arXiv subjects

Scott Aaronson

Publications and source records attributed to Scott Aaronson.

At least 37 records · Page 2Linked to original sources

arXiv2022

An Automated Approach to the Collatz Conjecture

We explore the Collatz conjecture and its variants through the lens of termination of string rewriting. We construct a rewriting system that simulates the iterated application of the Collatz function on strings corresponding to mixed binary-ternary representations of positive integers. We prove that the termination of this rewriting system is equivalent to the Collatz conjecture. We also prove that a previously studied rewriting system that simulates the Collatz function using unary representations does not admit termination proofs via natural matrix interpretations, even when used in conjunction with dependency pairs. To show the feasibility of our approach in proving mathematically interesting statements, we implement a minimal termination prover that uses natural/arctic matrix interpretations and we find automated proofs of nontrivial weakenings of the Collatz conjecture. Although we do not succeed in proving the Collatz conjecture, we believe that the ideas here represent an interesting new approach.

cs.LO↗

arXiv2022

Discrete Bulk Reconstruction

According to the AdS/CFT correspondence, the geometries of certain spacetimes are fully determined by quantum states that live on their boundaries -- indeed, by the von Neumann entropies of portions of those boundary states. This work investigates to what extent the geometries can be reconstructed from the entropies in polynomial time. Bouland, Fefferman, and Vazirani (2019) argued that the AdS/CFT map can be exponentially complex if one wants to reconstruct regions such as the interiors of black holes. Our main result provides a sort of converse: we show that, in the special case of a single 1D boundary, if the input data consists of a list of entropies of contiguous boundary regions, and if the entropies satisfy a single inequality called Strong Subadditivity, then we can construct a graph model for the bulk in linear time. Moreover, the bulk graph is planar, it has $O(N^2)$ vertices (the information-theoretic minimum), and it's ``universal,'' with only the edge weights depending on the specific entropies in question. From a combinatorial perspective, our problem boils down to an ``inverse'' of the famous min-cut problem: rather than being given a graph and asked to find a min-cut, here we're given the values of min-cuts separating various sets of vertices, and need to find a weighted undirected graph consistent with those values. Our solution to this problem relies on the notion of a ``bulkless'' graph, which might be of independent interest for AdS/CFT. We also make initial progress on the case of multiple 1D boundaries -- where the boundaries could be connected via wormholes -- including an upper bound of $O(N^4)$ vertices whenever a planar bulk graph exists (thus putting the problem into the complexity class $\mathsf{NP}$).

quant-ph↗

arXiv2022

How Much Structure Is Needed for Huge Quantum Speedups?

I survey, for a general scientific audience, three decades of research into which sorts of problems admit exponential speedups via quantum computers -- from the classics (like the algorithms of Simon and Shor), to the breakthrough of Yamakawa and Zhandry from April 2022. I discuss both the quantum circuit model, which is what we ultimately care about in practice but where our knowledge is radically incomplete, and the so-called oracle or black-box or query complexity model, where we've managed to achieve a much more thorough understanding that then informs our conjectures about the circuit model. I discuss the strengths and weaknesses of switching attention to sampling tasks, as was done in the recent quantum supremacy experiments. I make some skeptical remarks about widely-repeated claims of exponential quantum speedups for practical machine learning and optimization problems. Through many examples, I try to convey the "law of conservation of weirdness," according to which every problem admitting an exponential quantum speedup must have some unusual property to allow the amplitude to be concentrated on the unknown right answer(s).

quant-ph↗

arXiv2022

Learning Distributions over Quantum Measurement Outcomes

Shadow tomography for quantum states provides a sample efficient approach for predicting the properties of quantum systems when the properties are restricted to expectation values of $2$-outcome POVMs. However, these shadow tomography procedures yield poor bounds if there are more than 2 outcomes per measurement. In this paper, we consider a general problem of learning properties from unknown quantum states: given an unknown $d$-dimensional quantum state $ρ$ and $M$ unknown quantum measurements $\mathcal{M}_1,...,\mathcal{M}_M$ with $K\geq 2$ outcomes, estimating the probability distribution for applying $\mathcal{M}_i$ on $ρ$ to within total variation distance $ε$. Compared to the special case when $K=2$, we need to learn unknown distributions instead of values. We develop an online shadow tomography procedure that solves this problem with high success probability requiring $\tilde{O}(K\log^2M\log d/ε^4)$ copies of $ρ$. We further prove an information-theoretic lower bound that at least $Ω(\min\{d^2,K+\log M\}/ε^2)$ copies of $ρ$ are required to solve this problem with high success probability. Our shadow tomography procedure requires sample complexity with only logarithmic dependence on $M$ and $d$ and is sample-optimal for the dependence on $K$.

