arXiv subjects

Troy Lee

Publications and source records attributed to Troy Lee.

At least 19 recordsLinked to original sources

arXiv2023

On the quantum time complexity of divide and conquer

We initiate a systematic study of the time complexity of quantum divide and conquer algorithms for classical problems. We establish generic conditions under which search and minimization problems with classical divide and conquer algorithms are amenable to quantum speedup and apply these theorems to an array of problems involving strings, integers, and geometric objects. They include LONGEST DISTINCT SUBSTRING, KLEE'S COVERAGE, several optimization problems on stock transactions, and k-INCREASING SUBSEQUENCE. For most of these results, our quantum time upper bound matches the quantum query lower bound for the problem, up to polylogarithmic factors.

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arXiv2022

Cut query algorithms with star contraction

We study the complexity of determining the edge connectivity of a simple graph with cut queries. We show that (i) there is a bounded-error randomized algorithm that computes edge connectivity with $O(n)$ cut queries, and (ii) there is a bounded-error quantum algorithm that computes edge connectivity with $\~O(\sqrt{n})$ cut queries. We prove these results using a new technique called "star contraction" to randomly contract edges of a graph while preserving non-trivial minimum cuts. In star contraction vertices randomly contract an edge incident on a small set of randomly chosen vertices. In contrast to the related 2-out contraction technique of Ghaffari, Nowicki, and Thorup [SODA'20], star contraction only contracts vertex-disjoint star subgraphs, which allows it to be efficiently implemented via cut queries. The $O(n)$ bound from item (i) was not known even for the simpler problem of connectivity, and improves the $O(n\log^3 n)$ bound by Rubinstein, Schramm, and Weinberg [ITCS'18]. The bound is tight under the reasonable conjecture that the randomized communication complexity of connectivity is $\Omega(n\log n)$, an open question since the seminal work of Babai, Frankl, and Simon [FOCS'86]. The bound also excludes using edge connectivity on simple graphs to prove a superlinear randomized query lower bound for minimizing a symmetric submodular function. Item (ii) gives a nearly-quadratic separation with the randomized complexity and addresses an open question of Lee, Santha, and Zhang [SODA'21]. The algorithm can also be viewed as making $\~O(\sqrt{n})$ matrix-vector multiplication queries to the adjacency matrix. Finally, we demonstrate the use of star contraction outside of the cut query setting by designing a one-pass semi-streaming algorithm for computing edge connectivity in the vertex arrival setting. This contrasts with the edge arrival setting where two passes are required.

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arXiv2021

Finding the KT partition of a weighted graph in near-linear time

In a breakthrough work, Kawarabayashi and Thorup (J.~ACM'19) gave a near-linear time deterministic algorithm for minimum cut in a simple graph $G = (V,E)$. A key component is finding the $(1+\varepsilon)$-KT partition of $G$, the coarsest partition $\{P_1, \ldots, P_k\}$ of $V$ such that for every non-trivial $(1+\varepsilon)$-near minimum cut with sides $\{S, \bar{S}\}$ it holds that $P_i$ is contained in either $S$ or $\bar{S}$, for $i=1, \ldots, k$. Here we give a near-linear time randomized algorithm to find the $(1+\varepsilon)$-KT partition of a weighted graph. Our algorithm is quite different from that of Kawarabayashi and Thorup and builds on Karger's framework of tree-respecting cuts (J.~ACM'00). We describe applications of the algorithm. (i) The algorithm makes progress towards a more efficient algorithm for constructing the polygon representation of the set of near-minimum cuts in a graph. This is a generalization of the cactus representation initially described by Bencz\'ur (FOCS'95). (ii) We improve the time complexity of a recent quantum algorithm for minimum cut in a simple graph in the adjacency list model from $\widetilde O(n^{3/2})$ to $\widetilde O(\sqrt{mn})$. (iii) We describe a new type of randomized algorithm for minimum cut in simple graphs with complexity $O(m + n \log^6 n)$. For slightly dense graphs this matches the complexity of the current best $O(m + n \log^2 n)$ algorithm which uses a different approach based on random contractions. The key technical contribution of our work is the following. Given a weighted graph $G$ with $m$ edges and a spanning tree $T$, consider the graph $H$ whose nodes are the edges of $T$, and where there is an edge between two nodes of $H$ iff the corresponding 2-respecting cut of $T$ is a non-trivial near-minimum cut of $G$. We give a $O(m \log^4 n)$ time deterministic algorithm to compute a spanning forest of $H$.

