41,242 research outputs found

    Quantum singular value transformation and beyond: exponential improvements for quantum matrix arithmetics

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    Quantum computing is powerful because unitary operators describing the time-evolution of a quantum system have exponential size in terms of the number of qubits present in the system. We develop a new "Singular value transformation" algorithm capable of harnessing this exponential advantage, that can apply polynomial transformations to the singular values of a block of a unitary, generalizing the optimal Hamiltonian simulation results of Low and Chuang. The proposed quantum circuits have a very simple structure, often give rise to optimal algorithms and have appealing constant factors, while usually only use a constant number of ancilla qubits. We show that singular value transformation leads to novel algorithms. We give an efficient solution to a certain "non-commutative" measurement problem and propose a new method for singular value estimation. We also show how to exponentially improve the complexity of implementing fractional queries to unitaries with a gapped spectrum. Finally, as a quantum machine learning application we show how to efficiently implement principal component regression. "Singular value transformation" is conceptually simple and efficient, and leads to a unified framework of quantum algorithms incorporating a variety of quantum speed-ups. We illustrate this by showing how it generalizes a number of prominent quantum algorithms, including: optimal Hamiltonian simulation, implementing the Moore-Penrose pseudoinverse with exponential precision, fixed-point amplitude amplification, robust oblivious amplitude amplification, fast QMA amplification, fast quantum OR lemma, certain quantum walk results and several quantum machine learning algorithms. In order to exploit the strengths of the presented method it is useful to know its limitations too, therefore we also prove a lower bound on the efficiency of singular value transformation, which often gives optimal bounds.Comment: 67 pages, 1 figur

    A Nearly Optimal Lower Bound on the Approximate Degree of AC0^0

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    The approximate degree of a Boolean function f ⁣:{1,1}n{1,1}f \colon \{-1, 1\}^n \rightarrow \{-1, 1\} is the least degree of a real polynomial that approximates ff pointwise to error at most 1/31/3. We introduce a generic method for increasing the approximate degree of a given function, while preserving its computability by constant-depth circuits. Specifically, we show how to transform any Boolean function ff with approximate degree dd into a function FF on O(npolylog(n))O(n \cdot \operatorname{polylog}(n)) variables with approximate degree at least D=Ω(n1/3d2/3)D = \Omega(n^{1/3} \cdot d^{2/3}). In particular, if d=n1Ω(1)d= n^{1-\Omega(1)}, then DD is polynomially larger than dd. Moreover, if ff is computed by a polynomial-size Boolean circuit of constant depth, then so is FF. By recursively applying our transformation, for any constant δ>0\delta > 0 we exhibit an AC0^0 function of approximate degree Ω(n1δ)\Omega(n^{1-\delta}). This improves over the best previous lower bound of Ω(n2/3)\Omega(n^{2/3}) due to Aaronson and Shi (J. ACM 2004), and nearly matches the trivial upper bound of nn that holds for any function. Our lower bounds also apply to (quasipolynomial-size) DNFs of polylogarithmic width. We describe several applications of these results. We give: * For any constant δ>0\delta > 0, an Ω(n1δ)\Omega(n^{1-\delta}) lower bound on the quantum communication complexity of a function in AC0^0. * A Boolean function ff with approximate degree at least C(f)2o(1)C(f)^{2-o(1)}, where C(f)C(f) is the certificate complexity of ff. This separation is optimal up to the o(1)o(1) term in the exponent. * Improved secret sharing schemes with reconstruction procedures in AC0^0.Comment: 40 pages, 1 figur
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