Understanding Simon's Algorithm in Quantum Computation

Quantum gates: algebra on paper, rotations in your head. Quantum gates are the basic building blocks of quantum algorithms). They are best understood two ways: algebraically (matrices acting on amplitudes) and geometrically (rotations on the Bloch sphere). I made two illustrations to show that connection for the Pauli-X and Hadamard gates: X gate swaps amplitudes: X(α∣0⟩+β∣1⟩)=α∣1⟩+β∣0⟩. Geometrically it’s a 180° rotation about the x-axis on the Bloch sphere. H (Hadamard) maps Z ↔ X: it sends ∣0⟩→∣+⟩ and ∣1⟩→∣−⟩. Geometrically H is also a 180° rotation, but about the axis halfway between x and z (the unit vector (1,0,1)/√2. Short notice: Any single-qubit gate (up to an irrelevant global phase) is fully specified by an axis (a line through the centre of the Bloch sphere) and a rotation angle. Mathematically, single-qubit gates live in SU(2) group, so every single-qubit gate corresponds to a rotation (up to global phase). If you’re beginning with quantum computing, think of single-qubit gates as rotations makes it much easier to predict what happens to states and to reason about sequences of gates. Good to know: For a matrix to represent a valid quantum gate, it must be unitary (see the bottom-right corner of the illustration). #QuantumComputing #BlochSphere #QuantumGates #avenue78

🚀 Quantum Advantage Just Got More Practical One of the biggest criticisms of quantum machine learning has always been: How do you load massive classical datasets into a quantum computer efficiently? If data input and readout are too expensive, any theoretical speedup can disappear. A new result from https://lnkd.in/gn9ZuGQM suggests a major step forward. Instead of storing the full dataset in quantum memory, the data is processed on the fly through a sequence of carefully designed quantum operations — then immediately discarded. ⚡ This means: • No need to keep the full dataset stored in memory • Sampling cost scales much better than many expected bottlenecks • Potential to process exponentially larger datasets than classical methods in some settings. Why does this matter? Many promising quantum machine learning applications rely on solving large linear systems. That is exactly where Quantum Linear System Algorithms (QLSAs) come in. These new results explicitly highlight our optimal QLSAs https://lnkd.in/gFWKNzab as one of the strongest near-term uses of efficient classical-data loading. 📌 In other words: one of the main objections against practical quantum advantage may be weakening. Exciting times ahead for quantum algorithms, machine learning, and large-scale scientific computing. #QuantumComputing #QuantumAI #MachineLearning #QuantumMachineLearning #QuantumAlgorithms #QLSA #ArtificialIntelligence #FutureTech #Science #Innovation

> Sharing Resource < Great one: "Quantum computation at the edge of chaos" by Tomohiro Hashizume, Zhengjun Wang, Frank Schlawin, Dieter Jaksch Abstract: A key challenge in classical machine learning is to mitigate overparameterization by selecting sparse solutions. We translate this concept to the quantum domain, introducing quantum sparsity as a principle based on minimizing quantum information shared across multiple parties. This allows us to address fundamental issues in quantum data processing and convergence issues such as the barren plateau problem in Variational Quantum Algorithm (VQA). We propose a practical implementation of this principle using the topological Entanglement Entropy (TEE) as a cost function regularizer. A non-negative TEE is associated with states with a sparse structure in a suitable basis, while a negative TEE signals untrainable chaos. The regularizer, therefore, guides the optimization along the critical edge of chaos that separates these regimes. We link the TEE to structural complexity by analyzing quantum states encoding functions of tunable smoothness, deriving a quantum Nyquist-Shannon sampling theorem that bounds the resource requirements and error propagation in VQA. Numerically, our TEE regularizer demonstrates significantly improved convergence and precision for complex data encoding and ground-state search tasks. This work establishes quantum sparsity as a design principle for robust and efficient VQAs. Link: https://lnkd.in/eS_gVZhY #quantummachinelearning #quantumcomputing #research #paper