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arXiv2021

A sublinear query quantum algorithm for s-t minimum cut on dense simple graphs

An $s{\operatorname{-}}t$ minimum cut in a graph corresponds to a minimum weight subset of edges whose removal disconnects vertices $s$ and $t$. Finding such a cut is a classic problem that is dual to that of finding a maximum flow from $s$ to $t$. In this work we describe a quantum algorithm for the minimum $s{\operatorname{-}}t$ cut problem on undirected graphs. For an undirected graph with $n$ vertices, $m$ edges, and integral edge weights bounded by $W$, the algorithm computes with high probability the weight of a minimum $s{\operatorname{-}}t$ cut after $\widetilde O(\sqrt{m} n^{5/6} W^{1/3})$ queries to the adjacency list of $G$. For simple graphs this bound is always $\widetilde O(n^{11/6})$, even in the dense case when $m = \Omega(n^2)$. In contrast, a randomized algorithm must make $\Omega(m)$ queries to the adjacency list of a simple graph $G$ even to decide whether $s$ and $t$ are connected.

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arXiv2021

On the query complexity of connectivity with global queries

We study the query complexity of determining if a graph is connected with global queries. The first model we look at is matrix-vector multiplication queries to the adjacency matrix. Here, for an $n$-vertex graph with adjacency matrix $A$, one can query a vector $x \in \{0,1\}^n$ and receive the answer $Ax$. We give a randomized algorithm that can output a spanning forest of a weighted graph with constant probability after $O(\log^4(n))$ matrix-vector multiplication queries to the adjacency matrix. This complements a result of Sun et al.\ (ICALP 2019) that gives a randomized algorithm that can output a spanning forest of a graph after $O(\log^4(n))$ matrix-vector multiplication queries to the signed vertex-edge incidence matrix of the graph. As an application, we show that a quantum algorithm can output a spanning forest of an unweighted graph after $O(\log^5(n))$ cut queries, improving and simplifying a result of Lee, Santha, and Zhang (SODA 2021), which gave the bound $O(\log^8(n))$. In the second part of the paper, we turn to showing lower bounds on the linear query complexity of determining if a graph is connected. If $w$ is the weight vector of a graph (viewed as an $\binom{n}{2}$ dimensional vector), in a linear query one can query any vector $z \in \mathbb{R}^{n \choose 2}$ and receive the answer $\langle z, w\rangle$. We show that a zero-error randomized algorithm must make $\Omega(n)$ linear queries in expectation to solve connectivity. As far as we are aware, this is the first lower bound of any kind on the unrestricted linear query complexity of connectivity. We show this lower bound by looking at the linear query \emph{certificate complexity} of connectivity, and characterize this certificate complexity in a linear algebraic fashion.

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arXiv2021

An approximation algorithm for approximation rank

One of the strongest techniques available for showing lower bounds on quantum communication complexity is the logarithm of the approximation rank of the communication matrix--the minimum rank of a matrix which is entrywise close to the communication matrix. This technique has two main drawbacks: it is difficult to compute, and it is not known to lower bound quantum communication complexity with entanglement. Linial and Shraibman recently introduced a norm, called gamma_2^{alpha}, to quantum communication complexity, showing that it can be used to lower bound communication with entanglement. Here the parameter alpha is a measure of approximation which is related to the allowable error probability of the protocol. This bound can be written as a semidefinite program and gives bounds at least as large as many techniques in the literature, although it is smaller than the corresponding alpha-approximation rank, rk_alpha. We show that in fact log gamma_2^{alpha}(A)$ and log rk_{alpha}(A)$ agree up to small factors. As corollaries we obtain a constant factor polynomial time approximation algorithm to the logarithm of approximate rank, and that the logarithm of approximation rank is a lower bound for quantum communication complexity with entanglement.

