2,256 research outputs found

    Quantum speedup of Monte Carlo methods

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    Monte Carlo methods use random sampling to estimate numerical quantities which are hard to compute deterministically. One important example is the use in statistical physics of rapidly mixing Markov chains to approximately compute partition functions. In this work we describe a quantum algorithm which can accelerate Monte Carlo methods in a very general setting. The algorithm estimates the expected output value of an arbitrary randomised or quantum subrou-tine with bounded variance, achieving a near-quadratic speedup over the best possible classical algorithm. Combining the algorithm with the use of quantum walks gives a quantum speedup of the fastest known classical algorithms with rigorous performance bounds for computing partition functions, which use multiple-stage Markov chain Monte Carlo techniques. The quantum algo-rithm can also be used to estimate the total variation distance between probability distributions efficiently.

    Classical and Quantum Complexity of the Sturm-Liouville Eigenvalue Problem

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    We study the approximation of the smallest eigenvalue of a Sturm-Liouville problem in the classical and quantum settings. We consider a univariate Sturm-Liouville eigenvalue problem with a nonnegative function qq from the class C2([0,1])C^2([0,1]) and study the minimal number n(\e) of function evaluations or queries that are necessary to compute an \e-approximation of the smallest eigenvalue. We prove that n(\e)=\Theta(\e^{-1/2}) in the (deterministic) worst case setting, and n(\e)=\Theta(\e^{-2/5}) in the randomized setting. The quantum setting offers a polynomial speedup with {\it bit} queries and an exponential speedup with {\it power} queries. Bit queries are similar to the oracle calls used in Grover's algorithm appropriately extended to real valued functions. Power queries are used for a number of problems including phase estimation. They are obtained by considering the propagator of the discretized system at a number of different time moments. They allow us to use powers of the unitary matrix exp(12iM)\exp(\tfrac12 {\rm i}M), where MM is an n×nn\times n matrix obtained from the standard discretization of the Sturm-Liouville differential operator. The quantum implementation of power queries by a number of elementary quantum gates that is polylog in nn is an open issue.Comment: 33 page

    Quantum SDP-Solvers: Better upper and lower bounds

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    Brand\~ao and Svore very recently gave quantum algorithms for approximately solving semidefinite programs, which in some regimes are faster than the best-possible classical algorithms in terms of the dimension nn of the problem and the number mm of constraints, but worse in terms of various other parameters. In this paper we improve their algorithms in several ways, getting better dependence on those other parameters. To this end we develop new techniques for quantum algorithms, for instance a general way to efficiently implement smooth functions of sparse Hamiltonians, and a generalized minimum-finding procedure. We also show limits on this approach to quantum SDP-solvers, for instance for combinatorial optimizations problems that have a lot of symmetry. Finally, we prove some general lower bounds showing that in the worst case, the complexity of every quantum LP-solver (and hence also SDP-solver) has to scale linearly with mnmn when mnm\approx n, which is the same as classical.Comment: v4: 69 pages, small corrections and clarifications. This version will appear in Quantu

    Quantum Chebyshev's Inequality and Applications

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    In this paper we provide new quantum algorithms with polynomial speed-up for a range of problems for which no such results were known, or we improve previous algorithms. First, we consider the approximation of the frequency moments FkF_k of order k3k \geq 3 in the multi-pass streaming model with updates (turnstile model). We design a PP-pass quantum streaming algorithm with memory MM satisfying a tradeoff of P2M=O~(n12/k)P^2 M = \tilde{O}(n^{1-2/k}), whereas the best classical algorithm requires PM=Θ(n12/k)P M = \Theta(n^{1-2/k}). Then, we study the problem of estimating the number mm of edges and the number tt of triangles given query access to an nn-vertex graph. We describe optimal quantum algorithms that perform O~(n/m1/4)\tilde{O}(\sqrt{n}/m^{1/4}) and O~(n/t1/6+m3/4/t)\tilde{O}(\sqrt{n}/t^{1/6} + m^{3/4}/\sqrt{t}) queries respectively. This is a quadratic speed-up compared to the classical complexity of these problems. For this purpose we develop a new quantum paradigm that we call Quantum Chebyshev's inequality. Namely we demonstrate that, in a certain model of quantum sampling, one can approximate with relative error the mean of any random variable with a number of quantum samples that is linear in the ratio of the square root of the variance to the mean. Classically the dependency is quadratic. Our algorithm subsumes a previous result of Montanaro [Mon15]. This new paradigm is based on a refinement of the Amplitude Estimation algorithm of Brassard et al. [BHMT02] and of previous quantum algorithms for the mean estimation problem. We show that this speed-up is optimal, and we identify another common model of quantum sampling where it cannot be obtained. For our applications, we also adapt the variable-time amplitude amplification technique of Ambainis [Amb10] into a variable-time amplitude estimation algorithm.Comment: 27 pages; v3: better presentation, lower bound in Theorem 4.3 is ne