quant-ph↗

arXiv2022

A very preliminary analysis of DALL-E 2

The DALL-E 2 system generates original synthetic images corresponding to an input text as caption. We report here on the outcome of fourteen tests of this system designed to assess its common sense, reasoning and ability to understand complex texts. All of our prompts were intentionally much more challenging than the typical ones that have been showcased in recent weeks. Nevertheless, for 5 out of the 14 prompts, at least one of the ten images fully satisfied our requests. On the other hand, on no prompt did all of the ten images satisfy our requests.

cs.CV↗

arXiv2021

Quantum Approximate Counting, Simplified

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 $O\left( \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 Shor's 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.

quant-ph↗

arXiv2021

Open Problems Related to Quantum Query Complexity

I offer a case that quantum query complexity still has loads of enticing and fundamental open problems -- from relativized QMA versus QCMA and BQP versus IP, to time/space tradeoffs for collision and element distinctness, to polynomial degree versus quantum query complexity for partial functions, to the Unitary Synthesis Problem and more.

quant-ph↗

arXiv2020

Degree vs. Approximate Degree and Quantum Implications of Huang's Sensitivity Theorem

Based on the recent breakthrough of Huang (2019), we show that for any total Boolean function $f$, $\bullet \quad \mathrm{deg}(f) = O(\widetilde{\mathrm{deg}}(f)^2)$: The degree of $f$ is at most quadratic in the approximate degree of $f$. This is optimal as witnessed by the OR function. $\bullet \quad \mathrm{D}(f) = O(\mathrm{Q}(f)^4)$: The deterministic query complexity of $f$ is at most quartic in the quantum query complexity of $f$. This matches the known separation (up to log factors) due to Ambainis, Balodis, Belovs, Lee, Santha, and Smotrovs (2017). We apply these results to resolve the quantum analogue of the Aanderaa--Karp--Rosenberg conjecture. We show that if $f$ is a nontrivial monotone graph property of an $n$-vertex graph specified by its adjacency matrix, then $\mathrm{Q}(f)=Ω(n)$, which is also optimal. We also show that the approximate degree of any read-once formula on $n$ variables is $Θ(\sqrt{n})$.

quant-ph↗

arXiv2020

New Approaches for Quantum Copy-Protection

Quantum copy protection uses the unclonability of quantum states to construct quantum software that provably cannot be pirated. Copy protection would be immensely useful, but unfortunately little is known about how to achieve it in general. In this work, we make progress on this goal, by giving the following results: - We show how to copy protect any program that cannot be learned from its input/output behavior, relative to a classical oracle. This improves on Aaronson [CCC'09], which achieves the same relative to a quantum oracle. By instantiating the oracle with post-quantum candidate obfuscation schemes, we obtain a heuristic construction of copy protection. -We show, roughly, that any program which can be watermarked can be copy detected, a weaker version of copy protection that does not prevent copying, but guarantees that any copying can be detected. Our scheme relies on the security of the assumed watermarking, plus the assumed existence of public key quantum money. Our construction is general, applicable to many recent watermarking schemes.

cs.CR↗

arXiv2020

Quantum Versus Classical Proofs and Advice

This paper studies whether quantum proofs are more powerful than classical proofs, or in complexity terms, whether QMA=QCMA. We prove three results about this question. First, we give a "quantum oracle separation" between QMA and QCMA. More concretely, we show that any quantum algorithm needs $Ω(\sqrt{2^n/(m+1)})$ queries to find an $n$-qubit "marked state" $\lvertψ\rangle$, even if given an $m$-bit classical description of $\lvertψ\rangle$ together with a quantum black box that recognizes $\lvertψ\rangle$. Second, we give an explicit QCMA protocol that nearly achieves this lower bound. Third, we show that, in the one previously-known case where quantum proofs seemed to provide an exponential advantage, classical proofs are basically just as powerful. In particular, Watrous gave a QMA protocol for verifying non-membership in finite groups. Under plausible group-theoretic assumptions, we give a QCMA protocol for the same problem. Even with no assumptions, our protocol makes only polynomially many queries to the group oracle. We end with some conjectures about quantum versus classical oracles, and about the possibility of a classical oracle separation between QMA and QCMA.