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arXiv2020

On the cut dimension of a graph

Let $G = (V,w)$ be a weighted undirected graph with $m$ edges. The cut dimension of $G$ is the dimension of the span of the characteristic vectors of the minimum cuts of $G$, viewed as vectors in $\{0,1\}^m$. For every $n \ge 2$ we show that the cut dimension of an $n$-vertex graph is at most $2n-3$, and construct graphs realizing this bound. The cut dimension was recently defined by Graur et al.\ \cite{GPRW20}, who show that the maximum cut dimension of an $n$-vertex graph is a lower bound on the number of cut queries needed by a deterministic algorithm to solve the minimum cut problem on $n$-vertex graphs. For every $n\ge 2$, Graur et al.\ exhibit a graph on $n$ vertices with cut dimension at least $3n/2 -2$, giving the first lower bound larger than $n$ on the deterministic cut query complexity of computing mincut. We observe that the cut dimension is even a lower bound on the number of \emph{linear} queries needed by a deterministic algorithm to solve mincut, where a linear query can ask any vector $x \in \mathbb{R}^{\binom{n}{2}}$ and receives the answer $w^T x$. Our results thus show a lower bound of $2n-3$ on the number of linear queries needed by a deterministic algorithm to solve minimum cut on $n$-vertex graphs, and imply that one cannot show a lower bound larger than this via the cut dimension. We further introduce a generalization of the cut dimension which we call the $\ell_1$-approximate cut dimension. The $\ell_1$-approximate cut dimension is also a lower bound on the number of linear queries needed by a deterministic algorithm to compute minimum cut. It is always at least as large as the cut dimension, and we construct an infinite family of graphs on $n=3k+1$ vertices with $\ell_1$-approximate cut dimension $2n-2$, showing that it can be strictly larger than the cut dimension.

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arXiv2020

Quantum complexity of minimum cut

The minimum cut problem in an undirected and weighted graph $G$ is to find the minimum total weight of a set of edges whose removal disconnects $G$. We completely characterize the quantum query and time complexity of the minimum cut problem in the adjacency matrix model. If $G$ has $n$ vertices and edge weights at least $1$ and at most $\tau$, we give a quantum algorithm to solve the minimum cut problem using $\tilde O(n^{3/2}\sqrt{\tau})$ queries and time. Moreover, for every integer $1 \le \tau \le n$ we give an example of a graph $G$ with edge weights $1$ and $\tau$ such that solving the minimum cut problem on $G$ requires $\Omega(n^{3/2}\sqrt{\tau})$ many queries to the adjacency matrix of $G$. These results contrast with the classical randomized case where $\Omega(n^2)$ queries to the adjacency matrix are needed in the worst case even to decide if an unweighted graph is connected or not. In the adjacency array model, when $G$ has $m$ edges the classical randomized complexity of the minimum cut problem is $\tilde \Theta(m)$. We show that the quantum query and time complexity are $\tilde O(\sqrt{mn\tau})$ and $\tilde O(\sqrt{mn\tau} + n^{3/2})$, respectively, where again the edge weights are between $1$ and $\tau$. For dense graphs we give lower bounds on the quantum query complexity of $\Omega(n^{3/2})$ for $\tau > 1$ and $\Omega(\tau n)$ for any $1 \leq \tau \leq n$. Our query algorithm uses a quantum algorithm for graph sparsification by Apers and de Wolf (FOCS 2020) and results on the structure of near-minimum cuts by Kawarabayashi and Thorup (STOC 2015) and Rubinstein, Schramm and Weinberg (ITCS 2018). Our time efficient implementation builds on Karger's tree packing technique (STOC 1996).

