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Limitations of Quantum Advice and OneWay Communication
 Theory of Computing
, 2004
"... 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. ..."
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Cited by 59 (15 self)
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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.
NPcomplete problems and physical reality
 ACM SIGACT News Complexity Theory Column, March. ECCC
, 2005
"... Can NPcomplete problems be solved efficiently in the physical universe? I survey proposals including soap bubbles, protein folding, quantum computing, quantum advice, quantum adiabatic algorithms, quantummechanical nonlinearities, hidden variables, relativistic time dilation, analog computing, Mal ..."
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Cited by 55 (6 self)
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Can NPcomplete problems be solved efficiently in the physical universe? I survey proposals including soap bubbles, protein folding, quantum computing, quantum advice, quantum adiabatic algorithms, quantummechanical nonlinearities, hidden variables, relativistic time dilation, analog computing, MalamentHogarth spacetimes, quantum gravity, closed timelike curves, and “anthropic computing. ” The section on soap bubbles even includes some “experimental ” results. While I do not believe that any of the proposals will let us solve NPcomplete problems efficiently, I argue that by studying them, we can learn something not only about computation but also about physics. 1
On computation and communication with small bias
 In Proc. of the 22nd Conf. on Computational Complexity (CCC
, 2007
"... We present two results for computational models that allow error probabilities close to 1/2. First, most computational complexity classes have an analogous class in communication complexity. The class PP in fact has two, a version with weakly restricted bias called PP cc, and a version with unrestri ..."
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Cited by 47 (3 self)
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We present two results for computational models that allow error probabilities close to 1/2. First, most computational complexity classes have an analogous class in communication complexity. The class PP in fact has two, a version with weakly restricted bias called PP cc, and a version with unrestricted bias called UPP cc. Ever since their introduction by Babai, Frankl, and Simon in 1986, it has been open whether these classes are the same. We show that PP cc � UPP cc. Our proof combines a query complexity separation due to Beigel with a technique of Razborov that translates the acceptance probability of quantum protocols to polynomials. Second, we study how small the bias of minimaldegree polynomials that signrepresent Boolean functions needs to be. We show that the worstcase bias is at worst doubleexponentially small in the signdegree (which was very recently shown to be optimal by Podolski), while the averagecase bias can be made singleexponentially small in the signdegree (which we show to be close to optimal). 1
The Computational Complexity of Linear Optics
 in Proceedings of STOC 2011
"... We give new evidence that quantum computers—moreover, rudimentary quantumcomputers built entirely out of linearoptical elements—cannotbeefficientlysimulatedbyclassical computers. In particular, we define a model of computation in which identical photons are generated, sent through a linearoptical n ..."
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Cited by 34 (8 self)
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We give new evidence that quantum computers—moreover, rudimentary quantumcomputers built entirely out of linearoptical elements—cannotbeefficientlysimulatedbyclassical computers. In particular, we define a model of computation in which identical photons are generated, sent through a linearoptical 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 polynomialtime classical algorithm that samples from the same probability distribution as a linearoptical network, then P #P = BPP NP, 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 PermanentofGaussians Conjecture, which says that it is #Phard to approximate the permanent of a matrixAofindependentN (0,1)Gaussianentries, withhigh probability over A; and the Permanent AntiConcentration 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. For the 96page full version, see www.scottaaronson.com/papers/optics.pdf
Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy
, 2010
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The learnability of quantum states
 quantph/0608142
, 2006
"... Traditional quantum state tomography requires a number of measurements that grows exponentially with the number of qubits n. But using ideas from computational learning theory, we show that “for most practical purposes ” one can learn a state using a number of measurements that grows only linearly w ..."
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Cited by 22 (2 self)
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Traditional quantum state tomography requires a number of measurements that grows exponentially with the number of qubits n. But using ideas from computational learning theory, we show that “for most practical purposes ” one can learn a state using a number of measurements that grows only linearly with n. Besides possible implications for experimental physics, our learning theorem has two applications to quantum computing: first, a new simulation of quantum oneway communication protocols, and second, the use of trusted classical advice to verify untrusted quantum advice. 1
Bounds on the power of constantdepth quantum circuits. Preprint: quantph/0312209
 In Proc. 15th International Symposium on on Fundamentals of Computation Theory (FCT 2005), volume 3623 of Lecture Notes in Computer Science
, 2004
"... We show that if a language is recognized within certain error bounds by constantdepth quantum circuits over a finite family of gates, then it is computable in (classical) polynomial time. In particular, for 0 < ɛ ≤ δ ≤ 1, we define BQNC 0 ɛ,δ to be the class of languages recognized by constant d ..."
