Explore Research Products in the PAR
It may take a few hours for recently added research products to appear in PAR search results.


This content will become publicly available on August 31, 2027

Title: The power of quantum circuits in sampling
We give new evidence that quantum circuits are substantially more powerful than classical circuits. We show, relative to a random oracle, that polynomial-size quantum circuits can sample distributions that subexponential-size classical circuits cannot approximate even to TV distance $1-o(1)$. Prior work of Aaronson and Arkhipov (2011) showed such a separation for the case of exact sampling (i.e.~TV distance $$0$$), but separations for approximate sampling were only known for uniform algorithms. A key ingredient in our proof is a new hardness amplification lemma for the classical query complexity of the Yamakawa--Zhandry (2022) search problem. We show that the probability that any family of query algorithms collectively finds $$k$$ distinct solutions decays exponentially in $$k$$.  more » « less
Award ID(s):
2310818
PAR ID:
10677372
Author(s) / Creator(s):
; ; ; ;
Publisher / Repository:
The Theory of Quantum Computation, Communication and Cryptography (TQC)
Date Published:
Format(s):
Medium: X
Location:
Quebec, Canada
Sponsoring Org:
National Science Foundation
More Like this
  1. ; ; (, Theory of computing)
    null (Ed.)
    The approximate degree of a Boolean function f is the least degree of a real polynomial that approximates f pointwise to error at most 1/3. The approximate degree of f is known to be a lower bound on the quantum query complexity of f (Beals et al., FOCS 1998 and J. ACM 2001). We find tight or nearly tight bounds on the approximate degree and quantum query complexities of several basic functions. Specifically, we show the following. k-Distinctness: For any constant k, the approximate degree and quantum query complexity of the k-distinctness function is Ω(n3/4−1/(2k)). This is nearly tight for large k, as Belovs (FOCS 2012) has shown that for any constant k, the approximate degree and quantum query complexity of k-distinctness is O(n3/4−1/(2k+2−4)). Image size testing: The approximate degree and quantum query complexity of testing the size of the image of a function [n]→[n] is Ω~(n1/2). This proves a conjecture of Ambainis et al. (SODA 2016), and it implies tight lower bounds on the approximate degree and quantum query complexity of the following natural problems. k-Junta testing: A tight Ω~(k1/2) lower bound for k-junta testing, answering the main open question of Ambainis et al. (SODA 2016). Statistical distance from uniform: A tight Ω~(n1/2) lower bound for approximating the statistical distance of a distribution from uniform, answering the main question left open by Bravyi et al. (STACS 2010 and IEEE Trans. Inf. Theory 2011). Shannon entropy: A tight Ω~(n1/2) lower bound for approximating Shannon entropy up to a certain additive constant, answering a question of Li and Wu (2017). Surjectivity: The approximate degree of the surjectivity function is Ω~(n3/4). The best prior lower bound was Ω(n2/3). Our result matches an upper bound of O~(n3/4) due to Sherstov (STOC 2018), which we reprove using different techniques. The quantum query complexity of this function is known to be Θ(n) (Beame and Machmouchi, Quantum Inf. Comput. 2012 and Sherstov, FOCS 2015). Our upper bound for surjectivity introduces new techniques for approximating Boolean functions by low-degree polynomials. Our lower bounds are proved by significantly refining techniques recently introduced by Bun and Thaler (FOCS 2017). 
    more » « less
  2. ; ; (, Quantum)
    We give new quantum algorithms for evaluating composed functions whose inputs may be shared between bottom-level gates. Let f be an m -bit Boolean function and consider an n -bit function F obtained by applying f to conjunctions of possibly overlapping subsets of n variables. If f has quantum query complexity Q ( f ) , we give an algorithm for evaluating F using O ~ ( Q ( f ) ⋅ n ) quantum queries. This improves on the bound of O ( Q ( f ) ⋅ n ) that follows by treating each conjunction independently, and our bound is tight for worst-case choices of f . Using completely different techniques, we prove a similar tight composition theorem for the approximate degree of f .By recursively applying our composition theorems, we obtain a nearly optimal O ~ ( n 1 − 2 − d ) upper bound on the quantum query complexity and approximate degree of linear-size depth- d AC 0 circuits. As a consequence, such circuits can be PAC learned in subexponential time, even in the challenging agnostic setting. Prior to our work, a subexponential-time algorithm was not known even for linear-size depth-3 AC 0 circuits.As an additional consequence, we show that AC 0 ∘ ⊕ circuits of depth d + 1 require size Ω ~ ( n 1 / ( 1 − 2 − d ) ) ≥ ω ( n 1 + 2 − d ) to compute the Inner Product function even on average. The previous best size lower bound was Ω ( n 1 + 4 − ( d + 1 ) ) and only held in the worst case (Cheraghchi et al., JCSS 2018). 
    more » « less
  3. ; ; ; (, Quantum)
    null (Ed.)