quant-ph↗

arXiv2020

On the Hardness of Detecting Macroscopic Superpositions

When is decoherence "effectively irreversible"? Here we examine this central question of quantum foundations using the tools of quantum computational complexity. We prove that, if one had a quantum circuit to determine if a system was in an equal superposition of two orthogonal states (for example, the $|$Alive$\rangle$ and $|$Dead$\rangle$ states of Schrödinger's cat), then with only a slightly larger circuit, one could also $\mathit{swap}$ the two states (e.g., bring a dead cat back to life). In other words, observing interference between the $|$Alive$\rangle$and $|$Dead$\rangle$ states is a "necromancy-hard" problem, technologically infeasible in any world where death is permanent. As for the converse statement (i.e., ability to swap implies ability to detect interference), we show that it holds modulo a single exception, involving unitaries that (for example) map $|$Alive$\rangle$ to $|$Dead$\rangle$ but $|$Dead$\rangle$ to -$|$Alive$\rangle$. We also show that these statements are robust---i.e., even a $\mathit{partial}$ ability to observe interference implies partial swapping ability, and vice versa. Finally, without relying on any unproved complexity conjectures, we show that all of these results are quantitatively tight. Our results have possible implications for the state dependence of observables in quantum gravity, the subject that originally motivated this study.

quant-ph↗

arXiv2020

On the Quantum Complexity of Closest Pair and Related Problems

The closest pair problem is a fundamental problem of computational geometry: given a set of $n$ points in a $d$-dimensional space, find a pair with the smallest distance. A classical algorithm taught in introductory courses solves this problem in $O(n\log n)$ time in constant dimensions (i.e., when $d=O(1)$). This paper asks and answers the question of the problem's quantum time complexity. Specifically, we give an $\tilde{O}(n^{2/3})$ algorithm in constant dimensions, which is optimal up to a polylogarithmic factor by the lower bound on the quantum query complexity of element distinctness. The key to our algorithm is an efficient history-independent data structure that supports quantum interference. In $\mathrm{polylog}(n)$ dimensions, no known quantum algorithms perform better than brute force search, with a quadratic speedup provided by Grover's algorithm. To give evidence that the quadratic speedup is nearly optimal, we initiate the study of quantum fine-grained complexity and introduce the Quantum Strong Exponential Time Hypothesis (QSETH), which is based on the assumption that Grover's algorithm is optimal for CNF-SAT when the clause width is large. We show that the naïve Grover approach to closest pair in higher dimensions is optimal up to an $n^{o(1)}$ factor unless QSETH is false. We also study the bichromatic closest pair problem and the orthogonal vectors problem, with broadly similar results.

quant-ph↗

arXiv2020

Quantum Lower Bounds for Approximate Counting via Laurent Polynomials

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 $m = Ω(|S|)$ or $T=Ω\bigl(\sqrt{N/\left| S\right| } \bigr)$. 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 $\left| S\right\rangle = \frac{1}{\sqrt{\left| S\right| }} \sum_{i\in S}|i\rangle$ -- or even access to a unitary transformation that enables QSampling? We show that, even then, the algorithm needs either $Θ\bigl(\sqrt{N/\left| S\right| }\bigr)$ queries or else $Θ\bigl(\min \bigl\{\left| S\right| ^{1/3}, \sqrt{N/\left| S\right| }\bigr\}\bigr)$ QSamples or accesses to the unitary. Our lower bounds in both settings make essential use of Laurent polynomials, but in different ways.

quant-ph↗

arXiv2020

Quantum Implications of Huang's Sensitivity Theorem

Based on the recent breakthrough of Huang (2019), we show that for any total Boolean function $f$, the deterministic query complexity, $D(f)$, is at most quartic in the quantum query complexity, $Q(f)$: $D(f) = O(Q(f)^4)$. This matches the known separation (up to log factors) due to Ambainis, Balodis, Belovs, Lee, Santha, and Smotrovs (2017). We also use the result to resolve the quantum analogue of the Aanderaa-Karp-Rosenberg conjecture. We show that if $f$ is a nontrivial monotone graph property of an $n$-vertex graph specified by its adjacency matrix, then $Q(f) = Ω(n)$, which is also optimal.