    Assessing, testing, and challenging the computational power of quantum devices

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    Randomness is an intrinsic feature of quantum theory. The outcome of any measurement will be random, sampled from a probability distribution that is defined by the measured quantum state. The task of sampling from a prescribed probability distribution therefore seems to be a natural technological application of quantum devices. And indeed, certain random sampling tasks have been proposed to experimentally demonstrate the speedup of quantum over classical computation, so-called “quantum computational supremacy”. In the research presented in this thesis, I investigate the complexity-theoretic and physical foundations of quantum sampling algorithms. Using the theory of computational complexity, I assess the computational power of natural quantum simulators and close loopholes in the complexity-theoretic argument for the classical intractability of quantum samplers (Part I). In particular, I prove anticoncentration for quantum circuit families that give rise to a 2-design and review methods for proving average-case hardness. I present quantum random sampling schemes that are tailored to large-scale quantum simulation hardware but at the same time rise up to the highest standard in terms of their complexity-theoretic underpinning. Using methods from property testing and quantum system identification, I shed light on the question, how and under which conditions quantum sampling devices can be tested or verified in regimes that are not simulable on classical computers (Part II). I present a no-go result that prevents efficient verification of quantum random sampling schemes as well as approaches using which this no-go result can be circumvented. In particular, I develop fully efficient verification protocols in what I call the measurement-device-dependent scenario in which single-qubit measurements are assumed to function with high accuracy. Finally, I try to understand the physical mechanisms governing the computational boundary between classical and quantum computing devices by challenging their computational power using tools from computational physics and the theory of computational complexity (Part III). I develop efficiently computable measures of the infamous Monte Carlo sign problem and assess those measures both in terms of their practicability as a tool for alleviating or easing the sign problem and the computational complexity of this task. An overarching theme of the thesis is the quantum sign problem which arises due to destructive interference between paths – an intrinsically quantum effect. The (non-)existence of a sign problem takes on the role as a criterion which delineates the boundary between classical and quantum computing devices. I begin the thesis by identifying the quantum sign problem as a root of the computational intractability of quantum output probabilities. It turns out that the intricate structure of the probability distributions the sign problem gives rise to, prohibits their verification from few samples. In an ironic twist, I show that assessing the intrinsic sign problem of a quantum system is again an intractable problem

    Quantum Amplitude Amplification and Estimation

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    Consider a Boolean function χ:X{0,1}\chi: X \to \{0,1\} that partitions set XX between its good and bad elements, where xx is good if χ(x)=1\chi(x)=1 and bad otherwise. Consider also a quantum algorithm A\mathcal A such that A0=xXαxxA |0\rangle= \sum_{x\in X} \alpha_x |x\rangle is a quantum superposition of the elements of XX, and let aa denote the probability that a good element is produced if A0A |0\rangle is measured. If we repeat the process of running AA, measuring the output, and using χ\chi to check the validity of the result, we shall expect to repeat 1/a1/a times on the average before a solution is found. *Amplitude amplification* is a process that allows to find a good xx after an expected number of applications of AA and its inverse which is proportional to 1/a1/\sqrt{a}, assuming algorithm AA makes no measurements. This is a generalization of Grover's searching algorithm in which AA was restricted to producing an equal superposition of all members of XX and we had a promise that a single xx existed such that χ(x)=1\chi(x)=1. Our algorithm works whether or not the value of aa is known ahead of time. In case the value of aa is known, we can find a good xx after a number of applications of AA and its inverse which is proportional to 1/a1/\sqrt{a} even in the worst case. We show that this quadratic speedup can also be obtained for a large family of search problems for which good classical heuristics exist. Finally, as our main result, we combine ideas from Grover's and Shor's quantum algorithms to perform amplitude estimation, a process that allows to estimate the value of aa. We apply amplitude estimation to the problem of *approximate counting*, in which we wish to estimate the number of xXx\in X such that χ(x)=1\chi(x)=1. We obtain optimal quantum algorithms in a variety of settings.Comment: 32 pages, no figure
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