Author name cluster

Scott Aaronson

Possible papers associated with this exact author name in Arrow. This page groups case-insensitive exact name matches and is not a full identity disambiguation profile.

22 papers

2 author rows

STOC Conference 2023 Conference Paper

Certified Randomness from Quantum Supremacy

Scott Aaronson
Shih-Han Hung

We propose an application for near-term quantum devices: namely, generating cryptographically certified random bits, to use (for example) in proof-of-stake cryptocurrencies. Our protocol repurposes the existing “quantum supremacy” experiments, based on random circuit sampling, that Google and USTC have successfully carried out starting in 2019. We show that, whenever the outputs of these experiments pass the now-standard Linear Cross-Entropy Benchmark (LXEB), under plausible hardness assumptions they necessarily contain Ω( n ) min-entropy, where n is the number of qubits. To achieve a net gain in randomness, we use a small random seed to produce pseudorandom challenge circuits. In response to the challenge circuits, the quantum computer generates output strings that, after verification, can then be fed into a randomness extractor to produce certified nearly-uniform bits—thereby “bootstrapping” from pseudorandomness to genuine randomness. We prove our protocol sound in two senses: (i) under a hardness assumption called Long List Quantum Supremacy Verification, which we justify in the random oracle model, and (ii) unconditionally in the random oracle model against an eavesdropper who could share arbitrary entanglement with the device. (Note that our protocol’s output is unpredictable even to a computationally unbounded adversary who can see the random oracle.) Currently, the central drawback of our protocol is the exponential cost of verification, which in practice will limit its implementation to at most n ∼ 60 qubits, a regime where attacks are expensive but not impossible. Modulo that drawback, our protocol appears to be the only practical application of quantum computing that both requires a QC and is physically realizable today.

ICML Conference 2023 Conference Paper

Learning Distributions over Quantum Measurement Outcomes

Weiyuan Gong
Scott Aaronson

Shadow tomography for quantum states provides a sample efficient approach for predicting the measurement outcomes of quantum systems. However, these shadow tomography procedures yield poor bounds if there are more than two outcomes per measurement. In this paper, we consider a general problem of learning properties from quantum states: given an unknown $d$-dimensional quantum state $\rho$ 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 $\rho$ to within total variation distance $\epsilon$. Compared to the special case when $K=2$, we have to learn unknown distributions instead of values. Here, we propose an online shadow tomography procedure that solves this problem with high success probability requiring $\tilde{O}(K\log^2M\log d/\epsilon^4)$ copies of $\rho$. We further prove an information-theoretic lower bound showing that at least $\Omega(\min\{d^2, K+\log M\}/\epsilon^2)$ copies of $\rho$ 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 concerning the dependence on $K$.

STOC Conference 2021 Conference Paper

Degree vs. approximate degree and Quantum implications of Huang's sensitivity theorem

Scott Aaronson
Shalev Ben-David
Robin Kothari
Shravas Rao
Avishay Tal

Based on the recent breakthrough of Huang (2019), we show that for any total Boolean function f, deg(f) = O(adeg(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. D(f) = O(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 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(√n).

FOCS Conference 2019 Conference Paper

A Quantum Query Complexity Trichotomy for Regular Languages

Scott Aaronson
Daniel Grier
Luke Schaeffer

We present a trichotomy theorem for the quantum query complexity of regular languages. Every regular language has quantum query complexity Θ(1), Θ̃(√ n), or Θ(n). The extreme uniformity of regular languages prevents them from taking any other asymptotic complexity. This is in contrast to even the context-free languages, which we show can have query complexity Θ(n c ) for all computable c [1/2, 1]. Our result implies an equivalent trichotomy for the approximate degree of regular languages, and a dichotomy-either Θ(1) or Θ(n)-for sensitivity, block sensitivity, certificate complexity, deterministic query complexity, and randomized query complexity. The heart of the classification theorem is an explicit quantum algorithm which decides membership in any star-free language in Õ(√n) time. This well-studied family of the regular languages admits many interesting characterizations, for instance, as those languages expressible as sentences in first-order logic over the natural numbers with the less-than relation. Therefore, not only do the star-free languages capture functions such as OR, they can also express functions such as "there exist a pair of 2's such that everything between them is a 0. " Thus, we view the algorithm for star-free languages as a nontrivial generalization of Grover's algorithm which extends the quantum quadratic speedup to a much wider range of string-processing algorithms than was previously known. We show a variety of applications-new quantum algorithms for dynamic constant-depth Boolean formulas, balanced parentheses nested constantly many levels deep, binary addition, a restricted word break problem, and path-discovery in narrow grids-all obtained as immediate consequences of our classification theorem.