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We show that if a language is recognized within certain error bounds by constantdepth quantum circuits over a finite family of gates, then it is computable in (classical) polynomial time. In particular, for 0 < ɛ ≤ δ ≤ 1, we define BQNC 0 ɛ,δ to be the class of languages recognized by constant depth, polynomialsize quantum circuits with acceptance probability either < ɛ (for rejection) or ≥ δ (for acceptance). We show that BQNC 0 ɛ,δ ⊆ P, provided that 1 − δ ≤ 2 −2d (1 − ɛ), where d is the circuit depth. On the other hand, we adapt and extend ideas of Terhal & DiVincenzo [TD04] to show that, for any family F of quantum gates including Hadamard and CNOT gates, computing the acceptance probabilities of depthfive circuits over F is just as hard as computing these probabilities for arbitrary quantum circuits over F. In particular, this implies that NQNC 0 = NQACC = NQP = coC=P, where NQNC 0 is the constantdepth analog of the class NQP. This essentially refutes a conjecture of Green et al. that NQACC ⊆ TC 0 [GHMP02]. 1
The intersection of two halfspaces has high threshold degree
 In Proc. of the 50th Symposium on Foundations of Computer Science (FOCS
, 2009
"... Abstract. The threshold degree of a Boolean function f: {0, 1} n → {−1, +1} is the least degree of a real polynomial p such that f(x) ≡ sgn p(x). We construct two halfspaces on {0, 1} n whose intersection has threshold degree Θ ( √ n), an exponential improvement on previous lower bounds. This solv ..."
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Abstract. The threshold degree of a Boolean function f: {0, 1} n → {−1, +1} is the least degree of a real polynomial p such that f(x) ≡ sgn p(x). We construct two halfspaces on {0, 1} n whose intersection has threshold degree Θ ( √ n), an exponential improvement on previous lower bounds. This solves an open problem due to Klivans (2002) and rules out the use of perceptronbased techniques for PAC learning the intersection of two halfspaces, a central unresolved challenge in computational learning. We also prove that the intersection of two majority functions has threshold degree Ω(log n), which is tight and settles a conjecture of O’Donnell and Servedio (2003). Our proof consists of two parts. First, we show that for any nonconstant Boolean functions f and g, the intersection f(x) ∧ g(y) has threshold degree O(d) if and only if ‖f − F ‖ ∞ + ‖g − G‖ ∞ < 1 for some rational functions F, G of degree O(d). Second, we determine the least degree required for approximating a halfspace and a majority function to any given accuracy by rational functions. Our technique further allows us to obtain direct sum theorems for polynomial representations of composed Boolean functions. In particular, we give an improved lower bound on the approximate degree of the ANDOR tree. Key words. intersections of halfspaces, polynomial representations of Boolean functions, rational
Quantum Proofs for Classical Theorems
, 2009
"... Alongside the development of quantum algorithms and quantum complexity theory in recent years, quantum techniques have also proved instrumental in obtaining results in classical (nonquantum) areas. In this paper we survey these results and the quantum toolbox they use. ..."
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Cited by 15 (4 self)
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Alongside the development of quantum algorithms and quantum complexity theory in recent years, quantum techniques have also proved instrumental in obtaining results in classical (nonquantum) areas. In this paper we survey these results and the quantum toolbox they use.
Quantum Boolean Functions
, 2009
"... In this paper we introduce the study of quantum boolean functions, which are unitary operators f whose square is the identity: f² = I. We describe several generalisations of wellknown results in the theory of boolean functions, including quantum property testing; a quantum version of the Goldreich ..."
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Cited by 10 (4 self)
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In this paper we introduce the study of quantum boolean functions, which are unitary operators f whose square is the identity: f² = I. We describe several generalisations of wellknown results in the theory of boolean functions, including quantum property testing; a quantum version of the GoldreichLevin algorithm for finding the large Fourier coefficients of boolean functions; and two quantum versions of a theorem of Friedgut, Kalai and Naor on the Fourier spectra of boolean functions. In order to obtain one of these generalisations, we prove a quantum extension of the hypercontractive inequality of