Complexity Theory

Computational Complexity Theory is the mathematical study of the intrinsic power and limitations of computational resources like time, space, or randomness. The current workshop focused on recent developments in various sub-areas including arithmetic complexity, Boolean complexity, communication complexity, cryptography, probabilistic proof systems, pseudorandomness and randomness extraction. Many of the developments are related to diverse mathematical fields such as algebraic geometry, combinatorial number theory, probability theory, representation theory, and the theory of error-correcting codes

Broadcasting on Random Directed Acyclic Graphs

We study a generalization of the well-known model of broadcasting on trees. Consider a directed acyclic graph (DAG) with a unique source vertex $X$, and suppose all other vertices have indegree $d\geq 2$. Let the vertices at distance $k$ from $X$ be called layer $k$. At layer $0$, $X$ is given a random bit. At layer $k\geq 1$, each vertex receives $d$ bits from its parents in layer $k-1$, which are transmitted along independent binary symmetric channel edges, and combines them using a $d$-ary Boolean processing function. The goal is to reconstruct $X$ with probability of error bounded away from $1/2$ using the values of all vertices at an arbitrarily deep layer. This question is closely related to models of reliable computation and storage, and information flow in biological networks. In this paper, we analyze randomly constructed DAGs, for which we show that broadcasting is only possible if the noise level is below a certain degree and function dependent critical threshold. For $d\geq 3$, and random DAGs with layer sizes $\Omega(\log k)$ and majority processing functions, we identify the critical threshold. For $d=2$, we establish a similar result for NAND processing functions. We also prove a partial converse for odd $d\geq 3$ illustrating that the identified thresholds are impossible to improve by selecting different processing functions if the decoder is restricted to using a single vertex. Finally, for any noise level, we construct explicit DAGs (using expander graphs) with bounded degree and layer sizes $\Theta(\log k)$ admitting reconstruction. In particular, we show that such DAGs can be generated in deterministic quasi-polynomial time or randomized polylogarithmic time in the depth. These results portray a doubly-exponential advantage for storing a bit in DAGs compared to trees, where $d=1$ but layer sizes must grow exponentially with depth in order to enable broadcasting.Comment: 33 pages, double column format. arXiv admin note: text overlap with arXiv:1803.0752

From the Quantum Approximate Optimization Algorithm to a Quantum Alternating Operator Ansatz

The next few years will be exciting as prototype universal quantum processors emerge, enabling implementation of a wider variety of algorithms. Of particular interest are quantum heuristics, which require experimentation on quantum hardware for their evaluation, and which have the potential to significantly expand the breadth of quantum computing applications. A leading candidate is Farhi et al.'s Quantum Approximate Optimization Algorithm, which alternates between applying a cost-function-based Hamiltonian and a mixing Hamiltonian. Here, we extend this framework to allow alternation between more general families of operators. The essence of this extension, the Quantum Alternating Operator Ansatz, is the consideration of general parametrized families of unitaries rather than only those corresponding to the time-evolution under a fixed local Hamiltonian for a time specified by the parameter. This ansatz supports the representation of a larger, and potentially more useful, set of states than the original formulation, with potential long-term impact on a broad array of application areas. For cases that call for mixing only within a desired subspace, refocusing on unitaries rather than Hamiltonians enables more efficiently implementable mixers than was possible in the original framework. Such mixers are particularly useful for optimization problems with hard constraints that must always be satisfied, defining a feasible subspace, and soft constraints whose violation we wish to minimize. More efficient implementation enables earlier experimental exploration of an alternating operator approach to a wide variety of approximate optimization, exact optimization, and sampling problems. Here, we introduce the Quantum Alternating Operator Ansatz, lay out design criteria for mixing operators, detail mappings for eight problems, and provide brief descriptions of mappings for diverse problems.Comment: 51 pages, 2 figures. Revised to match journal pape

Applications of Derandomization Theory in Coding

Randomized techniques play a fundamental role in theoretical computer science and discrete mathematics, in particular for the design of efficient algorithms and construction of combinatorial objects. The basic goal in derandomization theory is to eliminate or reduce the need for randomness in such randomized constructions. In this thesis, we explore some applications of the fundamental notions in derandomization theory to problems outside the core of theoretical computer science, and in particular, certain problems related to coding theory. First, we consider the wiretap channel problem which involves a communication system in which an intruder can eavesdrop a limited portion of the transmissions, and construct efficient and information-theoretically optimal communication protocols for this model. Then we consider the combinatorial group testing problem. In this classical problem, one aims to determine a set of defective items within a large population by asking a number of queries, where each query reveals whether a defective item is present within a specified group of items. We use randomness condensers to explicitly construct optimal, or nearly optimal, group testing schemes for a setting where the query outcomes can be highly unreliable, as well as the threshold model where a query returns positive if the number of defectives pass a certain threshold. Finally, we design ensembles of error-correcting codes that achieve the information-theoretic capacity of a large class of communication channels, and then use the obtained ensembles for construction of explicit capacity achieving codes. [This is a shortened version of the actual abstract in the thesis.]Comment: EPFL Phd Thesi

When could NISQ algorithms start to create value in discrete manufacturing ?