At a deeper level, real-life procedures, for example, modeling of biological or chemical systems are not about 0 or 1, strictly. The above numbers are states that a classical computer uses to perform computational approaches in order to provide some results on a basic level or to apply findings in a real-world problem. Of course it is known that, matter or even living organisms, at a fundamental level, are composed of smaller units, atoms and molecules. These units exist and interact with each other and forge the nature around us -when it is possible or acceptable, depending on some properties or laws- by electrons. Electrons are described by Quantum Mechanics. The world of tiny particles or waves. Therefore, electrons fall under the category of dualism, a fundamental characteristic of Quantum Mechanics, and not the only. Quantum Mechanics follows a probabilistic model. The intersection of Quantum Mechanics and Informatics results in Quantum Computing. Quantum Computing provides more realistic results about the mentioned procedures, since computations take place based on the nature of the electrons, using Qubits. Isn't it fascinating!? The basic equation that represents a Quantum state is the Dirac Notation: • ∣ψ⟩ = α∣0⟩ + β∣1⟩ where ∣ψ⟩ is the quantum vector, ∣0⟩ and ∣1⟩ are the states, α and β are the complex probability amplitudes of being on that state, and ∣α∣^2 or ∣β∣^2 are the probabilities of measuring the system into ∣0⟩ or ∣1⟩. So, the system is in superposition and it collapses upon the measurement, providing the information on that point.

The April 6th edition of the Hilbert Space Post covers optimal inclusion of constraints in quantum optimization, quantum matrix multiplication, and parameter-free quantum optimizer. Here is the daily selection: 1️⃣ Scalable Determination of Penalization Weights for Constrained Optimizations on Approximate Solvers 🔗 https://lnkd.in/eZ_DCJTx 👨👩 Edoardo A., Fritz Schinkel, Stefan Walter, Martin K., Leandro Aolita, Ingo Roth 🔬 A new work tackles a key challenge in QUBO formulations: how to properly weight constraint penalties. The authors introduced a fast, principled method to choose these weights. The approach improves robustness and significantly speeds up optimization. 2️⃣ AQ-Stacker: An Adaptive Quantum Matrix Multiplication Algorithm with Scaling via Parallel Hadamard Stacking 🔗 https://lnkd.in/edHrGNr7 👨👩 Wladimir Silva 🔬 A new paper proposes a hybrid quantum–classical approach to matrix multiplication that uses Hadamard tests and QRAM to compute inner products more efficiently. It aims to improve scalability for large linear algebra tasks. 3️⃣RFOX (Rotated-Field Oscillatory eXchange) quantum algorithm: Towards Parameter-Free Quantum Optimizers 🔗 https://lnkd.in/eEYP8kMx 👨👩 Brian García Sarmina, Guo-Hua Sun and Shi-Hai Dong 🔬 RFOX is a parameter-free quantum optimization algorithm that combines non-stoquastic interactions with counter-diabatic terms to keep the energy gap nearly constant during evolution. As a result, it avoids the small-gap bottlenecks that limit traditional methods. That’s it for the daily selection. If you enjoyed it, please consider giving me a like or reposting to support our content. Thanks!

Quantum Algorithms Now Solve Complex Equations with Fewer Calculations Lower error rates in quantum linear system solvers have historically demanded increased computational complexity. New algorithms, however, achieve up to an order of magnitude improvement in accuracy compared to standard methods at a comparable polynomial degree, particularly when noise is present. Constrained optimal polynomials offer a framework inspired by classical numerical techniques to address this longstanding limitation. #quantum #quantumcomputing #technology https://lnkd.in/e7jhYsQg

Mixed States Algorithm for Quantum Matrix Integrals and Lyapunov Equations Quantinuum develops probabilistic quantum algorithm encoding functions into mixed states to solve Lyapunov equations and matrix inversions—an alternative to block encodings offering potential computational advantages. #QuantumAlgorithms #QuantumComputing #News #Informaq

