Understanding Quantum Complexity in Real-World Physics

We are pursuing quantum computing because there’s evidence that quantum can solve certain problems exponentially faster than any classical computer. I’m excited to share a new algorithm from our team with the potential for an exponential speedup in a real-world use case: simulating electric circuits. Circuits built from resistors, inductors, and capacitors — RLC circuits — show up across engineering, from power grids to analog filters to integrated circuit design. Predicting how voltages and currents evolve in these systems is routine. But as circuits grow large and complex, those simulations can become increasingly expensive on classical hardware. What makes RLC circuits so challenging to simulate is that they aren’t described by ordinary differential equations (ODEs), but by differential-algebraic equations (DAEs): systems that combine equations describing time evolution with constraints that must be satisfied at every instant. In the case of RLC circuits, we must solve Kirchhoff’s laws of charge and voltage conservation at every junction, but standard ODE solvers struggle to handle this mixed structure. A new paper authored by Arkopal Dutt, Anirban Chowdhury, Kristan Temme, and Hari Krovi, presents the first quantum algorithm tailored to DAEs of this kind. The approach separates the circuit’s state into two parts: one that evolves dynamically over time, and another that is fixed by the constraints. Each part is then handled with the appropriate technique. The result is an algorithm that prepares a quantum state encoding the circuit’s full time evolution, with a runtime that scales only polylogarithmically in the number of nodes — an exponential improvement over the polynomial worst-case scaling of classical methods. This speedup applies to well-conditioned networks where the circuit can be queried in superposition, meaning its structure is accessed as a function that returns entries on demand, rather than being read out element by element. From the quantum computer’s output state (the state encoding the full solution), physically meaningful quantities, like the energy stored in a set of capacitors or dissipated across a set of resistors, can be extracted directly. Interestingly, the authors also show that this energy-estimation task is as powerful as quantum computation itself: a quantum computer can solve it efficiently, and any problem that admits an efficient quantum solution can be reduced to an instance of it. In complexity-theoretic terms, this implies that, under standard assumptions, no classical algorithm can match a quantum computer on this task. Classical circuit simulation has been a workhorse of electronic design for decades. Demonstrating a provable quantum advantage on a problem this practical is an exciting step, and it lines up closely with IBM Quantum’s broader goal of identifying where quantum computing can deliver real value in engineering and industrial settings.   Full paper: https://lnkd.in/ekTFap64

The Schrödinger Equation Gets Practical: Quantum Algorithm Speeds Up Real-World Simulations Quantum computing has taken a major leap forward with a new algorithm designed to simulate coupled harmonic oscillators, systems that model everything from molecular vibrations to bridges and neural networks. By reformulating the dynamics of these oscillators into the Schrödinger equation and applying Hamiltonian simulation methods, researchers have shown that complex physical systems can be simulated exponentially faster on a quantum computer than with traditional algorithms. This breakthrough demonstrates not only a practical use of the Schrödinger equation but also the deep connection between quantum dynamics and classical mechanics. The study introduces two powerful quantum algorithms that reduce the required resources to only about log(N) qubits for N oscillators, compared to the massive computational demands of classical methods. This exponential speedup could transform fields such as engineering, chemistry, neuroscience, and material science, where coupled oscillators serve as the backbone of real-world modeling. By bridging theory and application, this research underscores how quantum computing is redefining problem-solving in physics and beyond. With proven exponential advantages and the ability to simulate systems once thought computationally impossible, this quantum algorithm marks a milestone in quantum simulation, Hamiltonian dynamics, and real-world physics applications. The findings point toward a future where quantum computers can accelerate scientific discovery, optimize engineering designs, and even open new frontiers in AI and computational neuroscience. #QuantumComputing #SchrodingerEquation #HamiltonianSimulation #QuantumAlgorithm #CoupledOscillators #QuantumPhysics #ComputationalScience #Neuroscience #Chemistry #Engineering

Simulating quantum chaos without chaos https://lnkd.in/eezQkfgU It took me a while to accept that the main result of this work is not wrong, which I still find surprising. Concretely, #quantumchaos is a quantum many-body phenomenon that is associated with a number of intricate properties, such as level repulsion in energy spectra or distinct scalings of out-of-time ordered correlation functions. In this work, we introduce a novel class of "pseudochaotic" quantum Hamiltonians that fundamentally challenges the conventional understanding of quantum chaos and its relationship to computational complexity. Our ensemble is #computationallyindistinguishable from the Gaussian unitary ensemble (#GUE) of strongly-interacting Hamiltonians, widely considered to be a quintessential model for quantum chaos. Surprisingly, despite this effective indistinguishability, our Hamiltonians lack all conventional signatures of chaos: it exhibits Poissonian level statistics, low operator complexity, and weak scrambling properties. This stark contrast between efficient computational indistinguishability and traditional chaos indicators calls into question fundamental assumptions about the nature of quantum chaos. We, furthermore, give an efficient quantum algorithm to simulate Hamiltonians from our ensemble, even though simulating Hamiltonians from the true GUE is known to require exponential time. Our work establishes fundamental limitations on #Hamiltonianlearning and testing protocols and derives stronger bounds on #entanglement and #magicstatedistillation. These results reveal a surprising separation between #computational and #informationtheoretic perspectives on quantum chaos, opening new avenues for research at the intersection of quantum chaos, computational complexity, and quantum information. Above all, it challenges conventional notions of what it fundamentally means to actually observe complex quantum systems. Warm thanks to Andi Gu, Yihui Quek, Susanne Yelin, and Lorenzo Leone for this fun, thought-provoking and wonderful Harvard University-Freie Universität Berlin-Helmholtz-Zentrum Berlin-collaboration. And thanks to our funders, in particular the Deutsche Forschungsgemeinschaft (DFG) - German Research Foundation, the Bundesministerium für Bildung und Forschung (Quantensysteme), the Munich Quantum Valley, MATH+, the QuantERA, BERLIN QUANTUM, and the European Research Council (ERC).

