Optimizing the Number of Gates in Quantum Search

$$In its usual form, Grover's quantum search algorithm uses $O(\sqrt{N})$ queries and $O(\sqrt{N} \log N)$ other elementary gates to find a solution in an $N$-bit database. Grover in 2002 showed how to reduce the number of other gates to $O(\sqrt{N}\log\log N)$ for the special case where the database has a unique solution, without significantly increasing the number of queries. We show how to reduce this further to $O(\sqrt{N}\log^{(r)} N)$ gates for any constant $r$, and sufficiently large $N$. This means that, on average, the gates between two queries barely touch more than a constant number of the $\log N$ qubits on which the algorithm acts. For a very large $N$ that is a power of 2, we can choose $r$ such that the algorithm uses essentially the minimal number $\frac{\pi}{4}\sqrt{N}$ of queries, and only $O(\sqrt{N}\log(\log^{\star} N))$ other gates.Comment: 11 pages LaTeX. Version 2: small improvements in the proof

QuantumSEA: In-Time Sparse Exploration for Noise Adaptive Quantum Circuits

Parameterized Quantum Circuits (PQC) have obtained increasing popularity thanks to their great potential for near-term Noisy Intermediate-Scale Quantum (NISQ) computers. Achieving quantum advantages usually requires a large number of qubits and quantum circuits with enough capacity. However, limited coherence time and massive quantum noises severely constrain the size of quantum circuits that can be executed reliably on real machines. To address these two pain points, we propose QuantumSEA, an in-time sparse exploration for noise-adaptive quantum circuits, aiming to achieve two key objectives: (1) implicit circuits capacity during training - by dynamically exploring the circuit's sparse connectivity and sticking a fixed small number of quantum gates throughout the training which satisfies the coherence time and enjoy light noises, enabling feasible executions on real quantum devices; (2) noise robustness - by jointly optimizing the topology and parameters of quantum circuits under real device noise models. In each update step of sparsity, we leverage the moving average of historical gradients to grow necessary gates and utilize salience-based pruning to eliminate insignificant gates. Extensive experiments are conducted with 7 Quantum Machine Learning (QML) and Variational Quantum Eigensolver (VQE) benchmarks on 6 simulated or real quantum computers, where QuantumSEA consistently surpasses noise-aware search, human-designed, and randomly generated quantum circuit baselines by a clear performance margin. For example, even in the most challenging on-chip training regime, our method establishes state-of-the-art results with only half the number of quantum gates and ~2x time saving of circuit executions. Codes are available at https://github.com/VITA-Group/QuantumSEA.Comment: IEEE International Conference on Quantum Computing and Engineering (QCE 2023

Hybrid Optimization Schemes for Quantum Control

Optimal control theory is a powerful tool for solving control problems in quantum mechanics, ranging from the control of chemical reactions to the implementation of gates in a quantum computer. Gradient-based optimization methods are able to find high fidelity controls, but require considerable numerical effort and often yield highly complex solutions. We propose here to employ a two-stage optimization scheme to significantly speed up convergence and achieve simpler controls. The control is initially parametrized using only a few free parameters, such that optimization in this pruned search space can be performed with a simplex method. The result, considered now simply as an arbitrary function on a time grid, is the starting point for further optimization with a gradient-based method that can quickly converge to high fidelities. We illustrate the success of this hybrid technique by optimizing a holonomic phasegate for two superconducting transmon qubits coupled with a shared transmission line resonator, showing that a combination of Nelder-Mead simplex and Krotov's method yields considerably better results than either one of the two methods alone.Comment: 17 pages, 5 figures, 2 table

Full-Stack, Real-System Quantum Computer Studies: Architectural Comparisons and Design Insights

In recent years, Quantum Computing (QC) has progressed to the point where small working prototypes are available for use. Termed Noisy Intermediate-Scale Quantum (NISQ) computers, these prototypes are too small for large benchmarks or even for Quantum Error Correction, but they do have sufficient resources to run small benchmarks, particularly if compiled with optimizations to make use of scarce qubits and limited operation counts and coherence times. QC has not yet, however, settled on a particular preferred device implementation technology, and indeed different NISQ prototypes implement qubits with very different physical approaches and therefore widely-varying device and machine characteristics. Our work performs a full-stack, benchmark-driven hardware-software analysis of QC systems. We evaluate QC architectural possibilities, software-visible gates, and software optimizations to tackle fundamental design questions about gate set choices, communication topology, the factors affecting benchmark performance and compiler optimizations. In order to answer key cross-technology and cross-platform design questions, our work has built the first top-to-bottom toolflow to target different qubit device technologies, including superconducting and trapped ion qubits which are the current QC front-runners. We use our toolflow, TriQ, to conduct {\em real-system} measurements on 7 running QC prototypes from 3 different groups, IBM, Rigetti, and University of Maryland. From these real-system experiences at QC's hardware-software interface, we make observations about native and software-visible gates for different QC technologies, communication topologies, and the value of noise-aware compilation even on lower-noise platforms. This is the largest cross-platform real-system QC study performed thus far; its results have the potential to inform both QC device and compiler design going forward.Comment: Preprint of a publication in ISCA 201