    Quantum computational supremacy arguments, which describe a way for a quantum computer to perform a task that cannot also be done by a classical computer, typically require some sort of computational assumption related to the limitations of classical computation. One common assumption is that the polynomial hierarchy ( P H ) does not collapse, a stronger version of the statement that P ≠ N P , which leads to the conclusion that any classical simulation of certain families of quantum circuits requires time scaling worse than any polynomial in the size of the circuits. However, the asymptotic nature of this conclusion prevents us from calculating exactly how many qubits these quantum circuits must have for their classical simulation to be intractable on modern classical supercomputers. We refine these quantum computational supremacy arguments and perform such a calculation by imposing fine-grained versions of the non-collapse conjecture. Our first two conjectures poly3-NSETH( a ) and per-int-NSETH( b ) take specific classical counting problems related to the number of zeros of a degree-3 polynomial in n variables over F 2 or the permanent of an n × n integer-valued matrix, and assert that any non-deterministic algorithm that solves them requires 2 c n time steps, where c ∈ { a , b } . A third conjecture poly3-ave-SBSETH( a ′ ) asserts a similar statement about average-case algorithms living in the exponential-time version of the complexity class S B P . We analyze evidence for these conjectures and argue that they are plausible when a = 1 / 2 , b = 0.999 and a ′ = 1 / 2 .Imposing poly3-NSETH(1/2) and per-int-NSETH(0.999), and assuming that the runtime of a hypothetical quantum circuit simulation algorithm would scale linearly with the number of gates/constraints/optical elements, we conclude that Instantaneous Quantum Polynomial-Time (IQP) circuits with 208 qubits and 500 gates, Quantum Approximate Optimization Algorithm (QAOA) circuits with 420 qubits and 500 constraints and boson sampling circuits (i.e. linear optical networks) with 98 photons and 500 optical elements are large enough for the task of producing samples from their output distributions up to constant multiplicative error to be intractable on current technology. Imposing poly3-ave-SBSETH(1/2), we additionally rule out simulations with constant additive error for IQP and QAOA circuits of the same size. Without the assumption of linearly increasing simulation time, we can make analogous statements for circuits with slightly fewer qubits but requiring 10 4 to 10 7 gates. 
    more » « less
  4. ; (, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)
    Saraf, Shubhangi (Ed.)
    The Forrelation problem is a central problem that demonstrates an exponential separation between quantum and classical capabilities. In this problem, given query access to n-bit Boolean functions f and g, the goal is to estimate the Forrelation function forr(f,g), which measures the correlation between g and the Fourier transform of f. In this work we provide a new linear algebraic perspective on the Forrelation problem, as opposed to prior analytic approaches. We establish a connection between the Forrelation problem and bent Boolean functions and through this connection, analyze an extremal version of the Forrelation problem where the goal is to distinguish between extremal instances of Forrelation, namely (f,g) with forr(f,g) = 1 and forr(f,g) = -1. We show that this problem can be solved with one quantum query and success probability one, yet requires Ω̃(2^{n/4}) classical randomized queries, even for algorithms with a one-third failure probability, highlighting the remarkable power of one exact quantum query. We also study a restricted variant of this problem where the inputs f,g are computable by small classical circuits and show classical hardness under cryptographic assumptions. 
    more » « less
  5. ; ; ; ; (, Proceedings of the ACM on Management of Data)
    Data summarization is a powerful approach to deal with large-scale data analytics, which has wide applications in web search, recommendation systems, approximate query processing, etc. It computes a small, compact summary that preserves vital properties of the original data. In this paper, we study the data summarization problem of conjunctive query results, i.e., computing a k-size subset of a conjunctive query output, for any given k>0, that optimizes a certain objective. More specifically, we are interested in two commonly studied objectives: cohesion, which measures the maximum distance between a tuple in the query result tuples and its closest tuple in the summary (k-center clustering); and diversity, which measures the pairwise distances between the summary items. A simple approach that computes the entire query output and then applies existing algorithms on top of these materialized tuples suffers from high computational complexity because the query output can be large, e.g., for a relational database of N tuples, the number of result tuples can be NO(1).We propose O(1)-approximation algorithms that compute well-representative summaries of size k in time O(N*kO(1)), or even O(N+ kO(1)) in some cases, without computing all result tuples. We also propose the first efficient (2+\eps)-approximation algorithm for the k-center clustering problem over relational data. Our main idea is to formulate a few oracles that enable us to access specific query result tuples with certain properties, to show how these oracles can be implemented efficiently, and to compute desired summaries with few invocations of these oracles. 
    more » « less
  • Citation Formats
  • MLA
  • APA
  • Chicago
  • BibTeX
  • Text Encoded Conversion
  • XML
  • Save / Share this Record
  • Send to Email