STOC Conference 2019 Conference Paper

Gentle measurement of quantum states and differential privacy

Scott Aaronson
Guy N. Rothblum

In differential privacy (DP) , we want to query a database about n users, in a way that “leaks at most ε 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 α,” 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 α-gentle for small α is also O ( α) -DP, and any product measurement that is ε-DP is also O ( ε√ 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 ρ, as well as known two-outcome measurements E 1 ,…, E m , shadow tomography asks us to estimate Pr[ E i accepts ρ], for every i ∈[ m ], by measuring few copies of ρ. 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 order ( log m ) 2 ( log d ) 2 copies of ρ, 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.

NeurIPS Conference 2018 Conference Paper

Online Learning of Quantum States

Scott Aaronson
Xinyi Chen
Elad Hazan
Satyen Kale
Ashwin Nayak

Suppose we have many copies of an unknown n-qubit state $\rho$. We measure some copies of $\rho$ 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 $\omega_t$ about the state $\rho$, using the outcomes of the previous measurements. We show that it is possible to do this in a way that guarantees that $|\trace(E_i \omega_t) - \trace(E_i\rho)|$, the error in our prediction for the next measurement, is at least $eps$ at most $O(n / eps^2) $\ 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 incur at most $O(\sqrt {Tn}) $ excess loss over the best possible state 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.

STOC Conference 2018 Conference Paper

Shadow tomography of quantum states

Scott Aaronson

We introduce the problem of *shadow tomography*: given an unknown D -dimensional quantum mixed state ρ, as well as known two-outcome measurements E 1 ,…, E M , estimate the probability that E i accepts ρ, to within additive error ε, for each of the M measurements. How many copies of ρ are needed to achieve this, with high probability? Surprisingly, we give a procedure that solves the problem by measuring only O ( ε −5 ·log 4 M ·log D ) copies. This means, for example, that we can learn the behavior of an arbitrary n -qubit state, on *all* accepting/rejecting circuits of some fixed polynomial size, by measuring only n O ( 1) copies of the state. This resolves an open problem of the author, which arose from his work on private-key quantum money schemes, but which also has applications to quantum copy-protected software, quantum advice, and quantum one-way communication. Recently, building on this work, Brandão et al. have given a different approach to shadow tomography using semidefinite programming, which achieves a savings in computation time.

TCS Journal 2018 Journal Article

The fewest clues problem

Erik D. Demaine
Fermi Ma
Ariel Schvartzman
Erik Waingarten
Scott Aaronson

When analyzing the computational complexity of well-known puzzles, most papers consider the algorithmic challenge of solving a given instance of (a generalized form of) the puzzle. We take a different approach by analyzing the computational complexity of designing a “good” puzzle. We assume a puzzle maker designs part of an instance, but before publishing it, wants to ensure that the puzzle has a unique solution. Given a puzzle, we introduce the FCP (fewest clues problem) version of the problem: Given an instance to a puzzle, what is the minimum number of clues we must add in order to make the instance uniquely solvable? We analyze this question for the Nikoli puzzles Sudoku, Shakashaka, and Akari. Solving these puzzles is NP-complete, and we show their FCP versions are Σ 2 P -complete. Along the way, we show that the FCP versions of Triangle Partition, Planar 1-in-3 SAT, and Latin Square are all Σ 2 P -complete. We show that even problems in P have difficult FCP versions, sometimes even Σ 2 P -complete, though “closed under cluing” problems are in the (presumably) smaller class NP; for example, FCP 2SAT is NP-complete.

STOC Conference 2017 Conference Paper

The computational complexity of ball permutations

Scott Aaronson
Adam Bouland
Greg Kuperberg
Saeed Mehraban

We define several models of computation based on permuting distinguishable particles (which we call balls) and characterize their computational complexity. In the quantum setting, we use the representation theory of the symmetric group to find variants of this model which are intermediate between BPP and DQC1 (the class of problems solvable with one clean qubit) and between DQC1 and BQP. Furthermore, we consider a restricted version of this model based on an exactly solvable scattering problem of particles moving on a line. Despite the simplicity of this model from the perspective of mathematical physics, we show that if we allow intermediate destructive measurements and specific input states, then the model cannot be efficiently simulated classically up to multiplicative error unless the polynomial hierarchy collapses. Finally, we define a classical version of this model in which one can probabilistically permute balls. We find this yields a complexity class which is intermediate between L and BPP, and that a nondeterministic version of this model is NP-complete.

