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34
Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy
, 2010
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Universal computation by multiparticle quantum walk. http://arxiv.org/abs/1205.3782v1
"... Quantum walk is a versatile and intuitive framework for developing quantum algorithms. Applications of quantum walk include an example of exponential speedup over classical computation Quantum walk can also be viewed as a model of computation. From this perspective it is natural to ask which quant ..."
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Quantum walk is a versatile and intuitive framework for developing quantum algorithms. Applications of quantum walk include an example of exponential speedup over classical computation Quantum walk can also be viewed as a model of computation. From this perspective it is natural to ask which quantum computations can be performed efficiently by quantum walk. This question was answered in reference In this work we consider a generalization of quantum walk to systems with more than one walker on a graph. A continuoustime, multiparticle quantum walk is generated by a timeindependent Hamiltonian with a term corresponding to a singleparticle quantum walk for each particle, along with a term corresponding to an interaction between particles. We show that multiparticle quantum walk is capable of universal quantum computation. Specifically, we show that any nqubit circuit with g gates can be simulated by the dynamics of O(n) particles that interact for a time poly(n, g) on an unweighted planar graph of maximum degree 4 with poly(n, g) vertices. We present explicit universal constructions based on the BoseHubbard model, fermions with nearestneighbor interactions, and distinguishable particles with nearestneighbor interactions. We also show that almost any interaction between indistinguishable particles can be used to perform universal computation. Because our graphs are exponentially smaller (as a function of n) than those used in the singleparticle universality construction, the multiparticle quantum walks we describe can be efficiently implemented using an architecture where vertices of the graph are represented by devices at different spatial locations. Recently, there has been considerable interest in experimentally implementing multiparticle quantum walk Performing universal computation using quantum walk is nontrivial since the model is highly restricted. Not only must the computation be performed without timedependent control, it must also be encoded entirely in the choice of a graph, with no ability to adjust edge weights to implement a desired gate. The previous singleparticle universality construction was based on scattering a quantum walker through subgraphs that implement gates, but scattering is substantially more complicated in manybody interacting systems. To overcome this challenge, we design a multiparticle quantum walk that is well approximated by independent oneand twoparticle scattering processes, and develop tools to make this approximation precise. We have demonstrated the computational power of a broad class of manybody systems in the absence of timedependent control. Since it is also possible to efficiently simulate a multiparticle quantum walk of the type we consider using a universal quantum computer, we have shown that this model exactly captures the power of quantum computation. Viewed from a different perspective, our result provides limitations on classical simulation of manybody interacting systems. For example, assuming quantum computers are more powerful than classical ones, our work implies that the dynamics of the BoseHubbard model on a sparse, planar graph cannot be efficiently simulated on a classical computer. Multiparticle quantum walk In a multiparticle quantum walk, the particles interact on a given simple graph G with vertex set V (G) and edge set E(G). The Hilbert space for m distinguishable particles on 1
Concentration and Moment Inequalities for Polynomials of Independent Random Variables
"... In this work we design a general method for proving moment inequalities for polynomials of independent random variables. Our method works for a wide range of random variables including Gaussian, Boolean, exponential, Poisson and many others. We apply our method to derive general concentration inequa ..."
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In this work we design a general method for proving moment inequalities for polynomials of independent random variables. Our method works for a wide range of random variables including Gaussian, Boolean, exponential, Poisson and many others. We apply our method to derive general concentration inequalities for polynomials of independent random variables. We show that our method implies concentration inequalities for some previously open problems, e.g. permanent of random symmetric matrices. We show that our concentration inequality is stronger than the wellknownconcentration inequalityduetoKimandVu[29]. The main advantage of our method in comparison with the existing ones is a wide range of random variables we can handle and bounds for previously intractable regimes of high degree polynomials and small expectations. On the negative side we show that even for boolean random variables each term in our concentration inequality is tight.
The equivalence of sampling and searching
 arXiv:1009.5104, ECCC
, 2010
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Multimode quantum interference of photons in multiport integrated devices
"... Photonics is a leading approach in realizing future quantum technologies and recently, optical waveguide circuits on silicon chips have demonstrated high levels of miniaturization and performance. Multimode interference (MMI) devices promise a straightforward implementation of compact and robust mul ..."
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Photonics is a leading approach in realizing future quantum technologies and recently, optical waveguide circuits on silicon chips have demonstrated high levels of miniaturization and performance. Multimode interference (MMI) devices promise a straightforward implementation of compact and robust multiport circuits. Here, we show quantum interference in a 2×2 MMI coupler with visibility of V = 95.6 ± 0.9%. We further demonstrate the operation of a 4×4 port MMI device with photon pairs, which exhibits complex quantum interference behaviour. We have developed a new technique to fully characterize such multiport devices, which removes the need for phasesensitive measurements and may find applications for a wide range of photonic devices. Our results show that MMI devices can operate in the quantum regime with high fidelity and promise substantial simplification and concatenation of photonic quantum circuits.
