#quantumsimulations

A New “Quantum Homotopy” Solves the Nonlinear Physics Divide

New York University (NYU) and Los Alamos National Laboratory (LANL) researchers have developed a quantum algorithm to solve nonlinear partial differential equations (PDEs), one of the hardest problems in science, which could revolutionize computational fluid dynamics.

The new approach, called “Quantum Homotopy,” can simulate complex flow problems that traditional supercomputers and quantum techniques have failed to solve. The study, led by Sachin S. Bharadwaj, Balasubramanya Nadiga, Stephan Eidenbenz, and Katepalli R. Sreenivasan, moves from theoretical “toy problems” to harsh, nonlinear reality needed for realistic scientific and engineering research.

Resolving Linear-Nonlinear Conflict

Computer scientists have struggled with a fundamental “mismatch” between quantum physics and the physical world for decades. Quantum computers are linear systems governed by the Schrödinger equation, making them ideal for linear operations. However, nonlinear equations, particularly the Navier-Stokes equations, govern the universe's most fundamental processes, such as water churning in a turbine, air turbulence over a jet wing, and plasma dynamics in fusion reactors.

Conventional quantum algorithms have long struggled with nonlinearities. Early methods like Carleman or Koopman embeddings performed best with “weak” nonlinearity. When solving complex or chaotic situations, these older methods sometimes saw exponential increases in processing cost or error, which is where a quantum advantage is most needed.

Power of Homotopy Analysis

The team invented the topologically based Homotopy Analysis Method (HAM) that continuously deforms mathematical objects. Researchers "stretch" a known, simple linear solution to match the intended difficult, nonlinear solution using this method.

By placing the difficult nonlinear equations in a truncated, high-dimensional linear space, the researchers solved a “quantum-unfriendly” nonlinear problem efficiently with quantum computing. Integrating this linearized system using a small finite-difference technique allows accurate quantum computing.

Unlike static linearization, the Quantum Homotopy (QH) is adaptive. It is more versatile than previous methods since its internal parameters can be adjusted based on the flow or nonlinearity being simulated.

Burger's Equation: proving robustness

The researchers employed the 1-D Burgers equation, a fluid dynamics benchmark, to test their innovative framework by simulating shock waves and dissipative turbulence. The method solved the equation with a Reynolds number of 100 better than previous methods. The results shocked.

The team showed that the allowable integration window parameter and a physically motivated nonlinearity metric similar to the flow Reynolds number were critically related. This option ensures that the algorithm remains accurate and computationally efficient as the problem becomes more complicated.

The researchers also linked the method to the “Kolmogorov scales,” the physical dimensions in which fluids dissipate. This provides a solid theoretical foundation for predicting when a quantum computer will outperform a traditional supercomputer in fluid simulations.

Aerospace to Green Energy

A robust, “near-optimal” quantum solution has many effects on nonlinear PDEs. Simulation computer costs often hinder engineering. To build a new airplane, thousands of hours of supercomputer work are needed, and various turbulent phenomena must be represented.

Quantum homotopy may allow:

Revolutionize Aerospace: Simulating high-speed “hypersonic” flows with high nonlinearity could transform the aerospace sector. Improve Climate Models: Forecasting marine and atmospheric dynamics at unprecedented resolutions can improve climate models.

Increasing energy transitions by improving nuclear fusion reactor and wind turbine design, where fluid flow and plasma are notoriously difficult to simulate.

Path to Quantum Advantage

Being “end-to-end” is one of Quantum Homotopy's most promising features. The method works on near-term quantum devices, which are prone to noise and decoherence, and is scalable for future fault-tolerant quantum computers.

A research team has found the first explicit sharp restrictions for the “nodes” of all eigenfunctions in Sturm Liouville Operators, a huge development for mathematical analysis and computational physics. This solution addresses a qualitative mathematical riddle that has persisted since the 1830s, according to Rohail T. It closes a two-century gap in scientific understanding by improving vibrating system and quantum state predicting accuracy.

Introduction to Sturm-Liouville Operators

Understanding Sturm-liouville Operators is necessary to understand this discovery. The theory of Joseph Liouville and Jacques Charles François Sturm defines a broad class of second-order linear ordinary differential equations. The mathematical foundation of conventional and quantum physics is provided by these operators. They replicate many physical processes, from the simple harmonics of a vibrating guitar string to the incredibly complicated energy levels of an electron in an atom.