STOC Conference 2016 Conference Paper

Separations in query complexity using cheat sheets

Scott Aaronson
Shalev Ben-David
Robin Kothari

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, which builds upon the techniques of Ambainis et al. [STOC 2016]. 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. [STOC 2016], to be shown in a unified manner.

STOC Conference 2015 Conference Paper

Forrelation: A Problem that Optimally Separates Quantum from Classical Computing

Scott Aaronson
Andris Ambainis

We achieve essentially the largest possible separation between quantum and classical query complexities. We do so using a property-testing problem called Forrelation, where one needs to decide whether one Boolean function is highly correlated with the Fourier transform of a second function. This problem can be solved using 1 quantum query, yet we show that any randomized algorithm needs Ω(√(N)log(N)) queries (improving an Ω(N 1/4 ) lower bound of Aaronson). Conversely, we show that this 1 versus Ω(√(N)) separation is optimal: indeed, any t-query quantum algorithm whatsoever can be simulated by an O(N 1-1/2t )-query randomized algorithm. Thus, resolving an open question of Buhrman et al. from 2002, there is no partial Boolean function whose quantum query complexity is constant and whose randomized query complexity is linear. We conjecture that a natural generalization of Forrelation achieves the optimal t versus Ω(N 1-1/2t ) separation for all t. As a bonus, we show that this generalization is BQP-complete. This yields what's arguably the simplest BQP-complete problem yet known, and gives a second sense in which Forrelation "captures the maximum power of quantum computation."

STOC Conference 2012 Conference Paper

Quantum money from hidden subspaces

Scott Aaronson
Paul F. Christiano

Forty years ago, Wiesner pointed out that quantum mechanics raises the striking possibility of money that cannot be counterfeited according to the laws of physics. We propose the first quantum money scheme that is (1) public-key---meaning that anyone can verify a banknote as genuine, not only the bank that printed it, and (2) cryptographically secure, under a "classical" hardness assumption that has nothing to do with quantum money. Our scheme is based on hidden subspaces, encoded as the zero-sets of random multivariate polynomials. A main technical advance is to show that the "black-box" version of our scheme, where the polynomials are replaced by classical oracles, is unconditionally secure. Previously, such a result had only been known relative to a quantum oracle (and even there, the proof was never published). Even in Wiesner's original setting---quantum money that can only be verified by the bank---we are able to use our techniques to patch a major security hole in Wiesner's scheme. We give the first private-key quantum money scheme that allows unlimited verifications and that remains unconditionally secure, even if the counterfeiter can interact adaptively with the bank. Our money scheme is simpler than previous public-key quantum money schemes, including a knot-based scheme of Farhi et al. The verifier needs to perform only two tests, one in the standard basis and one in the Hadamard basis---matching the original intuition for quantum money, based on the existence of complementary observables. Our security proofs use a new variant of Ambainis's quantum adversary method, and several other tools that might be of independent interest.

STOC Conference 2011 Conference Paper

The computational complexity of linear optics

Scott Aaronson
Alex Arkhipov

We give new evidence that quantum computers -- moreover, rudimentary quantum computers built entirely out of linear-optical elements -- cannot be efficiently simulated by classical computers. In particular, we define a model of computation in which identical photons are generated, sent through a linear-optical network, then nonadaptively measured to count the number of photons in each mode. This model is not known or believed to be universal for quantum computation, and indeed, we discuss the prospects for realizing the model using current technology. On the other hand, we prove that the model is able to solve sampling problems and search problems that are classically intractable under plausible assumptions. Our first result says that, if there exists a polynomial-time classical algorithm that samples from the same probability distribution as a linear-optical network, then P # P=BPP N P, and hence the polynomial hierarchy collapses to the third level. Unfortunately, this result assumes an extremely accurate simulation. Our main result suggests that even an approximate or noisy classical simulation would already imply a collapse of the polynomial hierarchy. For this, we need two unproven conjectures: the Permanent-of-Gaussians Conjecture, which says that it is #P-hard to approximate the permanent of a matrix A of independent N(0,1) Gaussian entries, with high probability over A; and the Permanent Anti-Concentration Conjecture, which says that |Per(A)|>=√(n!)poly(n) with high probability over A. We present evidence for these conjectures, both of which seem interesting even apart from our application. This paper does not assume knowledge of quantum optics. Indeed, part of its goal is to develop the beautiful theory of noninteracting bosons underlying our model, and its connection to the permanent function, in a self-contained way accessible to theoretical computer scientists.