Over the years, quantum computing has been judged mostly by its limitations — especially the gap between what today’s hardware can achieve and what classical algorithms can simulate. But the truth is more subtle and more exciting: the classical tools we rely on to simulate accurately quantum systems, like chemical compounds and materials, also have deep, well-known limitations. At Algorithmiq, we have been exploring how to turn this tension into something useful: a way to design and control information flow in artificial quantum materials, and to map out where classical methods begin to break while quantum methods provide reliable information. Why does this matter beyond physics? Because these simulations lies at the heart of the key industries driving the next decade: - catalytic processes for decarbonisation, - solid-state battery interfaces, - complex energy materials, - high-coherence quantum devices, - and next-generation computational chemistry. The challenge is that classical simulation becomes unreliable in precisely the regimes where these systems become most interesting — where disorder, interference, and entanglement govern their behaviour. We show that by pushing both quantum processors and classical algorithms into these hard regimes, we are beginning to see how quantum hardware can reveal properties impossible to discover with classical methods. Our initial evidence of quantum advantage for a useful use case is not just a scientific milestone — it is the early evidence of a technology crossing into real-world relevance. And challenges matter. They inspire people, create accountability, and accelerate progress. This is why I believe the Quantum Advantage Tracker, launched yesterday together with IBM Quantum, represents a turning point. It introduces the transparency, verification, and community benchmarking that every emerging technology needs to mature — and that investors rightly expect before deploying large-scale capital. We have published a detailed technical blog post explaining why information-flow modeling in artificial materials may become one of quantum computing’s most powerful use cases. 🔗 Link in the comments #QuantumComputing #QuantumAdvantage #InvestingInScience #DeepTech #MaterialsInnovation #Benchmarking #QDC2025 #QuantumMaterials #OpenScience

2D DIPOLAR SPIN ENSEMBLE OR LEVY FLIGHTS WHEN QUANTUM PARTICLES FLYING LIKE BEES Identifying universal properties of non-equilibrium quantum states is a major challenge in modern physics. A fascinating connection between many-body dynamics in a two-dimensional dipolar spin ensemble and Lévy flights lies in the statistical behavior and transport properties of the system. Lévy flights are a type of random walk characterized by step lengths that follow a heavy-tailed probability distribution, allowing for occasional long jumps. This behavior contrasts with classical Brownian motion, where step lengths are normally distributed. In 2D dipolar spin ensemble, the interactions between spins can lead to complex many-body dynamics, including anomalous transport phenomena. These dynamics can exhibit characteristics similar to Lévy flights, where the system's evolution involves non-local interactions or long-range correlations. Such behavior is often observed in systems with strong disorder or long-range interactions, where the transport deviates from classical diffusion and instead follows a pattern resembling Lévy flights. By studying these dynamics, researchers can gain insights into the fundamental properties of quantum systems. Quantum systems are incredibly complex, capable of assuming over two quadrillion states. Researchers from the Munich Quantum Valley and the University of Innsbruck have made a progress in addressing the quantum dynamics of 51 individually controlled ions by a long-range quantum magnet, realizing a long-range interacting spin chain. They demonstrated that the long-term behavior of such systems might be approximated using equations developed by the Bernoulli brothers in the 18th century to model fluid dynamics. Research team developed a protocol to measure space- and time-resolved spin correlations within an engineered infinite temperature state. This approach allows to experimentally demonstrate the emergence of hydrodynamics in a non-equilibrium quantum state. By measuring space- and time-resolved correlation functions in an infinite temperature state, study uncovered a diverse spectrum of hydrodynamic universality classes, spanning from normal diffusion to anomalous superdiffusion, characterized by Lévy flights. The transport coefficients from the hydrodynamic theory were extracted, which capture the microscopic properties of the system. The relaxation dynamics unfold in distinct stages. Initially, a local equilibrium is quickly established after just a few collisions with the abundant excitations in the infinite temperature state. This leads to a rapidly damped oscillation in the auto-correlation at short times. As time progresses, the system transitions into the hydrodynamic regime and ultimately moves toward global equilibrium, driven by the gradual rearrangement of spin excitations, which are restricted by the conserved magnetization. # DOI: 10.1126/science.abk2400