The important concepts in these mathematical systems are eigenvalues and eigenfunctions. Eigenfunctions and eigenvalues represent a system's possible states or modes and their energy levels or frequencies.

The “node” is the most elusive part of these solutions. Nodes are specific locations where eigenfunction solutions intersect zero. In a vibrating physical system, these nodes are “dead zones” that do not move.

The Node Mystery

Mathematicians have long recognised the features of nodes, such as the n-th eigenfunction having n−1 nodes, but pinpointing their particular positions remains challenging. Their positions were mostly understood through implicit relationships until recently, making them difficult to use in high-stakes circumstances like powerful quantum simulations or precise engineering.

Scientists struggled with node locations being considered “best guesses” rather than absolutes. The field used qualitative observations over 200 years. Researchers often used complex, time-consuming computer simulations to verify a system's stability because there were no obvious formulas.

Breakthrough: Guesswork to Formulas

Recent research by Jifeng Chu, Shuyuan Guo, Gang Meng, and Meirong Zhang has progressed from implicit estimates to mathematical certainty. Their primary contribution was to treat these nodes as “nonlinear functionals” of the system's potential energy landscape rather than fixed placements. The researchers used optimisation to find the minimum and maximum positions for each system node. The researchers focused on potentials inside L 1 balls, a mathematical framework for systems with limited “strength” or “energy”.

The allegation that the group used advanced inverse spectral theory and “strong continuity” of nodes with regard to potential to reach their conclusions. These constraints are first described as “elementary functions”. This lets physicists calculate node boundaries without substantial computational models.

One key mathematical obstacle the team overcame was the absence of local compactness in the space of potentials, a recurrent problem in these arguments. They solved this difficulty and determined accurate node bounds by evaluating L p norm limit situations as they approached L 1.

Changing Quantum Engineering and Technology

Quantum technology development is notably affected by this discovery. In quantum physics, wave function nodes are regions of zero probability density, or places where particles cannot exist. The industry needs accurate node placement to build high-precision sensors and stronger quantum computers.

The identify many critical areas where this revelation will have an immediate impact:

Modern sensors use high-resolution quantum sensing to detect minute quantum state changes. Researchers uncovered the potential for sensors to detect temperature changes as small as 4×10−6 °C by using nodal point boundaries to create more accurate quantum activity maps.

Mechanical and aeronautical engineers must determine nodes to reduce material fatigue. Engineers may now place crucial components in “quiet zones” where a system won't shake using these clear boundaries to ensure long-term durability and safety.

Inverse Spectral Theory: The “can you hear the shape of a drum?” conundrum involves reasoning backwards to determine an object's internal structure from its vibrations. These new explicit bounds help reconstructive mathematics by allowing more exact inferences of a system's intrinsic features from observable nodes.

To conclude

This latest technique turns qualitative observations into quantitative certainty. The paper provides explicit bounds for nodes in measure differential equations to better understand the relationship between nodes and the potential energy landscape.

As quantum computing spreads from labs to industry, “hard numbers” are preferred above “soft estimates”. This discovery opens the door to new algorithms in structural engineering, quantum chemistry, and materials research.

A powerful new algorithm developed by researchers from Spain, Sweden, and the Netherlands will improve the precision and efficiency of preparing the ground state of a complicated quantum system. This advances quantum computing and simulation, which are growing rapidly.

The new algorithm, Multiple-Time Quantum Imaginary Time Evolution (MT-QITE), improves on QITE. It promises to open new capabilities on near-term quantum hardware by improving the quality of a prepared state's match to the genuine ground state and reducing the costly measurement overhead that plagues quantum simulations.

Repeatedly evolving a quantum system in imaginary time yields a more faithful ground state while reducing the measurement burden, according to Evert van Nieuwenburg from Leiden University, Mats Granath from Gothenburg University, and Julio Del Castillo from Universidad Nacional de Educación

Importantly, this novel technique is deterministic without complex, system-specific assumptions. The team also shows that this unique technology is easily parallelizable, benefiting long-distance systems.

Quantum Physics' Foundation: The Ground State Challenge

Precision ground state calculations are the computational goal of quantum chemistry and materials research. Ground states, the lowest energy states of quantum systems, are any substance or molecule's natural resting place. Its properties affect everything from a medicinal molecule's chemical reactivity to a material's magnetic and superconducting properties.

Before synthesising a new material in a lab, scientists can accurately simulate its ground state and forecast its properties. This speeds up discovery. Typical supercomputers cannot solve systems with more than a few dozen interacting particles because they require exponentially more processing resources. Intractable because the system's Hamiltonian, which describes its energy, is complex.

