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1. Depth-First Search in Directed Graphs, Revisited

   Allender, Eric; Chauhan, Archit; Datta, Samir (January 2020, Electronic colloquium on computational complexity)

   We present an algorithm for constructing a depth-first search tree in planar digraphs; the algorithm can be implemented in the complexity class UL, which is contained in nondeterministic logspace NL, which in turn lies in NC^2. Pior to this (for more than a quarter-century), the fastest uniform deterministic parallel algorithm for this problem was O(log^10 n) (corresponding to the complexity class AC^10 ⊆ NC^11). We also consider the problem of computing depth-first search trees in other classes of graphs, and obtain additional new upper bounds. 

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2. Exponential separation between shallow quantum circuits and unbounded fan-in shallow classical circuits

   https://doi.org/10.1145/3313276.3316404  

   Watts, Adam Bene; Kothari, Robin; Schaeffer, Luke; Tal, Avishay (June 2019, Proceedings of the 51st Annual ACM SIGACT Symposium on Theory of Computing)

   Recently, Bravyi, Gosset, and Konig (Science, 2018) exhibited a search problem called the 2D Hidden Linear Function (2D HLF) problem that can be solved exactly by a constant-depth quantum circuit using bounded fan-in gates (or QNC^0 circuits), but cannot be solved by any constant-depth classical circuit using bounded fan-in AND, OR, and NOT gates (or NC^0 circuits). In other words, they exhibited a search problem in QNC^0 that is not in NC^0. We strengthen their result by proving that the 2D HLF problem is not contained in AC^0, the class of classical, polynomial-size, constant-depth circuits over the gate set of unbounded fan-in AND and OR gates, and NOT gates. We also supplement this worst-case lower bound with an average-case result: There exists a simple distribution under which any AC^0 circuit (even of nearly exponential size) has exponentially small correlation with the 2D HLF problem. Our results are shown by constructing a new problem in QNC^0, which we call the Parity Halving Problem, which is easier to work with. We prove our AC^0 lower bounds for this problem, and then show that it reduces to the 2D HLF problem. 

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3. A global algorithm for AC optimal power flow based on successive linear conic optimization

   https://doi.org/10.1109/PESGM.2017.8273957  

   Barati, Masoud; Kargarian, Amin (July 2017, 2017 IEEE Power & Energy Society General Meeting)

   Newly, there has been significant research interest in the exact solution of the AC optimal power flow (AC-OPF) problem. A semideflnite relaxation solves many OPF problems globally. However, the real problem exists in which the semidefinite relaxation fails to yield the global solution. The appropriation of relaxation for AC-OPF depends on the success or unfulflllment of the SDP relaxation. This paper demonstrates a quadratic AC-OPF problem with a single negative eigenvalue in objective function subject to linear and conic constraints. The proposed solution method for AC-OPF model covers the classical AC economic dispatch problem that is known to be NP-hard. In this paper, by combining successive linear conic optimization (SLCO), convex relaxation and line search technique, we present a global algorithm for AC-OPF which can locate a globally optimal solution to the underlying AC-OPF within given tolerance of global optimum solution via solving linear conic optimization problems. The proposed algorithm is examined on modified IEEE 6-bus test system. The promising numerical results are described. 

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4. Quantum algorithms and approximating polynomials for composed functions with shared inputs

   https://doi.org/10.22331/q-2021-09-16-543  

   Bun, Mark; Kothari, Robin; Thaler, Justin (September 2021, Quantum)

   We give new quantum algorithms for evaluating composed functions whose inputs may be shared between bottom-level gates. Let f be an m -bit Boolean function and consider an n -bit function F obtained by applying f to conjunctions of possibly overlapping subsets of n variables. If f has quantum query complexity Q ( f ) , we give an algorithm for evaluating F using O ~ ( Q ( f ) ⋅ n ) quantum queries. This improves on the bound of O ( Q ( f ) ⋅ n ) that follows by treating each conjunction independently, and our bound is tight for worst-case choices of f . Using completely different techniques, we prove a similar tight composition theorem for the approximate degree of f .By recursively applying our composition theorems, we obtain a nearly optimal O ~ ( n 1 − 2 − d ) upper bound on the quantum query complexity and approximate degree of linear-size depth- d AC 0 circuits. As a consequence, such circuits can be PAC learned in subexponential time, even in the challenging agnostic setting. Prior to our work, a subexponential-time algorithm was not known even for linear-size depth-3 AC 0 circuits.As an additional consequence, we show that AC 0 ∘ ⊕ circuits of depth d + 1 require size Ω ~ ( n 1 / ( 1 − 2 − d ) ) ≥ ω ( n 1 + 2 − d ) to compute the Inner Product function even on average. The previous best size lower bound was Ω ( n 1 + 4 − ( d + 1 ) ) and only held in the worst case (Cheraghchi et al., JCSS 2018). 

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5. Shallow-Depth Quantum Circuit for an Unstructured Database Search

   https://doi.org/10.3390/quantum6040037  

   Zhan, Junpeng (October 2024, Quantum Reports)

   Grover’s search algorithm (GSA) offers quadratic speedup in searching unstructured databases but suffers from exponential circuit depth complexity. Here, we present two quantum circuits called HX and Ry layers for the searching problem. Remarkably, both circuits maintain a fixed circuit depth of two and one, respectively, irrespective of the number of qubits used. When the target element’s position index is known, we prove that either circuit, combined with a single multi-controlled X gate, effectively amplifies the target element’s probability to over 0.99 for any qubit number greater than seven. To search unknown databases, we use the depth-1 Ry layer as the ansatz in the Variational Quantum Search (VQS), whose efficacy is validated through numerical experiments on databases with up to 26 qubits. The VQS with the Ry layer exhibits an exponential advantage, in circuit depth, over the GSA for databases of up to 26 qubits. 

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