This thesis consists of two parts. In Part I we present a new class of norm discretization inequalities suited for low-degree polynomials in many dimensions, with applications to discrete harmonic analysis and to quantum and classical learning theory.
\r\n\r\nDiscretization inequalities (of Bernstein type) control the supremum norm of polynomials f by their supremum norms over certain finite subsets T of the domain. Unlike earlier multivariate Bernstein-type discretization inequalities we establish dimension-free comparisons for simple and generic T, such as product sets T=S\u2081 \u00d7 \u22c5\u22c5\u22c5 \u00d7 S\u2099 for S\u2c7c's consisting of well-spread points in R or C, in exchange for a constant that grows with deg(f).
\r\n\r\nOur results also introduce the notion of \"individual degree\"\u2014the maximum degree of f in any one variable\u2014as a fundamental parameter for discretization inequalities: we show for the first time that dimension-free discretizations of the uniform norm are possible for T with cardinality independent of deg(f), provided f has bounded individual degree.
\r\n\r\nOur work offers a new, high-dimensional perspective on discretization inequalities and yields several new results in analysis on the hypergrid (i.e., products of cyclic groups), including Bohnenblust\u2013Hille-type inequalities, dimension-free supremum norm bounds on level-k Fourier projections, and junta theorems. These estimates in turn provide the key analytic tools for extending recent breakthroughs in learning low-degree functions to the hypergrid and to its quantum analogue, local observables on K-level qudit systems.
\r\n\r\nIn Part II we apply ideas from analysis of Boolean functions to study other aspects of (quantum) computation: circuit complexity and property testing.
\r\n\r\nFirst, we introduce and study a deceptively simple model of constant-depth quantum circuits and begin the project of proving bounds on its capabilities, ultimately drawing on connections to nonlocal games and notions of approximate degree.
\r\n\r\nSecond, we introduce a new access model for property testing, quantum data, which allows for ultrafast testing algorithms where classical data provably yields no fast testers\u2014such as for monotonicity, symmetry, and triangle-freeness.
", "doi": "10.7907/grjv-rz74", "publication_date": "2025", "thesis_type": "phd", "thesis_year": "2025" }, { "id": "thesis:17285", "collection": "thesis", "collection_id": "17285", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:05282025-173306123", "primary_object_url": { "basename": "Thesis.pdf", "content": "final", "filesize": 2587888, "license": "other", "mime_type": "application/pdf", "url": "/17285/1/Thesis.pdf", "version": "v4.0.0" }, "type": "thesis", "title": "Advancing Applications of Quantum Computers in Quantum Simulation, Optimization, Learning, and Topological Data Analysis", "author": [ { "family_name": "King", "given_name": "William Robert", "orcid": "0000-0002-8152-6340", "clpid": "King-William-Robert" } ], "thesis_advisor": [ { "family_name": "Schulman", "given_name": "Leonard J.", "orcid": "0000-0001-9901-2797", "clpid": "Schulman-L-J" } ], "thesis_committee": [ { "family_name": "Vidick", "given_name": "Thomas Georges", "orcid": "0000-0002-6405-365X", "clpid": "Vidick-T" }, { "family_name": "Schulman", "given_name": "Leonard J.", "orcid": "0000-0001-9901-2797", "clpid": "Schulman-L-J" }, { "family_name": "Preskill", "given_name": "John P.", "orcid": "0000-0002-2421-4762", "clpid": "Preskill-J" }, { "family_name": "Huang", "given_name": "Hsin-Yuan (Robert)", "orcid": "0000-0001-5317-2613", "clpid": "Huang-Hsin-Yuan" }, { "family_name": "Umans", "given_name": "Christopher M.", "orcid": "0000-0002-6390-9401", "clpid": "Umans-C-M" } ], "local_group": [ { "literal": "div_eng" } ], "abstract": "This thesis investigates novel directions for harnessing the potential of quantum computers in future applications. It is structured into three sections.
\r\n\r\nQuantum Simulation.
\r\nWe address two key questions: what systems exhibit quantum advantage in predicting ground state properties, and how can we reduce the cost of quantum simulations? For the former, we find that strongly interacting fermionic systems have promising characteristics for quantum advantage. For the latter, we develop an improved method for compiling block encodings using sum-of-squares optimization.
Learning with Entangled Measurements.
\r\nWe explore the benefits of leveraging entangled measurements on quantum states stored in quantum memory. These learning algorithms can be applied to the readout stage of quantum simulations, or to learn from quantum data from nature.
