Applications of Quantum Variational Methods in Research

⚛️ A Review of Variational Quantum Algorithms: Insights into Fault-Tolerant Quantum Computing 📜 Variational quantum algorithms (VQAs) have established themselves as a central computational paradigm in the Noisy Intermediate-Scale Quantum (NISQ) era. By coupling parameterized quantum circuits (PQCs) with classical optimization, they operate effectively under strict hardware limitations. However, as quantum architectures transition toward early fault-tolerant (EFT) and ultimate fault-tolerant (FT) regimes, the foundational principles and long-term viability of VQAs require systematic reassessment. This review offers an insightful analysis of VQAs and their progression toward the fault-tolerant regime. We deconstruct the core algorithmic framework by examining ansatz design and classical optimization strategies, including cost function formulation, gradient computation, and optimizer selection. Concurrently, we evaluate critical training bottlenecks, notably barren plateaus (BPs), alongside established mitigation strategies. The discussion then explores the EFT phase, detailing how the integration of quantum error mitigation and partial error correction can sustain algorithmic performance. Addressing the FT phase, we analyze the inherent challenges confronting current hybrid VQA models. Furthermore, we synthesize recent VQA applications across diverse domains, including many-body physics, quantum chemistry, machine learning, and mathematical optimization. Ultimately, this review outlines a theoretical roadmap for adapting quantum algorithms to future hardware generations, elucidating how variational principles can be systematically refined to maintain their relevance and efficiency within an error-corrected computational environment. ℹ️ Zhirao Wang et al - 2026

Recently the team published a paper in Nature Computational Science in collaboration with researchers from Los Alamos National Lab and the University of Basel. The paper was on provable bounds for noise-free expectation values computed from noisy samples. This calibration started in the optimization working group. The paper discusses how the “Layer Fidelity” or how effective two qubit error as measured by the “Error Per Layered Gate” can be used to quantify the impact of hardware noise on sampling-based quantum (optimization) algorithms. Each one of our devices reports this number in the resource tab of the IBM Quantum Platform (https://lnkd.in/eRd2yKwB). The paper allows you to estimate the number of additional shots required to compensate for the impact of noise. It turns out that by using this method it is much cheaper than mitigating the noise when requiring unbiased estimators of expectation values (sqrt(gamma) vs gamma^2). These insights allowed us to prove that the Conditional Value at Risk (CvaR) – an alternative loss function suggested in 2019 and widely used to train variational algorithms, borrowed from mathematical finance – leads to provable bounds on expectation values using only noisy samples. The theoretical insights have been demonstrated on two use cases using up to 127 qubits: estimation of state fidelity (as required, e.g. to evaluate quantum kernels) and optimization (QAOA). In both cases, the team see a good agreement between the theory and experiment. Read the paper here https://lnkd.in/ehyz4GCJ

Much-lamented limitation of quantum optimization research is the near-exclusive focus on binary problems that map naturally to qubits. We show that extending the analysis techniques to integer problems may unlock new opportunities for beating classical solvers. In a new paper, we extend the path-integral analysis of introduced for the analysis of QAOA on qubits by Farhi, Goldstone, Gutman, and Zhou [arXiv:1910.08187] to qudits. This gives an iterative formula for computing QAOA energy for a broad range of integer problems on graphs of any size. Evaluating this formula for Max-k-Cut on large graphs, we see that QAOA with p ≤ 4 beats the best classical solver with provable guarantees (SDP-based Frieze-Jerrum algorithm). Not content with easy wins, we introduce a better classical heuristic, improving classical state-of-the-art for Max-k-Cut. We predict that to beat this new heuristic, QAOA depth of p ≤ 20 is sufficient, though studying QAOA at such high depth is difficult with our techniques. While we do not beat all classical solvers, our results show clear promise of QAOA applied to integer problems. Paper: https://lnkd.in/e9wfJPER Congratulations to Anuj Apte, Sami Boulebnane, Yuwei Jin, Sivaprasad Omanakuttan, and Michael A Perlin on this great result.