STOC Conference 2010 Conference Paper

A full characterization of quantum advice

Scott Aaronson
Andrew Drucker

We prove the following surprising result: given any quantum state rho on n qubits, there exists a local Hamiltonian H on poly(n) qubits (e.g., a sum of two-qubit interactions), such that any ground state of H can be used to simulate rho on all quantum circuits of fixed polynomial size. In terms of complexity classes, this implies that BQP/qpoly is contained in QMA/poly, which supersedes the previous result of Aaronson that BQP/qpoly is contained in PP/poly. Indeed, we can exactly characterize quantum advice, as equivalent in power to untrusted quantum advice combined with trusted classical advice. Proving our main result requires combining a large number of previous tools -- including a result of Alon et al. on learning of real-valued concept classes, a result of Aaronson on the learnability of quantum states, and a result of Aharonov and Regev on "QMA+ super-verifiers" -- and also creating some new ones. The main new tool is a so-called majority-certificates lemma, which is closely related to boosting in machine learning, and which seems likely to find independent applications. In its simplest version, this lemma says the following. Given any set S of Boolean functions on n variables, any function f in S can be expressed as the pointwise majority of m=O(n) functions f1,...,fm in S, such that each fi is the unique function in S compatible with O(log|S|) input/output constraints.

STOC Conference 2010 Conference Paper

BQP and the polynomial hierarchy

Scott Aaronson

The relationship between BQP and PH has been an open problem since the earliest days of quantum computing. We present evidence that quantum computers can solve problems outside the entire polynomial hierarchy, by relating this question to topics in circuit complexity, pseudorandomness, and Fourier analysis. First, we show that there exists an oracle relation problem (i.e., a problem with many valid outputs) that is solvable in BQP, but not in PH. This also yields a non-oracle relation problem that is solvable in quantum logarithmic time, but not in AC0. Second, we show that an oracle decision problem separating BQP from PH would follow from the Generalized Linial-Nisan Conjecture, which we formulate here and which is likely of independent interest. The original Linial-Nisan Conjecture (about pseudorandomness against constant-depth circuits) was recently proved by Braverman, after being open for twenty years.

STOC Conference 2008 Conference Paper

Algebrization: a new barrier in complexity theory

Scott Aaronson
Avi Wigderson

Any proof of P!=NP will have to overcome two barriers: relativization and natural proofs. Yet over the last decade, we have seen circuit lower bounds (for example, that PP does not have linear-size circuits) that overcome both barriers simultaneously. So the question arises of whether there is a third barrier to progress on the central questions in complexity theory. In this paper we present such a barrier, which we call algebraic relativization or algebrization. The idea is that, when we relativize some complexity class inclusion, we should give the simulating machine access not only to an oracle A, but also to a low-degree extension of A over a finite field or ring. We systematically go through basic results and open problems in complexity theory to delineate the power of the new algebrization barrier. First, we show that all known non-relativizing results based on arithmetization -- both inclusions such as IP=PSPACE and MIP=NEXP, and separations such as MAEXP not in P/poly -- do indeed algebrize. Second, we show that almost all of the major open problems -- including P versus NP, P versus RP, and NEXP versus P/poly -- will require non-algebrizing techniques. In some cases algebrization seems to explain exactly why progress stopped where it did: for example, why we have superlinear circuit lower bounds for PromiseMA but not for NP. Our second set of results follows from lower bounds in a new model of algebraic query complexity, which we introduce in this paper and which is interesting in its own right. Some of our lower bounds use direct combinatorial and algebraic arguments, while others stem from a surprising connection between our model and communication complexity. Using this connection, we are also able to give an MA-protocol for the Inner Product function with O(sqrt(n) log n) communication (essentially matching a lower bound of Klauck).

FOCS Conference 2008 Conference Paper

The Polynomial Method in Quantum and Classical Computing

Scott Aaronson

In 1889, A. A. Markov proved a powerful result about low-degree real polynomials: roughly speaking, that such polynomials cannot have a sharp jump followed by a long, relatively flat part. A century later, this result - as well as other results from the field of approximation theory - came to play a surprising role in classical and quantum complexity theory. In this article, the author tries to tell this story in an elementary way, beginning with classic results in approximation theory and ending with some recent applications.