Topological Data Analysis.
\r\n\r\nUsing complexity-theoretic insights, we demonstrate that certain problems in topological data analysis possess a quantum mechanical structure, suggesting opportunities for quantum algorithms in this area.
The evolution of the human brain was one of the milestones in the history of information after the emergence of life. The underlying biological, chemical, and physical processes of the brain have amazed scientists for a long time. It is still a mystery how the human brain computes a simple arithmetical operation like 2 + 2 = 4. This enigma has spurred investigations into understanding the intrinsic architecture of the brain.
\r\n \r\nThis thesis delves into two primary models for brain architecture: Feedforward Neural Networks and Nearest Neighbor (NN) Representations. Both models are treated under the hypothesis that our brain does not work with \"large\" numbers and expressive power is derived from connectivity. Thus, when examining a network or, more precisely, a single neuron model, we strive to minimize the bit resolution of weights, potentially increasing depth or circuit complexity.
\r\n \r\nFor the NN representations, the memory is defined by a set of vectors in R\u207f (that we call anchors), computation is performed by convergence from an input vector to a nearest neighbor anchor, and the output is a label associated with an anchor. Limited bit resolution in the anchor entries may result in an increase of the size of the NN representation.
\r\n\r\nIn the digital age, computers universally employ the binary numeral system, ensuring the enduring relevance of Boolean functions. This study specifically explores the trade-off between resolution and size for the computation models for Boolean functions. It is established that \"low resolution\" models may require a polynomial or even an exponential increase in the size complexity of the \"high resolution\" model, potentially making the practical implementation infeasible. Building upon prior research, our goal is to optimize these blow-ups by narrowing the gaps between theoretical upper and lower bounds under various constraints. Additionally, we aim to establish connections between NN representations and neural network models by providing explicit NN representations for well-known Boolean functions in Circuit Complexity Theory.
", "doi": "10.7907/trse-ff38", "publication_date": "2024", "thesis_type": "phd", "thesis_year": "2024" }, { "id": "thesis:16064", "collection": "thesis", "collection_id": "16064", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:06022023-202038500", "primary_object_url": { "basename": "Caltech_Thesis (8).pdf", "content": "final", "filesize": 1383310, "license": "other", "mime_type": "application/pdf", "url": "/16064/1/Caltech_Thesis (8).pdf", "version": "v4.0.0" }, "type": "thesis", "title": "Revocable Cryptography in a Quantum World", "author": [ { "family_name": "Poremba", "given_name": "Alexander Mario", "orcid": "0000-0002-7330-1539", "clpid": "Poremba-Alexander-Mario" } ], "thesis_advisor": [ { "family_name": "Vidick", "given_name": "Thomas G.", "orcid": "0000-0002-6405-365X", "clpid": "Vidick-T" } ], "thesis_committee": [ { "family_name": "Mahadev", "given_name": "Urmila", "clpid": "Mahadev-Urmila" }, { "family_name": "Preskill", "given_name": "John P.", "orcid": "0000-0002-2421-4762", "clpid": "Preskill-J" }, { "family_name": "Umans", "given_name": "Christopher M.", "orcid": "0000-0002-6390-9401", "clpid": "Umans-C-M" }, { "family_name": "Vidick", "given_name": "Thomas G.", "orcid": "0000-0002-6405-365X", "clpid": "Vidick-T" } ], "local_group": [ { "literal": "div_eng" } ], "abstract": "Quantum cryptography leverages unique features of quantum mechanics in order to construct cryptographic primitives which are oftentimes impossible for digital computers. Cryptographic applications of quantum computers therefore have the potential for useful quantum advantage---entirely without computational speed-ups. Can we use the power of quantum states to address fundamental limitations in the world of classical cryptography, such as the intricate problem of ``revoking'' information from an untrusted party? This thesis undertakes a systematic study of how to delegate and revoke privileges in a world in which quantum computers become widely available. As part of a single framework we call revocable cryptography, we show how to revoke programs, encrypted data, and even cryptographic keys under standard assumptions.
