Quantum Singular Value Transformation & Its Algorithmic Applications
András Gilyén
Abstract:
In this dissertation we study how efficiently quantum computers can solve various problems, and how large speedups can be achieved compared to classical computers. In particular we develop a generic quantum algorithmic framework that we call "quantum singular value transformation", and show how it unifies a large number of prominent quantum algorithms. Then we show several problems where quantum singular value transformation leads to new quantum algorithms or improves various aspects of earlier approaches.
In Chapter 2 we develop a new quantum singular value transformation algorithm capable of working with exponentially large matrices, that can apply polynomial transformations to the singular values of a block of a unitary. The proposed quantum circuits have a very simple structure, often give rise to optimal algorithms and have appealing constant factors, while typically only using a constant number of ancilla qubits. We prove several properties of quantum singular value transformation, including its robustness.
In Chapter 3 we show that quantum singular value transformation leads to novel algorithms. We propose a new method for singular value estimation, and 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. We also show that quantum singular value transformation leads to a unified framework of quantum algorithms incorporating a variety of quantum speed-ups. Using this framework we can describe many quantum algorithms on a high level, hopefully making them accessible to researchers even outside the quantum algorithm community. We illustrate this by showing how our meta-algorithm generalizes a number of prominent quantum algorithms, and quickly derive the following algorithms: optimal Hamiltonian simulation, implementing the Moore-Penrose pseudoinverse (i.e., the HHL algorithm) 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 bound on the efficiency of quantum singular value transformation, which often gives optimal lower bounds.
In Chapter 4 we develop an improved quantum algorithm for computing the gradient of a multivariate real-valued function f : R^d → R by evaluating it at only a logarithmic number of points in superposition. Our algorithm is an improved version of Jordan's gradient computation algorithm [Jor05], providing an approximation of the gradient ∇f with quadratically better dependence on the evaluation accuracy of f, for an important class of smooth functions. Furthermore, we show that most objective functions arising from a class of quantum optimization procedures satisfy the necessary smoothness conditions, hence our algorithm improves the runtime of prior approaches for training quantum auto-encoders, variational quantum eigensolvers (VQE), and quantum approximate optimization algorithms (QAOA). Finally we prove that in a continuous phase-query model, our gradient computation algorithm has essentially optimal query complexity, for a class of smooth functions. For our lower bound we derive a continuous input version of the so-called hybrid method.
In Chapter 5 we study to what extent quantum algorithms can speed up solving convex optimization problems. Following the classical literature we assume access to a convex set via various oracles, and we examine the efficiency of reductions between the different oracles. In particular, we show how a separation oracle can be implemented using ~O(1) quantum queries to a membership oracle, which is an exponential quantum speed-up over the Ω(n) membership queries that are needed classically. We show that a quantum computer can very efficiently compute an approximate subgradient of a convex Lipschitz function. Combining this with a simplification of recent classical work of Lee, Sidford, and Vempala [LSV18] gives our efficient separation oracle. This in turn implies, via a known algorithm, that ~O(n) quantum queries to a membership oracle suffice to implement an optimization oracle (the best known classical upper bound on the number of membership queries is quadratic). We also prove several lower bounds: Ω(√n) quantum separation (or membership) queries are needed for optimization if the algorithm knows an interior point of the convex set, and Ω(n) quantum separation queries are needed if it does not.
In Chapter 6 we take a new perspective on quantum SDP-solvers, introducing several new techniques, and improve on all prior quantum algorithms for SDP-solving. Our new input model generalizes all prior input models, and assumes that the input matrices are provided as block-encodings. In this model we give a ~O((√m+γ√n)αγ^4) algorithm, where n is the size of the matrices, m is the number of constraints, γ is the reciprocal of the scale-invariant relative precision parameter, and α is a normalization factor of the input matrices. In particular for the standard sparse-matrix access model, the above result gives a quantum algorithm where α equals the sparsity s. We also improve on recent results of Brandão et al. [BKL+18], who consider the special case when the input matrices are proportional to mixed quantum states that one can query. For this model Brandão et al. [BKL+18] showed that the dependence on n can be replaced by a polynomial dependence on both the rank and the trace of the input matrices. We remove the dependence on the rank and hence require only a dependence on the trace of the input matrices. We also show an application to the problem of shadow tomography, recently introduced by Aaronson [Aar18]. Finally we prove a new ~Ω(αγ√m) lower bound for solving LPs and SDPs in the quantum operator model, which also implies a lower bound for the model of Brandão et al. [BKL+18].
In Chapter 7 we study constructive versions of the Lovász Local Lemma (LLL) and its quantum generalization. The quantum Lovász Local Lemma can be stated in terms of frustration-free local Hamiltonians: these Hamiltonians have the property that their ground state minimizes the energy of all local terms simultaneously. We improve on the previous constructive quantum results by designing an algorithm that works efficiently for non-commuting terms as well, assuming that the system is "uniformly" gapped, by which we mean that the system and all its subsystems have an energy gap that is at least inverse polynomially large. We generalize and simplify Moser's classical "compression argument", and derive a non-commutative quantum version of the famous Moser-Tardos resampling algorithm. Finally, in the so-called variable version of the classical LLL we find optimal bounds for the "guaranteed-to-be-feasible" probabilities on cyclic dependency graphs, and show that this region is always strictly larger than in the generic non-variable version, where Shearer's bound is optimal. This in turn shows a separation between the variable version of the classical and the quantum LLL.