Generalizing and Derandomizing Gurvits’s Approximation Algorithm for the Permanent
"... Around 2002, Leonid Gurvits gave a striking randomized algorithm to approximate the permanent of an n × n matrix A. The algorithm runs in O ( n 2 /ε 2) time, and approximates Per(A) to within ±ε‖A ‖ n additive error. A major advantage of Gurvits’s algorithm is that it works for arbitrary matrices, n ..."
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Around 2002, Leonid Gurvits gave a striking randomized algorithm to approximate the permanent of an n × n matrix A. The algorithm runs in O ( n 2 /ε 2) time, and approximates Per(A) to within ±ε‖A ‖ n additive error. A major advantage of Gurvits’s algorithm is that it works for arbitrary matrices, not just for nonnegative matrices. This makes it highly relevant to quantum optics, where the permanents of boundednorm complex matrices play a central role. Indeed, the existence of Gurvits’s algorithm is why, in their recent work on the hardness of quantum optics, Aaronson and Arkhipov (AA) had to talk about sampling problems rather than estimation problems. In this paper, we improve Gurvits’s algorithm in two ways. First, using an idea from quantum optics, we generalize the algorithm so that it yields a better approximation when the matrix A has either repeated rows or repeated columns. Translating back to quantum optics, this lets us classically estimate the probability of any outcome of an AAtype experiment—even an outcome involving multiple photons “bunched ” in the same mode—at least as well as that probability can be estimated by the experiment itself. (This does not, of course, let us solve the AA sampling problem.) It also yields a general upper bound on the probabilities of “bunched” outcomes, which resolves a conjecture of Gurvits and might be of independent physical interest. Second, we use εbiased sets to derandomize Gurvits’s algorithm, in the special case where the matrix A is nonnegative. More interestingly, we generalize the notion of εbiased sets to the complex numbers, construct “complex εbiased sets, ” then use those sets to derandomize even our generalization of Gurvits’s algorithm to the multirow/multicolumn case (again for nonnegative A). Whether Gurvits’s algorithm can be derandomized for general A remains an outstanding problem. 1
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
Verified Delegated Quantum Computing with One Pure Qubit
, 2014
"... While building a universal quantum computer remains challenging, devices of restricted power such as the socalled one pure qubit model have attracted considerable attention. An important step in the construction of these limited quantum computational devices is the understanding of whether the ver ..."
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While building a universal quantum computer remains challenging, devices of restricted power such as the socalled one pure qubit model have attracted considerable attention. An important step in the construction of these limited quantum computational devices is the understanding of whether the verification of the computation within these models could be also performed in the restricted scheme. Encoding via blindness (a cryptographic protocol for delegated computing) has proven successful for the verification of universal quantum computation with a restricted verifier. In this paper, we present the adaptation of this approach to the one pure qubit model, and present the first feasible scheme for the verification of delegated one pure qubit model of quantum computing. 1
Unconditionally verifiable blind computation
, 2014
"... Blind Quantum Computing (BQC) allows a client to have a server carry out a quantum computation for them such that the client’s input, output and computation remain private. A desirable property for any BQC protocol is verification, whereby the client can verify with high probability whether the serv ..."
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Blind Quantum Computing (BQC) allows a client to have a server carry out a quantum computation for them such that the client’s input, output and computation remain private. A desirable property for any BQC protocol is verification, whereby the client can verify with high probability whether the server has followed the instructions of the protocol, or if there has been some deviation resulting in a corrupted output state. A verifiable BQC protocol can be viewed as an interactive proof system leading to consequences for complexity theory. The authors, together with Broadbent, previously proposed a universal and unconditionally secure BQC scheme where the client only needs to be able to prepare single qubits in separable states randomly chosen from a finite set and send them to the server, who has the balance of the required quantum computational resources. In this paper we extend that protocol with new functionality allowing blind computational basis measurements, which we use to construct a new verifiable BQC protocol based on a new class of resource states. We rigorously prove that the probability of failing to detect an incorrect output is exponentially small in a security parameter, while resource overhead remains polynomial in this parameter. The new resource state allows entangling gates to be performed between arbitrary pairs of logical qubits with only constant overhead. This is a significant improvement on the original scheme, which required that all computations to be performed must first be put into a nearest neighbour form, incurring linear overhead in the number of qubits. Such an improvement has important consequences for efficiency and faulttolerance thresholds. 1