\r\n\r\nIn the first part of this thesis, we focus on the following question: can we use the no-cloning principle of quantum mechanics and encode a program in such a way that it can be evaluated, yet it cannot be pirated? Naturally, we would also like to ensure that, once the program is ``returned,'' the recipient loses its ability to evaluate it. While this quantum notion of secure software leasing (SSL) was shown to be impossible for general programs by Ananth and La Placa (Eurocrypt 2021), their work left open the possibility that it is achievable for more primitive classes of programs. We construct an SSL scheme for a large class of evasive functions known as compute-and-compare programs---a more expressive generalization of point functions. Our scheme can be instantiated with any cryptographic hash function, and we prove its security in the quantum random oracle model. As a complementary result, we also construct a quantum copy-protection scheme for multi-bit point functions, which achieves a related but stronger notion of software protection previously introduced by Aaronson (CCC 2009).
\r\n\r\nIn the second part of this thesis, we ask: is it possible to provably delete information by leveraging the laws of quantum mechanics? We revisit a cryptographic notion called certified deletion, which was proposed by Broadbent and Islam (TCC 2020). While this remarkable notion allows a classical verifier to be convinced that quantum ciphertext has been deleted by an untrusted party, it offers no additional layer of functionality. We use Gaussian superpositions over lattices to construct the first fully homomorphic encryption scheme with certified deletion -- a protocol which allows an untrusted quantum server to compute on encrypted data and to also prove data deletion to a client. Our scheme has the desirable property that verification of a deletion certificate is public; meaning anyone can verify whether deletion has taken place. Assuming the quantum subexponential hardness of the learning with errors problem (Regev, STOC 2005), we can prove that our scheme achieves a particularly strong information-theoretic deletion guarantee; namely, once a valid deletion certificate is presented, the plaintext remains hidden even if the adversary is subsequently allowed to run in unbounded time.
\r\n\r\nIn the final part of this thesis, we ask: is it possible to revoke a crytographic key by using the power of quantum information? We give an affirmative answer to this question and design cryptosystems with key-revocation capabilities; specifically, we consider schemes with the guarantee that, once the secret key (represented as a quantum state) is successfully revoked from a user, they no longer have the ability to perform the same functionality as before. We define and construct several fundamental cryptographic primitives with key-revocation capabilities, namely pseudorandom functions, secret-key and public-key encryption, and even fully homomorphic encryption, assuming the subexponential hardness of the learning with errors problem. Central to all our constructions is our approach for making the Dual-Regev encryption scheme (Gentry, Peikert and Vaikuntanathan, STOC 2008) revocable.
", "doi": "10.7907/y62s-j417", "publication_date": "2023", "thesis_type": "phd", "thesis_year": "2023" }, { "id": "thesis:10241", "collection": "thesis", "collection_id": "10241", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:06012017-013622968", "primary_object_url": { "basename": "Guo_Zeyu_2017.pdf", "content": "final", "filesize": 1548403, "license": "other", "mime_type": "application/pdf", "url": "/10241/1/Guo_Zeyu_2017.pdf", "version": "v4.0.0" }, "type": "thesis", "title": "P-Schemes and Deterministic Polynomial Factoring Over Finite Fields", "author": [ { "family_name": "Guo", "given_name": "Zeyu", "orcid": "0000-0001-7893-4346", "clpid": "Guo-Zeyu" } ], "thesis_advisor": [ { "family_name": "Umans", "given_name": "Christopher M.", "orcid": "0000-0002-6390-9401", "clpid": "Umans-C-M" } ], "thesis_committee": [ { "family_name": "Umans", "given_name": "Christopher M.", "orcid": "0000-0002-6390-9401", "clpid": "Umans-C-M" }, { "family_name": "Schulman", "given_name": "Leonard J.", "orcid": "0000-0001-9901-2797", "clpid": "Schulman-L-J" }, { "family_name": "Vidick", "given_name": "Thomas G.", "orcid": "0000-0002-6405-365X", "clpid": "Vidick-T" }, { "family_name": "Huang", "given_name": "Ming-Deh", "clpid": "Huang-Ming-Deh" } ], "local_group": [ { "literal": "div_eng" } ], "abstract": "We introduce a family of mathematical objects called P-schemes, where P is a poset of subgroups of a finite group G. A P-scheme is a collection of partitions of the right coset spaces H\\G, indexed by H\u2208P, that satisfies a list of axioms. These objects generalize the classical notion of association schemes [BI84] as well as the notion of m-schemes [IKS09].
\r\n\r\nBased on P-schemes, we develop a unifying framework for the problem of deterministic factoring of univariate polynomials over finite field under the generalized Riemann hypothesis (GRH). More specifically, our results include the following:
\r\n\r\nWe show an equivalence between m-scheme as introduced in [IKS09] and P-schemes in the special setting that G is an multiply transitive permutation group and P is a poset of pointwise stabilizers, and therefore realize the theory of m-schemes as part of the richer theory of P-schemes.