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arXiv2020

Quantum algorithms for graph problems with cut queries

Let $G$ be an $n$-vertex graph with $m$ edges. When asked a subset $S$ of vertices, a cut query on $G$ returns the number of edges of $G$ that have exactly one endpoint in $S$. We show that there is a bounded-error quantum algorithm that determines all connected components of $G$ after making $O(\log(n)^6)$ many cut queries. In contrast, it follows from results in communication complexity that any randomized algorithm even just to decide whether the graph is connected or not must make at least $Ω(n/\log(n))$ many cut queries. We further show that with $O(\log(n)^8)$ many cut queries a quantum algorithm can with high probability output a spanning forest for $G$. En route to proving these results, we design quantum algorithms for learning a graph using cut queries. We show that a quantum algorithm can learn a graph with maximum degree $d$ after $O(d \log(n)^2)$ many cut queries, and can learn a general graph with $O(\sqrt{m} \log(n)^{3/2})$ many cut queries. These two upper bounds are tight up to the poly-logarithmic factors, and compare to $Ω(dn)$ and $Ω(m/\log(n))$ lower bounds on the number of cut queries needed by a randomized algorithm for the same problems, respectively. The key ingredients in our results are the Bernstein-Vazirani algorithm, approximate counting with "OR queries", and learning sparse vectors from inner products as in compressed sensing.

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arXiv2020

The quantum query complexity of composition with a relation

The negative weight adversary method, $\mathrm{ADV}^\pm(g)$, is known to characterize the bounded-error quantum query complexity of any Boolean function $g$, and also obeys a perfect composition theorem $\mathrm{ADV}^\pm(f \circ g^n) = \mathrm{ADV}^\pm(f) \mathrm{ADV}^\pm(g)$. Belovs gave a modified version of the negative weight adversary method, $\mathrm{ADV}_{rel}^\pm(f)$, that characterizes the bounded-error quantum query complexity of a relation $f \subseteq \{0,1\}^n \times [K]$, provided the relation is efficiently verifiable. A relation is efficiently verifiable if $\mathrm{ADV}^\pm(f_a) = o(\mathrm{ADV}_{rel}^\pm(f))$ for every $a \in [K]$, where $f_a$ is the Boolean function defined as $f_a(x) = 1$ if and only if $(x,a) \in f$. In this note we show a perfect composition theorem for the composition of a relation $f$ with a Boolean function $g$ \[ \mathrm{ADV}_{rel}^\pm(f \circ g^n) = \mathrm{ADV}_{rel}^\pm(f) \mathrm{ADV}^\pm(g) \enspace . \] For an efficiently verifiable relation $f$ this means $Q(f \circ g^n) = Θ( \mathrm{ADV}_{rel}^\pm(f) \mathrm{ADV}^\pm(g) )$.

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arXiv2018

A composition theorem for randomized query complexity via max conflict complexity

Let $R_ε(\cdot)$ stand for the bounded-error randomized query complexity with error $ε> 0$. For any relation $f \subseteq \{0,1\}^n \times S$ and partial Boolean function $g \subseteq \{0,1\}^m \times \{0,1\}$, we show that $R_{1/3}(f \circ g^n) \in Ω(R_{4/9}(f) \cdot \sqrt{R_{1/3}(g)})$, where $f \circ g^n \subseteq (\{0,1\}^m)^n \times S$ is the composition of $f$ and $g$. We give an example of a relation $f$ and partial Boolean function $g$ for which this lower bound is tight. We prove our composition theorem by introducing a new complexity measure, the max conflict complexity $\bar χ(g)$ of a partial Boolean function $g$. We show $\bar χ(g) \in Ω(\sqrt{R_{1/3}(g)})$ for any (partial) function $g$ and $R_{1/3}(f \circ g^n) \in Ω(R_{4/9}(f) \cdot \bar χ(g))$; these two bounds imply our composition result. We further show that $\bar χ(g)$ is always at least as large as the sabotage complexity of $g$, introduced by Ben-David and Kothari.

