Quantum Probability Oracles & Multidimensional Amplitude Estimation

We give a multidimensional version of amplitude estimation. Let p be an n-dimensional probability distribution which can be sampled from using a quantum circuit U_p. We show that all coordinates of p can be estimated up to error ε per coordinate using Õ(1/(ε)) applications of U_p and its inverse. This generalizes the normal amplitude estimation algorithm, which solves the problem for n = 2. Our results also imply a Õ(n/ε) query algorithm for ℓ1-norm (the total variation distance) estimation and a Õ(√n/ε) query algorithm for ℓ2-norm. We also show that these results are optimal up to logarithmic factors

A quantum view on convex optimization

In this dissertation we consider quantum algorithms for convex optimization. We start by considering a black-box setting of convex optimization. In this setting we show that quantum computers require exponentially fewer queries to a membership oracle for a convex set in order to implement a separation oracle for that set. We do so by proving that Jordan's quantum gradient algorithm can also be applied to find sub-gradients of convex Lipschitz functions, even though these functions might not even be differentiable. As a corollary we get a quadraticly faster algorithm for convex optimization using membership queries. As a second set of results we give sub-linear time quantum algorithms for semidefinite optimization by speeding up the iterations of the Arora-Kale algorithm. For the problem of finding approximate Nash equilibria for zero-sum games we then give specific algorithms that improve the error-dependence and only depend on the sparsity of the game, not it's size. These last results yield improved algorithms for linear programming as a corollary. We also show several lower bounds in these settings, matching the upper bounds in most or all parameters

Quantum SDP-Solvers: Better upper and lower bounds

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 $n$ of the problem and the number $m$ 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 $mn$ when $m\approx n$, which is the same as classical.Comment: v4: 69 pages, small corrections and clarifications. This version will appear in Quantu

Improvements in Quantum SDP-Solving with Applications

Following the first paper on quantum algorithms for SDP-solving by Brandão and Svore [Brandão and Svore, 2017] in 2016, rapid developments have been made on quantum optimization algorithms. In this paper we improve and generalize all prior quantum algorithms for SDP-solving and give a simpler and unified framework. We take a new perspective on quantum SDP-solvers and introduce several new techniques. One of these is the quantum operator input model, which generalizes the different input models used in previous work, and essentially any other reasonable input model. This new model assumes that the input matrices are embedded in a block of a unitary operator. In this model we give a O~((sqrt{m}+sqrt{n}gamma)alpha gamma^4) algorithm, where n is the size of the matrices, m is the number of constraints, gamma is the reciprocal of the scale-invariant relative precision parameter, and alpha is a normalization factor of the input matrices. In particular for the standard sparse-matrix access, the above result gives a quantum algorithm where alpha=s. We also improve on recent results of Brandão et al. [Fernando G. S. L. Brandão et al., 2018], who consider the special case w

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

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

Quantum algorithms for matrix scaling and matrix balancing

Matrix scaling and matrix balancing are two basic linear-algebraic problems with a wide variety of applications, such as approximating the permanent, and pre-conditioning linear systems to make them more numerically stable. We study the power and limitations of quantum algorithms for these problems. We provide quantum implementations of two classical (in both senses of the word) methods: Sinkhorn's algorithm for matrix scaling and Osborne's algorithm for matrix balancing. Using amplitude estimation as our main tool, our quantum implementations both run in time Õ(√mn/∈4) for scaling or balancing an n×n matrix (given by an oracle) with m non-zero entries to within ℓ1-error ∈. Their classical analogs use time Õ(m/∈2), and every classical algorithm for scaling or balancing with small constant ∈ requires Ω(m) queries to the entries of the input matrix. We thus achieve a polynomial speed-up in terms of n, at the expense of a worse polynomial dependence on the obtained ℓ1-error ∈. Even for constant ∈ these problems are already non-trivial (and relevant in applications). Along the way, we extend the classical analysis of Sinkhorn's and Osborne's algorithm to allow for errors in the computation of marginals. We also adapt an improved analysis of Sinkhorn's algorithm for entrywise-positive matrices to the ℓ1-setting, obtaining an Õ(n1.5/∈3)-time quantum algorithm for ∈-ℓ1-scaling. We also prove a lower bound, showing our quantum algorithm for matrix scaling is essentially optimal for constant ∈: every quantum algorithm for matrix scaling that achieves a constant ℓ1-error w.r.t. uniform marginals needs Ω(√ mn) queries