\r\n\r\nWe give a generic deterministic algorithm that computes the factorization of the input polynomial \u0192(X) \u2208 Fq[X] given a \"lifted polynomial\" \u0192~(X) of \u0192(X) and a collection F of \"effectively constructible\" subfields of the splitting field of \u0192~(X) over a certain base field. It is routine to compute \u0192~(X) from \u0192(X) by lifting the coefficients of \u0192(X) to a number ring. The algorithm then successfully factorizes \u0192(X) under GRH in time polynomial in the size of \u0192~(X) and F, provided that a certain condition concerning P-schemes is satisfied, for P being the poset of subgroups of the Galois group G of \u0192~(X) defined by F via the Galois correspondence. By considering various choices of G, P and verifying the condition, we are able to derive the main results of known (GRH-based) deterministic factoring algorithms [Hua91a; Hua91b; Ron88; Ron92; Evd92; Evd94; IKS09] from our generic algorithm in a uniform way.
\r\n \r\nWe investigate the schemes conjecture in [IKS09] and formulate analogous conjectures associated with various families of permutation groups, each of which has applications on deterministic polynomial factoring. Using a technique called induction of P-schemes, we establish reductions among these conjectures and show that they form a hierarchy of relaxations of the original schemes conjecture.
\r\n\r\nWe connect the complexity of deterministic polynomial factoring with the complexity of the Galois group G of \u0192~(X). Specifically, using techniques from permutation group theory, we obtain a (GRH-based) deterministic factoring algorithm whose running time is bounded in terms of the noncyclic composition factors of G. In particular, this algorithm runs in polynomial time if G is in \u0393k for some k=2O(\u221a(log n), where \u0393k denotes the family of finite groups whose noncyclic composition factors are all isomorphic of subgroups of the symmetric group of degree k. Previously, polynomial-time algorithms for \u0393k were known only for bounded k.
\r\n\r\nWe discuss various aspects of the theory of P-schemes, including techniques of constructing new P-schemes from old ones, P-schemes for symmetric groups and linear groups, orbit P-schemes, etc. For the closely related theory of m-schemes, we provide explicit constructions of strongly antisymmetric homogeneous m-schemes for m\u22643. We also show that all antisymmetric homogeneous orbit 3-schemes have a matching for m\u22653, improving a result in [IKS09] that confirms the same statement for m\u22654.
\r\n\r\nIn summary, our framework reduces the algorithmic problem of deterministic polynomial factoring over finite fields to a combinatorial problem concerning P-schemes, allowing us to not only recover most of the known results but also discover new ones. We believe progress in understanding P-schemes associated with various families of permutation groups will shed some light on the ultimate goal of solving deterministic polynomial factoring over finite fields in polynomial time.
", "doi": "10.7907/Z94F1NSG", "publication_date": "2017", "thesis_type": "phd", "thesis_year": "2017" }, { "id": "thesis:9861", "collection": "thesis", "collection_id": "9861", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:06082016-032155301", "type": "thesis", "title": "Blackbox Reconstruction of Depth Three Circuits with Top Fan-In Two", "author": [ { "family_name": "Sinha", "given_name": "Gaurav", "orcid": "0000-0002-3590-9543", "clpid": "Sinha-Gaurav" } ], "thesis_advisor": [ { "family_name": "Rains", "given_name": "Eric M.", "orcid": "0000-0002-9915-0919", "clpid": "Rains-E-M" } ], "thesis_committee": [ { "family_name": "Rains", "given_name": "Eric M.", "orcid": "0000-0002-9915-0919", "clpid": "Rains-E-M" }, { "family_name": "Schulman", "given_name": "Leonard J.", "orcid": "0000-0001-9901-2797", "clpid": "Schulman-L-J" }, { "family_name": "Umans", "given_name": "Christopher M.", "orcid": "0000-0002-6390-9401", "clpid": "Umans-C-M" }, { "family_name": "Katz", "given_name": "Nets H.", "orcid": "0000-0002-6239-5429", "clpid": "Katz-N-H" } ], "local_group": [ { "literal": "div_pma" } ], "abstract": "Reconstruction of arithmetic circuits has been heavily studied in the past few years and has connections to proving lower bounds and deterministic identity testing. In\r\nthis thesis we present a polynomial time randomized algorithm for reconstructing \u03a3\u03a0\u03a3(2) circuits over characteristic zero fields F i.e. depth\u22123 circuits with fan-in 2 at the top addition gate and having coefficients from a field of characteristic zero.