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arXiv2018

Bounding quantum-classical separations for classes of nonlocal games

We bound separations between the entangled and classical values for several classes of nonlocal $t$-player games. Our motivating question is whether there is a family of $t$-player XOR games for which the entangled bias is $1$ but for which the classical bias goes down to $0$, for fixed $t$. Answering this question would have important consequences in the study of multi-party communication complexity, as a positive answer would imply an unbounded separation between randomized communication complexity with and without entanglement. Our contribution to answering the question is identifying several general classes of games for which the classical bias can not go to zero when the entangled bias stays above a constant threshold. This rules out the possibility of using these games to answer our motivating question. A previously studied set of XOR games, known not to give a positive answer to the question, are those for which there is a quantum strategy that attains value 1 using a so-called Schmidt state. We generalize this class to mod-$m$ games and show that their classical value is always at least $\frac{1}{m} + \frac{m-1}{m} t^{1-t}$. Secondly, for free XOR games, in which the input distribution is of product form, we show $β(G) \geq β^*(G)^{2^t}$ where $β(G)$ and $β^*(G)$ are the classical and entangled biases of the game respectively. We also introduce so-called line games, an example of which is a slight modification of the Magic Square game, and show that they can not give a positive answer to the question either. Finally we look at two-player unique games and show that if the entangled value is $1-ε$ then the classical value is at least $1-\mathcal{O}(\sqrt{ε\log k})$ where $k$ is the number of outputs in the game. Our proofs use semidefinite-programming techniques, the Gowers inverse theorem and hypergraph norms.

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arXiv2018

Two new results about quantum exact learning

We present two new results about exact learning by quantum computers. First, we show how to exactly learn a $k$-Fourier-sparse $n$-bit Boolean function from $O(k^{1.5}(\log k)^2)$ uniform quantum examples for that function. This improves over the bound of $\widetilde{\Theta}(kn)$ uniformly random \emph{classical} examples (Haviv and Regev, CCC'15). Additionally, we provide a possible direction to improve our $\widetilde{O}(k^{1.5})$ upper bound by proving an improvement of Chang's lemma for $k$-Fourier-sparse Boolean functions. Second, we show that if a concept class $\mathcal{C}$ can be exactly learned using $Q$ quantum membership queries, then it can also be learned using $O\left(\frac{Q^2}{\log Q}\log|\mathcal{C}|\right)$ \emph{classical} membership queries. This improves the previous-best simulation result (Servedio and Gortler, SICOMP'04) by a $\log Q$-factor.

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arXiv2018

Strategies for quantum races

We initiate the study of quantum races, games where two or more quantum computers compete to solve a computational problem. While the problem of dueling algorithms has been studied for classical deterministic algorithms, the quantum case presents additional sources of uncertainty for the players. The foremost among these is that players do not know if they have solved the problem until they measure their quantum state. This question of `when to measure?' presents a very interesting strategic problem. We develop a game-theoretic model of a multiplayer quantum race, and find an approximate Nash equilibrium where all players play the same strategy. In the two-party case, we further show that this strategy is nearly optimal in terms of payoff among all symmetric Nash equilibria. A key role in our analysis of quantum races is played by a more tractable version of the game where there is no payout on a tie; for such races we completely characterize the Nash equilibria in the two-party case. One application of our results is to the stability of the Bitcoin protocol when mining is done by quantum computers. Bitcoin mining is a race to solve a computational search problem, with the winner gaining the right to create a new block. Our results inform the strategies that eventual quantum miners should use, and also indicate that the collision probability---the probability that two miners find a new block at the same time---would not be too high in the case of quantum miners. Such collisions are undesirable as they lead to forking of the Bitcoin blockchain.

