liao, and Qian (2019) extended the argument for random functions and quantum advice, but the lower bound remains $ST^2 = tildeOmega(N)$. In this work, we prove that even with quantum advice, $ST + T^2 = tildeOmega(N)$ is required for an algorithm to invert random functions. This demonstrates that Grovers search is optimal for $S = tilde O(sqrt{N})$, ruling out any substantial speed-up for Grovers search even with quantum advice. Further improvements to our bounds would imply new classical circuit lower bounds, as shown by Corrigan-Gibbs and Kogan (2019). To prove this result, we develop a general framework for establishing quantum time-space lower bounds. We further demonstrate the power of our framework by proving quantum time-space lower bounds for Yaos box problem and salted cryptography.
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.
We examine the number T of queries that a quantum network requires to compute several Boolean functions on {0,1}^N in the black-box model. We show that, in the black-box model, the exponential quantum speed-up obtained for partial functions (i.e. problems involving a promise on the input) by Deutsch and Jozsa and by Simon cannot be obtained for any total function: if a quantum algorithm computes some total Boolean function f with bounded-error using T black-box queries then there is a classical deterministic algorithm that computes f exactly with O(T^6) queries. We also give asymptotically tight characterizations of T for all symmetric f in the exact, zero-error, and bounded-error settings. Finally, we give new precise bounds for AND, OR, and PARITY. Our results are a quantum extension of the so-called polynomial method, which has been successfully applied in classical complexity theory, and also a quantum extension of results by Nisan about a polynomial relationship between randomized and deterministic decision tree complexity.
In this work, we initiate a formal study of probably approximately correct (PAC) learning under evasion attacks, where the adversarys goal is to emph{misclassify} the adversarially perturbed sample point $widetilde{x}$, i.e., $h(widetilde{x}) eq c(widetilde{x})$, where $c$ is the ground truth concept and $h$ is the learned hypothesis. Previous work on PAC learning of adversarial examples have all modeled adversarial examples as corrupted inputs in which the goal of the adversary is to achieve $h(widetilde{x}) eq c(x)$, where $x$ is the original untampered instance. These two definitions of adversarial risk coincide for many natural distributions, such as images, but are incomparable in general. We first prove that for many theoretically natural input spaces of high dimension $n$ (e.g., isotropic Gaussian in dimension $n$ under $ell_2$ perturbations), if the adversary is allowed to apply up to a sublinear $o(||x||)$ amount of perturbations on the test instances, PAC learning requires sample complexity that is exponential in $n$. This is in contrast with results proved using the corrupted-input framework, in which the sample complexity of robust learning is only polynomially more. We then formalize hybrid attacks in which the evasion attack is preceded by a poisoning attack. This is perhaps reminiscent of trapdoor attacks in which a poisoning phase is involved as well, but the evasion phase here uses the error-region definition of risk that aims at misclassifying the perturbed instances. In this case, we show PAC learning is sometimes impossible all together, even when it is possible without the attack (e.g., due to the bounded VC dimension).
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.
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