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Electronic Colloquium on Computational Complexity

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All reports by Author Wei Zhan:

TR23-196 | 7th December 2023
Huacheng Yu, Wei Zhan

Sampling, Flowers and Communication

Given a distribution over $[n]^n$ such that any $k$ coordinates need $k/\log^{O(1)}n$ bits of communication to sample, we prove that any map that samples this distribution from uniform cells requires locality $\Omega(\log(n/k)/\log\log(n/k))$. In particular, we show that for any constant $\delta>0$, there exists $\varepsilon=2^{-\Omega(n^{1-\delta})}$ such that $\Omega(\log n/\log\log n)$ non-adaptive ... more >>>

TR23-107 | 20th July 2023
Uma Girish, Ran Raz, Wei Zhan

Quantum Logspace Computations are Verifiable

In this note, we observe that quantum logspace computations are verifiable by classical logspace algorithms, with unconditional security. More precisely, every language in BQL has an information-theoretically secure) streaming proof with a quantum logspace prover and a classical logspace verifier. The prover provides a polynomial-length proof that is streamed to ... more >>>

TR23-096 | 28th June 2023
Huacheng Yu, Wei Zhan

Randomized vs. Deterministic Separation in Time-Space Tradeoffs of Multi-Output Functions

We prove the first polynomial separation between randomized and deterministic time-space tradeoffs of multi-output functions. In particular, we present a total function that on the input of $n$ elements in $[n]$, outputs $O(n)$ elements, such that:

- There exists a randomized oblivious algorithm with space $O(\log n)$, time $O(n\log n)$ ... more >>>

TR23-040 | 28th March 2023
Edward Pyne, Ran Raz, Wei Zhan

Certified Hardness vs. Randomness for Log-Space

Let $\mathcal{L}$ be a language that can be decided in linear space and let $\epsilon >0$ be any constant. Let $\mathcal{A}$ be the exponential hardness assumption that for every $n$, membership in $\mathcal{L}$ for inputs of length~$n$ cannot be decided by circuits of size smaller than $2^{\epsilon n}$.
We ... more >>>

TR23-018 | 1st March 2023
Qipeng Liu, Ran Raz, Wei Zhan

Memory-Sample Lower Bounds for Learning with Classical-Quantum Hybrid Memory

In a work by Raz (J. ACM and FOCS 16), it was proved that any algorithm for parity learning on $n$ bits requires either $\Omega(n^2)$ bits of classical memory or an exponential number (in~$n$) of random samples. A line of recent works continued that research direction and showed that for ... more >>>

TR22-043 | 2nd April 2022
Uma Girish, Kunal Mittal, Ran Raz, Wei Zhan

Polynomial Bounds On Parallel Repetition For All 3-Player Games With Binary Inputs

We prove that for every 3-player (3-prover) game $\mathcal G$ with value less than one, whose query distribution has the support $\mathcal S = \{(1,0,0), (0,1,0), (0,0,1)\}$ of hamming weight one vectors, the value of the $n$-fold parallel repetition $\mathcal G^{\otimes n}$ decays polynomially fast to zero; that is, there ... more >>>

TR22-039 | 14th March 2022
Uma Girish, Justin Holmgren, Kunal Mittal, Ran Raz, Wei Zhan

Parallel Repetition For All 3-Player Games Over Binary Alphabet

We prove that for every 3-player (3-prover) game, with binary questions and answers and value less than $1$, the value of the $n$-fold parallel repetition of the game decays polynomially fast to $0$. That is, for every such game, there exists a constant $c>0$, such that the value of the ... more >>>

TR21-101 | 13th July 2021
Uma Girish, Justin Holmgren, Kunal Mittal, Ran Raz, Wei Zhan

A Parallel Repetition Theorem for the GHZ Game: A Simpler Proof

Revisions: 1

We give a new proof of the fact that the parallel repetition of the (3-player) GHZ game reduces the value of the game to zero polynomially quickly. That is, we show that the value of the $n$-fold GHZ game is at most $n^{-\Omega(1)}$. This was first established by Holmgren and ... more >>>

TR20-101 | 7th July 2020
Uma Girish, Ran Raz, Wei Zhan

Lower Bounds for XOR of Forrelations

The Forrelation problem, first introduced by Aaronson [AA10] and Aaronson and Ambainis [AA15], is a well studied computational problem in the context of separating quantum and classical computational models. Variants of this problem were used to give tight separations between quantum and classical query complexity [AA15]; the first separation between ... more >>>

TR20-087 | 8th June 2020
Uma Girish, Ran Raz, Wei Zhan

Quantum Logspace Algorithm for Powering Matrices with Bounded Norm

Revisions: 2

We give a quantum logspace algorithm for powering contraction matrices, that is, matrices with spectral norm at most 1. The algorithm gets as an input an arbitrary $n\times n$ contraction matrix $A$, and a parameter $T \leq poly(n)$ and outputs the entries of $A^T$, up to (arbitrary) polynomially small additive ... more >>>

TR19-162 | 15th November 2019
Ran Raz, Wei Zhan

The Random-Query Model and the Memory-Bounded Coupon Collector

We study a new model of space-bounded computation, the {\it random-query} model. The model is based on a branching-program over input variables $x_1,\ldots,x_n$. In each time step, the branching program gets as an input a random index $i \in \{1,\ldots,n\}$, together with the input variable $x_i$ (rather than querying an ... more >>>

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