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arXiv2018

On the randomised query complexity of composition

Let $f\subseteq\{0,1\}^n\times\Xi$ be a relation and $g:\{0,1\}^m\to\{0,1,*\}$ be a promise function. This work investigates the randomised query complexity of the relation $f\circ g^n\subseteq\{0,1\}^{m\cdot n}\times\Xi$, which can be viewed as one of the most general cases of composition in the query model (letting $g$ be a relation seems to result in a rather unnatural definition of $f\circ g^n$). We show that for every such $f$ and $g$, $$\mathcal R(f\circ g^n) \in \Omega(\mathcal R(f)\cdot\sqrt{\mathcal R(g)}),$$ where $\mathcal R$ denotes the randomised query complexity. On the other hand, we demonstrate a relation $f_0$ and a promise function $g_0$, such that $\mathcal R(f_0)\in\Theta(\sqrt n)$, $\mathcal R(g_0)\in\Theta(n)$ and $\mathcal R(f_0\circ g_0^n)\in\Theta(n)$ $-$ that is, our composition statement is tight. To the best of our knowledge, there was no known composition theorem for the randomised query complexity of relations or promise functions (and for the special case of total functions our lower bound gives multiplicative improvement of $\sqrt{\log n}$).

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arXiv2017

Quantum attacks on Bitcoin, and how to protect against them

The key cryptographic protocols used to secure the internet and financial transactions of today are all susceptible to attack by the development of a sufficiently large quantum computer. One particular area at risk are cryptocurrencies, a market currently worth over 150 billion USD. We investigate the risk of Bitcoin, and other cryptocurrencies, to attacks by quantum computers. We find that the proof-of-work used by Bitcoin is relatively resistant to substantial speedup by quantum computers in the next 10 years, mainly because specialized ASIC miners are extremely fast compared to the estimated clock speed of near-term quantum computers. On the other hand, the elliptic curve signature scheme used by Bitcoin is much more at risk, and could be completely broken by a quantum computer as early as 2027, by the most optimistic estimates. We analyze an alternative proof-of-work called Momentum, based on finding collisions in a hash function, that is even more resistant to speedup by a quantum computer. We also review the available post-quantum signature schemes to see which one would best meet the security and efficiency requirements of blockchain applications.

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arXiv2017

Quadratically Tight Relations for Randomized Query Complexity

Let $f:\{0,1\}^n \rightarrow \{0,1\}$ be a Boolean function. The certificate complexity $C(f)$ is a complexity measure that is quadratically tight for the zero-error randomized query complexity $R_0(f)$: $C(f) \leq R_0(f) \leq C(f)^2$. In this paper we study a new complexity measure that we call expectational certificate complexity $EC(f)$, which is also a quadratically tight bound on $R_0(f)$: $EC(f) \leq R_0(f) = O(EC(f)^2)$. We prove that $EC(f) \leq C(f) \leq EC(f)^2$ and show that there is a quadratic separation between the two, thus $EC(f)$ gives a tighter upper bound for $R_0(f)$. The measure is also related to the fractional certificate complexity $FC(f)$ as follows: $FC(f) \leq EC(f) = O(FC(f)^{3/2})$. This also connects to an open question by Aaronson whether $FC(f)$ is a quadratically tight bound for $R_0(f)$, as $EC(f)$ is in fact a relaxation of $FC(f)$. In the second part of the work, we upper bound the distributed query complexity $D^μ_ε(f)$ for product distributions $μ$ by the square of the query corruption bound ($\mathrm{corr}_ε(f)$) which improves upon a result of Harsha, Jain and Radhakrishnan [2015]. A similar statement for communication complexity is open.

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arXiv2017

A Composition Theorem for Randomized Query Complexity

Let the randomized query complexity of a relation for error probability $ε$ be denoted by $R_ε(\cdot)$. We prove that for any relation $f \subseteq \{0,1\}^n \times \mathcal{R}$ and Boolean function $g:\{0,1\}^m \rightarrow \{0,1\}$, $R_{1/3}(f\circ g^n) = Ω(R_{4/9}(f)\cdot R_{1/2-1/n^4}(g))$, where $f \circ g^n$ is the relation obtained by composing $f$ and $g$. We also show that $R_{1/3}\left(f \circ \left(g^\oplus_{O(\log n)}\right)^n\right)=Ω(\log n \cdot R_{4/9}(f) \cdot R_{1/3}(g))$, where $g^\oplus_{O(\log n)}$ is the function obtained by composing the xor function on $O(\log n)$ bits and $g^t$.

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