\r\n\r\nThe algorithm needs only a black-box query access to the polynomial f \u2208 F[x1,...,xn] of degree d, computable by a \u03a3\u03a0\u03a3(2) circuit C. In addition, we assume that the\r\n\"simple rank\" of this polynomial (essential number of variables after removing the g.c.d. of the two multiplication gates) is bigger than a fixed constant. Our algorithm runs in time polynomial in n and d and with high probability returns an equivalent \u03a3\u03a0\u03a3(2) circuit.
\r\n\r\nThe problem of reconstructing \u03a3\u03a0\u03a3(2) circuits over finite fields was first proposed by Shpilka [27]. The generalization to \u03a3\u03a0\u03a3(k) circuits, k = O(1) (over finite\r\nfields) was addressed by Karnin and Shpilka in [18]. The techniques in these previous involve iterating over all objects of certain kinds over the ambient field and thus\r\nthe running time depends on the size of the field F. Their reconstruction algorithm uses lower bounds on the lengths of linear locally decodable codes with 2 queries.
\r\n\r\nIn our setting, such ideas immediately pose a problem and we need new techniques.
\r\n\r\nOur main techniques are based on the use of quantitative Sylvester Gallai theorems from the work of Barak et.al. [3] to find a small collection of \"nice\" subspaces to\r\nproject onto. The heart of this work lies in subtle applications of the quantitative Sylvester Gallai theorems to prove why projections w.r.t. the \"nice\" subspaces can\r\nbe \u201dglued\u201d. We also use Brill\u2019s equations from [9] to construct a small set of candidate linear forms (containing linear forms from both gates). Another important\r\ntechnique which comes very handy is the polynomial time randomized algorithm for factoring multivariate polynomials given by Kaltofen [17].
", "doi": "10.7907/Z92N507D", "publication_date": "2016", "thesis_type": "phd", "thesis_year": "2016" }, { "id": "thesis:8502", "collection": "thesis", "collection_id": "8502", "cite_using_url": "https://resolver.caltech.edu/CaltechTHESIS:06072014-074257676", "type": "thesis", "title": "Protein Structure Refinement Algorithms", "author": [ { "family_name": "Chitsaz", "given_name": "Mohsen", "clpid": "Chitsaz-Mohsen" } ], "thesis_advisor": [ { "family_name": "Mayo", "given_name": "Stephen L.", "orcid": "0000-0002-9785-5018", "clpid": "Mayo-S-L" } ], "thesis_committee": [ { "family_name": "Goddard", "given_name": "William A., III", "orcid": "0000-0003-0097-5716", "clpid": "Goddard-W-A-III" }, { "family_name": "Bjorkman", "given_name": "Pamela Jane", "orcid": "0000-0002-2277-3990", "clpid": "Bjorkman-P-J" }, { "family_name": "Umans", "given_name": "Christopher M.", "orcid": "0000-0002-6390-9401", "clpid": "Umans-C-M" }, { "family_name": "Mayo", "given_name": "Stephen L.", "orcid": "0000-0002-9785-5018", "clpid": "Mayo-S-L" } ], "local_group": [ { "literal": "div_bbe" } ], "abstract": "Protein structure prediction has remained a major challenge in structural biology for more than half a century. Accelerated and cost efficient sequencing technologies have allowed researchers to sequence new organisms and discover new protein sequences. Novel protein structure prediction technologies will allow researchers to study the structure of proteins and to determine their roles in the underlying biology processes and develop novel therapeutics.
\r\n\r\nDifficulty of the problem stems from two folds: (a) describing the energy landscape that corresponds to the protein structure, commonly referred to as force field problem; and (b) sampling of the energy landscape, trying to find the lowest energy configuration that is hypothesized to be the native state of the structure in solution. The two problems are interweaved and they have to be solved simultaneously. This thesis is composed of three major contributions. In the first chapter we describe a novel high-resolution protein structure refinement algorithm called GRID. In the second chapter we present REMCGRID, an algorithm for generation of low energy decoy sets. In the third chapter, we present a machine learning approach to ranking decoys by incorporating coarse-grain features of protein structures.
\r\n", "doi": "10.7907/7731-QM74", "publication_date": "2014", "thesis_type": "phd", "thesis_year": "2014" } ]