A source is compressible if we can efficiently compute short
descriptions of strings in the support and efficiently
recover the strings from the descriptions. In this paper, we
present a technique for proving lower bounds on
compressibility in an oracle setting, which yields the
following results:

- We ... more >>>

Although a quantum state requires exponentially many classical bits to describe, the laws of quantum mechanics impose severe restrictions on how that state can be accessed. This paper shows in three settings that quantum messages have only limited advantages over classical ones.
First, we show that BQP/qpoly is contained in ... more >>>

We explore the space "just above" BQP by defining a complexity class PDQP (Product Dynamical Quantum Polynomial time) which is larger than BQP but does not contain NP relative to an oracle. The class is defined by imagining that quantum computers can perform measurements that do not collapse the ... more >>>

We present a distribution $D$ over inputs in $\{-1,1\}^{2N}$, such that:
(1) There exists a quantum algorithm that makes one (quantum) query to the input, and runs in time $O(\log N)$, that distinguishes between $D$ and the uniform distribution with advantage $\Omega(1/\log N)$.
(2) No Boolean circuit of $\mathrm{quasipoly}(N)$ ... more >>>

After presentations of the oracle separation of BQP and PH result by Raz and Tal [ECCC TR18-107], several people
(e.g. Ryan O’Donnell, James Lee, Avishay Tal) suggested that the proof may be simplified by
stochastic calculus. In this short note, we describe such a simplification.

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This paper proves new limitations on the power of quantum computers to solve approximate counting---that is, multiplicatively estimating the size of a nonempty set $S\subseteq [N]$.

Given only a membership oracle for $S$, it is well known that approximate counting takes $\Theta(\sqrt{N/|S|})$ quantum queries. But what if a quantum algorithm ... more >>>

A PAC learning model involves two worst-case requirements: a learner must learn all functions in a class on all example distributions. However, basing the hardness of learning on NP-hardness has remained a key challenge for decades. In fact, recent progress in computational complexity suggests the possibility that a weaker assumption ... more >>>

Kumar (CCC, 2023) used a novel switching lemma to prove exponential-size lower bounds for a circuit class $GC^0$ that not only contains $AC^0$ but can---with a single gate---compute functions that require exponential-size $TC^0$ circuits. Their main result was that switching-lemma lower bounds for $AC^0$ lift to $GC^0$ with no loss ... more >>>