Our paper "Equivariant Quantum Approximate Optimization Algorithm" is finally out in IEEE Transactions on Quantum Engineering! Many combinatorial optimization problems have natural symmetries, but standard QAOA mixer doesn't take advantage of them. We introduce new symmetry aware mixers that are tailored to the structure of the problem and can be efficiently implemented as quantum circuits. Through numerical experiments on the coloring and partitioning problems on graphs, we demonstrate that these mixers outperform the standard approach. We also explain why warm start QAOA often fails to improve results, identifying a fundamental limitation that prevents guaranteed convergence. Congratulations and huge thanks to our co-authors Boris Tsvelikhovskiy and Yuri Alexeev! Check our paper at https://lnkd.in/e3xAAPNW #QuantumComputing #QAOA #QuantumOptimization #IEEETQE #QuantumEngineering #HybridAlgorithms #Research #VariationalQuantumAlgorithms

#variational #QML #strikesback #discretely In the recent work (#link in the first comment) we show that adaptivity in variational quantum algorithms can give an exponential advantage over non-adaptive approaches. By framing circuit recompilation as a discrete optimization problem, we identify a setting where hybrid quantum–classical loops are not just helpful, but necessary. Big thanks to my collaborators — 𝗭𝗼ë 𝗛𝗼𝗹𝗺𝗲𝘀, for super-energising discussions and pushing this through, and Chukwudubem Umeano, for helping to compile this piece together. Careful, long read below ❗️ 𝗔𝗱𝘃𝗮𝗻𝘁𝗮𝗴𝗲 𝗳𝗼𝗿 𝗗𝗶𝘀𝗰𝗿𝗲𝘁𝗲 𝗩𝗮𝗿𝗶𝗮𝘁𝗶𝗼𝗻𝗮𝗹 𝗤𝘂𝗮𝗻𝘁𝘂𝗺 𝗔𝗹𝗴𝗼𝗿𝗶𝘁𝗵𝗺𝘀 𝗶𝗻 𝗖𝗶𝗿𝗰𝘂𝗶𝘁 𝗥𝗲𝗰𝗼𝗺𝗽𝗶𝗹𝗮𝘁𝗶𝗼𝗻 The question of whether variational quantum algorithms really need the hybrid quantum–classical loop has been a matter of debate in the last two years. The challenge comes from the ever-growing power of surrogating quantum models, and advances in classical simulation methods for problems with structure. This has been framed as a question to the community: 𝘋𝘰𝘦𝘴 𝘵𝘩𝘦 𝘱𝘳𝘰𝘷𝘢𝘣𝘭𝘦 𝘢𝘣𝘴𝘦𝘯𝘤𝘦 𝘰𝘧 𝘣𝘢𝘳𝘳𝘦𝘯 𝘱𝘭𝘢𝘵𝘦𝘢𝘶𝘴 𝘪𝘮𝘱𝘭𝘺 𝘴𝘪𝘮𝘶𝘭𝘢𝘣𝘪𝘭𝘪𝘵𝘺? Counterexamples exist, but most of them correspond to specific Boolean function learning or Shor-type algorithms in disguise. What we tried to do is to identify a setting where adaptivity appears more naturally, and relies on a different structure. The starting point is a simple observation: in classical discrete optimization, exponential separations between adaptive and non-adaptive search are known. If this logic carries over, then there should be quantum problems where adaptivity gives the same kind of exponential edge. In our preprint, we propose such a problem — quantum circuit recompilation. The task is to recover a “hidden” circuit structure, starting with quantum data (a state) prepared with randomly placed T-gates (“puzzles”) embedded in partially random unitaries. The ansatz has the reverse structure, but we don’t know where the gates are, and there are exponentially many options. We show that discrete quantum optimizers can efficiently unravel these puzzles, finding the right locations with a quadratic number of circuit evaluations. In contrast, non-adaptive searches face an exponentially hard landscape: non-separable, highly entangled, and full of magic. Importantly, we also find that optimization remains feasible under noise, using finite-shot samplers, even as the solution space grows exponentially. This points to a very particular (and largely overlooked) role for discrete optimization in variational quantum algorithms. It also raises a bigger question: what new problems can benefit from this kind of adaptivity? While we remain open to new discoveries, we already see clear connections to fault-tolerant protocols with partial structure, and to blind quantum computing scenarios. Grateful for the support from EPSRC and the QCi3 Hub!