Are quantum advantages in discrete manufacturing achievable in the near term? As manufacturing-relevant NISQ algorithms, we identified Quantum Annealing (QA) and the Quantum Approximate Optimization Algorithm (QAOA) for combinatorial optimization as well as Derivative Quantum Circuits (DQC) for solving non-linear PDEs. While there is evidence for QAOA's outperformance, this requires post-NISQ circuit depths. In the case of QA, there is up to now no unquestionable evidence for advantage compared to classical computation. Yet different protocols could lead to finding such instances. Together with a well-chosen quantum feature map, DQC are a promising concept. Further investigations for higher dimensional problems and improvements in training could follow.Comment: 39 pages (thesis

On computationally efficient learning for stabilizers and beyond

Artificial intelligence, big data, machine learning, neural networks – look up any recent research proposal and with good probability at least one of these phrases will appear. It’s no secret that learning has taken this era of computer science by storm in our attempt to create software that perform extremely complicated tasks. As one of the most accurate models of our physical world currently known, it then makes sense to think about what kinds of quantum systems can or cannot be learned. As with many problems in quantum information and quantum computing, the simplest non-trivial versions of these problems start with the stabilizer formalism. In this dissertation, we examine learning problems centered around the stabilizer formalism in various different models from a theoretical standpoint using the tools of computer science and quantum information. Specifically, our focus will be on computational complexity, rather than sample complexity. We begin by looking at learning in the tomographical sense. Here, one has black-box access to copies of an unknown quantum state |ψ⟩ and want to learn properties of the state or outright given an approximation of |ψ⟩. In this setting, [Mon17] gave an efficient learning algorithm for stabilizer states. The key algorithmic tool was Bell difference sampling, which allows one to sample from the stabilizer group of a stabilizer state. [GNW21] extended the analysis of Bell difference sampling beyond just stabilizer states. Throughout Part I we turn to Bell difference sampling to improve upon learning algorithms for states with only a few (i.e., either O(log n) or strictly less than n depending on context) T gates. By using symplectic Fourier analysis, which is the generalization of Boolean Fourier analysis for a symplectic vector space over [superscript 2n over subscript 2], we derive powerful tools to understand the Bell difference sampling distribution. With these tools we first give a tolerant property testing algorithm for stabilizer states. That is, we give an algorithm that distinguishes whether a state is ε1 close to some stabilizer state or ε2 far from all stabilizer states for certain parameter regimes of ε1 and ε2. We use our improved knowledge of Bell difference sampling to improve upon the completeness and soundness analysis of the property tester given by [GNW21], which is not tolerant. A second application is stabilizer fidelity estimation and approximation. Given a state |ψ⟩ that is O(1) close to a stabilizer state, we output such a stabilizer state in time 2 [superscript O(n)]. This beats the previous 2 [superscript O(n²)] brute force search algorithm. Having such a stabilizer state also lets us figure out how close |ψ⟩ is to being stabilizer. A third application is extending Montanaro’s learning algorithm to the output of Clifford + O(log n) non-Clifford gate circuits. More generally, our algorithm interpolates between Montanaro’s algorithm and pure state tomography algorithms with runtime that is poly(n)∗exp(t) where t is the number of non-Clifford gates. This asymptotically matches the runtime of classical simulation algorithms for such circuits. A key algorithmic step in this work is the ability to “compress” the “stabilizer-ness” of a state onto a few qubits, allowing the “non-stabilizer-ness” to be brute-forced on the remaining qubits. Our final application is pseudorandomness lower bounds. Introduced by [JLS18], a pseudorandom quantum state ensemble is a set of quantum states that are computationally indistinguishable from Haar random. By re-purposing algorithms from above, we produce a test that behaves differently when given a state produced by less than n T gates in a Clifford + T circuit versus being given a Haar random state. We note that this is tight assuming the existence of linear-time quantum-secure One-Way Functions. Pivoting now, we also study the stabilizer formalism in the PAC learning framework proposed by [Val84]. Here one does not have control over the measurements, but must make do regardless (within information theoretic limits). We analyze the problem in two ways. First we show that, unlike stabilizer states, learning the associated Clifford unitaries in the proper PAC model is NP-hard. This is done by a reduction to the problem of finding a full rank matrix in an affine subspace of matrices over ₂. The second is studying stabilizer states in the presence of noise. We utilize the Statistical Query framework, a popular modification to the PAC learning framework that is inherently tolerant to noise. There, we also show hardness in this framework by a reduction to Learning Parities with Noise. This gives evidence that even in the PAC model stabilizer states are hard to learn with noise.Computer Science