quant-ph↗

arXiv2020

On the Classical Hardness of Spoofing Linear Cross-Entropy Benchmarking

Recently, Google announced the first demonstration of quantum computational supremacy with a programmable superconducting processor. Their demonstration is based on collecting samples from the output distribution of a noisy random quantum circuit, then applying a statistical test to those samples called Linear Cross-Entropy Benchmarking (Linear XEB). This raises a theoretical question: how hard is it for a classical computer to spoof the results of the Linear XEB test? In this short note, we adapt an analysis of Aaronson and Chen [2017] to prove a conditional hardness result for Linear XEB spoofing. Specifically, we show that the problem is classically hard, assuming that there is no efficient classical algorithm that, given a random n-qubit quantum circuit C, estimates the probability of C outputting a specific output string, say 0^n, with variance even slightly better than that of the trivial estimator that always estimates 1/2^n. Our result automatically encompasses the case of noisy circuits.

quant-ph↗

arXiv2019

Online Learning of Quantum States

Suppose we have many copies of an unknown $n$-qubit state $ρ$. We measure some copies of $ρ$ using a known two-outcome measurement $E_{1}$, then other copies using a measurement $E_{2}$, and so on. At each stage $t$, we generate a current hypothesis $σ_{t}$ about the state $ρ$, using the outcomes of the previous measurements. We show that it is possible to do this in a way that guarantees that $|\operatorname{Tr}(E_{i} σ_{t}) - \operatorname{Tr}(E_{i}ρ) |$, the error in our prediction for the next measurement, is at least $\varepsilon$ at most $\operatorname{O}\!\left(n / \varepsilon^2 \right) $ times. Even in the "non-realizable" setting---where there could be arbitrary noise in the measurement outcomes---we show how to output hypothesis states that do significantly worse than the best possible states at most $\operatorname{O}\!\left(\sqrt {Tn}\right) $ times on the first $T$ measurements. These results generalize a 2007 theorem by Aaronson on the PAC-learnability of quantum states, to the online and regret-minimization settings. We give three different ways to prove our results---using convex optimization, quantum postselection, and sequential fat-shattering dimension---which have different advantages in terms of parameters and portability.

quant-ph↗

arXiv2019

Separations in query complexity using cheat sheets

We show a power 2.5 separation between bounded-error randomized and quantum query complexity for a total Boolean function, refuting the widely believed conjecture that the best such separation could only be quadratic (from Grover's algorithm). We also present a total function with a power 4 separation between quantum query complexity and approximate polynomial degree, showing severe limitations on the power of the polynomial method. Finally, we exhibit a total function with a quadratic gap between quantum query complexity and certificate complexity, which is optimal (up to log factors). These separations are shown using a new, general technique that we call the cheat sheet technique. The technique is based on a generic transformation that converts any (possibly partial) function into a new total function with desirable properties for showing separations. The framework also allows many known separations, including some recent breakthrough results of Ambainis et al., to be shown in a unified manner.

quant-ph↗

arXiv2019

Gentle Measurement of Quantum States and Differential Privacy

In differential privacy (DP), we want to query a database about n users, in a way that "leaks at most eps about any individual user," even conditioned on any outcome of the query. Meanwhile, in gentle measurement, we want to measure n quantum states, in a way that "damages the states by at most alpha," even conditioned on any outcome of the measurement. In both cases, we can achieve the goal by techniques like deliberately adding noise to the outcome before returning it. This paper proves a new and general connection between the two subjects. Specifically, we show that on products of n quantum states, any measurement that is alpha-gentle for small alpha is also O(alpha)-DP, and any product measurement that is eps-DP is also O(eps*sqrt(n))-gentle. Illustrating the power of this connection, we apply it to the recently studied problem of shadow tomography. Given an unknown d-dimensional quantum state rho, as well as known two-outcome measurements E_1,...,E_m, shadow tomography asks us to estimate Pr[E_i accepts rho], for every i in [m], by measuring few copies of rho. Using our connection theorem, together with a quantum analog of the so-called private multiplicative weights algorithm of Hardt and Rothblum, we give a protocol to solve this problem using O((log m)^2 (log d)^2) copies of rho, compared to Aaronson's previous bound of ~O((log m)^4 (log d)). Our protocol has the advantages of being online (that is, the E_i's are processed one at a time), gentle, and conceptually simple. Other applications of our connection include new lower bounds for shadow tomography from lower bounds on DP, and a result on the safe use of estimation algorithms as subroutines inside larger quantum algorithms.

quant-ph↗