🧵 New preprint out! "Obstructions to universality in globally controlled qubit graphs" In globally controlled architectures, subset of qubits are driven by the same field — no individual addressing. The question is: which interaction graphs allow universal quantum computation under these constraints? Hu et al. [arXiv:2508.19075] conjectured that universality holds if and only if a control term breaks all graph automorphisms. We show this is false. We construct explicit 7- and 9-qubit counterexamples: graphs with trivial automorphism group that are provably non-universal. The obstruction comes from hidden symmetries — operators that commute with the full Hamiltonian but have nothing to do with the graph structure. We also show that breaking individual graph symmetries is not enough: linear combinations of them can survive, and that's sufficient to prevent universality. The takeaway: graph automorphisms capture only part of the relevant symmetry structure. The correct criterion for universality involves adjoint symmetries — a stronger and harder-to-check condition. 👉 Pre-print: https://lnkd.in/dDbmiyHf — with Roberto Gargiulo, Vittorio Giovannetti and Robert Zeier Open questions remain on how to characterize these hidden symmetries systematically, and whether graphs relevant for near-term hardware fall into the universal or non-universal class. 🚨 If you work on quantum control, globally driven architectures, or Lie algebraic methods and find these questions interesting — we’d love to hear from you.

🔒 The Eras of Computation - And Why It Matters for Cryptographic Governance We talk about "classical vs quantum" as if that is the whole story. It is not. Computation has at least two distinct eras. We are in the second. The question nobody is asking is whether a third even exists. 𝐄𝐫𝐚 1: 𝐂𝐥𝐚𝐬𝐬𝐢𝐜𝐚𝐥 Deterministic machines built on stable bits. Silicon, CMOS, Von Neumann. Everything predictable, everything repeatable. Every cryptographic system in production today was designed for this world. 𝐄𝐫𝐚 2: 𝐂𝐥𝐚𝐬𝐬𝐢𝐜𝐚𝐥 + 𝐐𝐮𝐚𝐧𝐭𝐮𝐦 (𝐭𝐨𝐝𝐚𝐲) Quantum is not a computer. It is an accelerator strapped to a classical control stack. Every qubit pulse, every timing sequence, every error correction cycle, all classical. We call it "quantum computing", but the reality is: Classical → Quantum → Classical A sandwich, not a replacement. The classical layer is not temporary scaffolding. Under current physics, it is structural. Measurement collapses state. No-cloning forbids copying. Decoherence destroys persistence. Probabilistic evolution breaks determinism. Quantum cannot supervise itself. Classical is not going anywhere. This has direct consequences for governance. If your cryptographic transition plan assumes quantum will eventually "replace" classical, you are planning for a world that does not exist. Migration planning, lifecycle management and architectural dependency mapping all start from this fact. 𝐈𝐬 𝐭𝐡𝐞𝐫𝐞 𝐚𝐧 𝐄𝐫𝐚 3? A PhD quantum mathematician recently put a question to the SITG-Consulting team that we have not been able to dismiss. If computation moved from purely classical to classical-plus-quantum, is there a theoretical threshold where the classical control layer dissolves entirely? Or where something beyond both classical and quantum emerges? Current physics says no. Measurement, cloning, decoherence and probabilistic evolution are hard constraints. The classical layer is load-bearing. But 30 years ago, fault-tolerant quantum error correction was considered implausible. The constraints were real. The assumption that they were permanent was not. None of this changes the immediate operational priority. Migrate. Govern your cryptographic lifecycle. Treat PQC as an engineering problem, not a theoretical one. But the question stands. If you are working at the boundary of quantum information theory, we would welcome the conversation. #Quantum #PQC #Qubits #Classical Brian C.

I am excited that my first paper in quantum optimization entitled "Quantum Bridge Analytics: A QUBO Approach to the Uncapacitated Facility Location Problem", co-authored with Rick Hennig, Gary Kochenberger and Fred Glover, has been published in Annals of Operations Research: https://lnkd.in/gj9wTezy We apply quantum bridge analytics (QBA) and quadratic unconstrained binary optimization (QUBO) approach for the well-known uncapacitated facility location problem (UFLP) in supply chain network design, using Gurobi’s QUBO solver, D-Wave’s hybrid classical-quantum solver, and the proprietary next generation quantum (NGQ) solver. A comprehensive computational study is performed on the UFLP benchmark instances with comparisons to four recent algorithms for UFLP in the literature. The QBA/QUBO approach achieved high-quality results for problems of all sizes, solving all the small-size and some medium-size instances with up to 10,000 decision variables to optimality in less than 20 s, and achieving less than 3% optimality gap for large instances with more than 250,000 decision variables in less than 10 min on average. Computational results also show that in addition to problem size, the polyhedral properties of a UFLP instance may significantly impact the efficiency of quantum annealing based algorithm.