STOC Conference 2005 Conference Paper

The complexity of agreement

Scott Aaronson

A celebrated 1976 theorem of Aumann asserts that Bayesian agents with common priors can never "agree to disagree": if their opinions about any topic are common knowledge, then those opinions must be equal. But two key questions went unaddressed: first, can the agents reach agreement after a conversation of reasonable length? Second, can the computations needed for that conversation be performed efficiently? This paper answers both questions in the affirmative, thereby strengthening Aumann's original conclusion.We show that for two agents with a common prior to agree within ε about the expectation of a [0,1] variable with high probability over their prior, it suffices for them to exchange O(1/ε 2 ) bits. This bound is completely independent of the number of bits n of relevant knowledge that the agents have. We also extend the bound to three or more agents; and we give an example where the "standard protocol" (which consists of repeatedly announcing one's current expectation) nearly saturates the bound, while a new "attenuated protocol" does better. Finally, we give a protocol that would cause two Bayesians to agree within ε after exchanging O(1/ε 2 ) messages, and that can be simulated by agents with limited computational resources. By this we mean that, after examining the agents' knowledge and a transcript of their conversation, no one would be able to distinguish the agents from perfect Bayesians. The time used by the simulation procedure is exponential in 1/ε 6 but not in n.

STOC Conference 2004 Conference Paper

Lower bounds for local search by quantum arguments

Scott Aaronson

The problem of finding a local minimum of a black-box function is central for understanding local search as well as quantum adiabatic algorithms. For functions on the Boolean hypercube 0,1 n , we show a lower bound of Ω(2 n /4 / n ) on the number of queries needed by a quantum computer to solve this problem. More surprisingly, our approach, based on Ambainis's quantum adversary method, also yields a lower bound of Ω(2 n /2 / n 2 ) on the problem's classical randomized query complexity. This improves and simplifies a 1983 result of Aldous. Finally, in both the randomized and quantum cases, we give the first nontrivial lower bounds for finding local minima on grids of constant dimension d ≥3.

STOC Conference 2004 Conference Paper

Multilinear formulas and skepticism of quantum computing

Scott Aaronson

Several researchers, including Leonid Levin, Gerard 't Hooft, and Stephen Wolfram, have argued that quantum mechanics will break down before the factoring of large numbers becomes possible. If this is true, then there should be a natural set of quantum states that can account for all quantum computing experiments performed to date, but not for Shor's factoring algorithm. We investigate as a candidate the set of states expressible by a polynomial number of additions and tensor products. Using a recent lower bound on multilinear formula size due to Raz, we then show that states arising in quantum error-correction require n Ω(log n) additions and tensor products even to approximate, which incidentally yields the first superpolynomial gap between general and multilinear formula size of functions. More broadly, we introduce a complexity classification of pure quantum states, and prove many basic facts about this classification. Our goal is to refine vague ideas about a breakdown of quantum mechanics into specific hypotheses that might be experimentally testable in the near future.

FOCS Conference 2003 Conference Paper

Quantum Search of Spatial Regions

Scott Aaronson
Andris Ambainis

Can Grover's quantum search algorithm speed up search of a physical region - for example a 2D grid of size /spl radic/n x /spl radic/n? The problem is that /spl radic/n time seems to be needed for each query, just to move amplitude across the grid. Here we show that this problem can be surmounted, refuting a claim to the contrary by Benioff. In particular, we show how to search a d-dimensional hypercube in time 0(/spl radic/n) for d /spl ges/ 3, or 0(/spl radic/n log/sup 3/ n) for d = 2. More generally, we introduce a model of quantum query complexity on graphs, motivated by fundamental physical limits on information storage, particularly the holographic principle from black hole thermodynamics. Our results in this model include almost-tight upper and lower bounds for many search tasks; a generalized algorithm that works for any graph with good expansion properties, not just hypercubes; and relationships among several notions of locality for unitary matrices acting on graphs. As an application of our results, we give an 0(/spl radic/n)-qubit communication protocol for the disjointness problem, which improves an upper bound of Hoyer and de Wolf and matches a lower bound of Razborov.

STOC Conference 2002 Conference Paper

Quantum lower bound for the collision problem

Scott Aaronson

(MATH) The collision problem is to decide whether a function X : { 1,…, n } → { 1, …, n } is one-to-one or two-to-one, given that one of these is the case. We show a lower bound of Ω( n 1/5 ) on the number of queries needed by a quantum computer to solve this problem with bounded error probability. The best known upper bound is O ( n 1/3 ), but obtaining any lower bound better than Ω(1) was an open problem since 1997. Our proof uses the polynomial method augmented by some new ideas. We also give a lower bound of Ω( n 1/7 ) for the problem of deciding whether two sets are equal or disjoint on a constant fraction of elements. Finally we give implications of these results for quantum complexity theory.