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25
Improved simulation of stabilizer circuits
 Phys. Rev. Lett
"... The GottesmanKnill theorem says that a stabilizer circuit—that is, a quantum circuit consisting solely of CNOT, Hadamard, and phase gates—can be simulated efficiently on a classical computer. This paper improves that theorem in several directions. • By removing the need for Gaussian elimination, we ..."
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The GottesmanKnill theorem says that a stabilizer circuit—that is, a quantum circuit consisting solely of CNOT, Hadamard, and phase gates—can be simulated efficiently on a classical computer. This paper improves that theorem in several directions. • By removing the need for Gaussian elimination, we make the simulation algorithm much faster at the cost of a factor2 increase in the number of bits needed to represent a state. We have implemented the improved algorithm in a freelyavailable program called CHP (CNOTHadamardPhase), which can handle thousands of qubits easily. • We show that the problem of simulating stabilizer circuits is complete for the classical complexity class ⊕L, which means that stabilizer circuits are probably not even universal for classical computation. • We give efficient algorithms for computing the inner product between two stabilizer states, putting any nqubit stabilizer circuit into a “canonical form ” that requires at most O ( n 2 /log n) gates, and other useful tasks. • We extend our simulation algorithm to circuits acting on mixed states, circuits containing a limited number of nonstabilizer gates, and circuits acting on general tensorproduct initial states but containing only a limited number of measurements. 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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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
Simulating quantum computation by contracting tensor networks
 SIAM Journal on Computing
, 2005
"... The treewidth of a graph is a useful combinatorial measure of how close the graph is to a tree. We prove that a quantum circuit with T gates whose underlying graph has treewidth d can be simulated deterministically in T O(1) exp[O(d)] time, which, in particular, is polynomial in T if d = O(logT). Am ..."
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The treewidth of a graph is a useful combinatorial measure of how close the graph is to a tree. We prove that a quantum circuit with T gates whose underlying graph has treewidth d can be simulated deterministically in T O(1) exp[O(d)] time, which, in particular, is polynomial in T if d = O(logT). Among many implications, we show efficient simulations for quantum formulas, defined and studied by Yao (Proceedings of the 34th Annual Symposium on Foundations of Computer Science, 352–361, 1993), and for logdepth circuits whose gates apply to nearby qubits only, a natural constraint satisfied by most physical implementations. We also show that oneway quantum computation of Raussendorf and Briegel (Physical Review Letters, 86:5188– 5191, 2001), a universal quantum computation scheme with promising physical implementations, can be efficiently simulated by a randomized algorithm if its quantum resource is derived from a smalltreewidth graph.
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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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
A simple proof that Toffoli and Hadamard are quantum universal
 IN QUANTPH/0301040
, 2003
"... Recently Shi [15] proved that Toffoli and Hadamard are universal for quantum computation. This is perhaps the simplest universal set of gates that one can hope for, conceptually; It shows that one only needs to add the Hadamard gate to make a ’classical ’ set of gates quantum universal. In this note ..."
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Recently Shi [15] proved that Toffoli and Hadamard are universal for quantum computation. This is perhaps the simplest universal set of gates that one can hope for, conceptually; It shows that one only needs to add the Hadamard gate to make a ’classical ’ set of gates quantum universal. In this note we give a few lines proof of this fact relying on Kitaev’s universal set of gates [11], and discuss the meaning of the result.
1 QUANTUM COMPUTING AND THE ENTANGLEMENT FRONTIER
, 2014
"... Quantum information science explores the frontier of highly complex quantum states, the “entanglement frontier. ” This study is motivated by the observation (widely believed but unproven) that classical systems cannot simulate highly entangled quantum systems efficiently, and we hope to hasten the d ..."
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Quantum information science explores the frontier of highly complex quantum states, the “entanglement frontier. ” This study is motivated by the observation (widely believed but unproven) that classical systems cannot simulate highly entangled quantum systems efficiently, and we hope to hasten the day when well controlled quantum systems can perform tasks surpassing what can be done in the classical world. One way to achieve such “quantum supremacy ” would be to run an algorithm on a quantum computer which solves a problem with a superpolynomial speedup relative to classical computers, but there may be other ways that can be achieved sooner, such as simulating exotic quantum states of strongly correlated matter. To operate a large scale quantum computer reliably we will need to overcome the debilitating effects of decoherence, which might be done using “standard ” quantum hardware protected by quantum errorcorrecting codes, or by exploiting the nonabelian quantum statistics of anyons realized in solid state systems, or by combining both methods. Only by challenging the entanglement frontier will we learn whether Nature provides extravagant resources far beyond what the classical world would allow. Rapporteur talk at the 25th Solvay Conference on Physics
BosonSampling Is Far From Uniform
"... BosonSampling, which we proposed three years ago, is a scheme for using linearoptical networks to solve sampling problems that appear to be intractable for a classical computer. In a recent manuscript, Gogolin et al. claimed that even an ideal BosonSampling device’s output would be “operationally i ..."
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BosonSampling, which we proposed three years ago, is a scheme for using linearoptical networks to solve sampling problems that appear to be intractable for a classical computer. In a recent manuscript, Gogolin et al. claimed that even an ideal BosonSampling device’s output would be “operationally indistinguishable ” from a purely random string, at least “without detailed a priori knowledge”—or at any rate, that telling the two apart might itself be a hard computational problem. In this paper, we first answer these claims—explaining why the first is based on a definition of “a priori knowledge ” so strange that, were it adopted, almost no quantum algorithm could be distinguished from a pure randomnumber source; while the second is neither new nor a practical obstacle to interesting BosonSampling experiments (for reasons discussed in our original paper, which Gogolin et al. fail to acknowledge). However, we then go further, and address some interesting research questions inspired by Gogolin et al.’s mistaken arguments. We prove that, provided the number of output modes is at least quadratically greater than the number of photons, with high probability over a Haarrandom matrix A, the BosonSampling distribution induced by A is far from the uniform
Quantum information processing with lowdimensional systems,” in Quantum information processing: From theory to experiment, edby
, 2006
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