Only quantum computers can solve this classical bottleneck. Standard Quantum Imaginary Time Evolution (QITE) prepares ground states more simply and predictably than Variational Quantum Eigensolvers (VQE), which use sophisticated classical optimisation loops.

Power and Risk of Imaginary Time

Classical computational physics uses imaginary time evolution. In imaginary time, a quantum system "thermalises" or relaxes into the ground state, unlike real time evolution, which causes a complicated, oscillatory journey. This technique strongly filters out unwanted, higher-energy components.

QITE turns this elegant theory into quantum hardware by projecting continuous imaginary time development onto several executable quantum circuits. Conventional QITE implementation is theoretically sound and predictable, yielding reliable results without probabilistic luck, but it has a huge measurement overhead. Running the program requires repeated system measurements. Today's Noisy Intermediate-Scale Quantum (NISQ) devices' limited coherence time and resources are quickly consumed by the amount of measurements needed to maintain accuracy for sophisticated Hamiltonians, especially those representing non-local or long-range interactions. Precision is often costly.

MT-QITE multiplying fidelity and efficiency

Del Castillo, Granath, and van Nieuwenburg's innovative MT-QITE algorithm solves this resource problem quickly. The key finding is surprisingly simple: carefully dividing the total imaginary time into several smaller evolution steps improves execution efficiency and result quality compared to a single, drawn-out computational run. A geometric interpretation of the algorithm is included in this updated implementation to better understand its behaviour and optimisation potential.

This multistep approach produces amazing results. Researchers showed that they could boost ground state fidelity by one to two orders of magnitude compared to regular QITE after the same computational steps. Due to this massive accuracy increase, the quantum states will more accurately match the physical ground state. For this fidelity improvement, the measurement budget is greatly lowered.

The algorithm adjusts step size to accelerate and improve convergence to the ground state. This adaptive nature uses multiple imaginary time steps to decrease quantum operations, speeding up and improving the process's dependability on quantum technology.

Deterministic, Parallel, Partition-Friendly

In addition to performance indicators, MT-QITE includes other architectural aspects that make it effective for NISQ computing.

MT-QITE remains deterministic. In a field that relies on statistical sampling and probabilistic outcomes, scientific certainty and reproducibility require a procedure that converges without system-specific assumptions.

Second, algorithm parallelisation is simple. MT-QITE takes advantage of parallelisation, which breaks down difficult calculations into smaller pieces that may be run simultaneously. This is very useful for simulating complicated systems, especially ones with long-distance interactions.

Hamiltonian partitioning had an unexpected benefit for the group. Even in complex systems with non-local interactions, partitioning the Hamiltonian into simpler terms helped. This discovery considerably helps computation by breaking down a monolithic simulation problem into concurrent tasks, reducing resource requirements. The researchers want to reduce quantum operations to make the technology suitable for current and future quantum computers.

Enabling Quantum Simulation Breakthroughs

This research solves a major physics and chemistry problem to improve material property and molecular behaviour calculations. By using quantum computation, MT-QITE can speed up these computations faster than standard computers. The researchers extensively tested the technique to prepare ground states with improved fidelity and cheaper measurement costs, enabling quantum simulations of complex physical systems.

TWA Quantum

The capacity to simulate complicated quantum systems on platforms ranging from powerful supercomputers to everyday consumer laptops has increased recently thanks to a major breakthrough made by physicists at the University at Buffalo (UB). This finding primarily focusses on the expansion and simplification of a well-known computing method known as the truncated Wigner approximation (TWA). The TWA serves as a kind of "physics shortcut" to make quantum mathematics easier to understand.

The complexity of quantum mechanics drove this progress. Scientists studying quantum physics encounter minuscule particles that can interact in over a trillion ways. Physics has usually used powerful supercomputers or AI to mimic quantum systems and their infinite states due to their mind-bending complexity.

The physics world has long recognised that many of these problems might be solved without massive quantities of computing power, but it was difficult to make this a reality.

The Semiclassical Shortcut

The TWA, or shortened Wigner approximation, has been a computationally accessible method since the 1970s. It belongs to the field of semiclassical physics and is categorised as a middle-ground calculation method. Semiclassical physics is crucial because it is difficult to solve every quantum system precisely, and the computing power required grows exponentially with system complexity.

Instead of aiming for flawless solutions, the semiclassical approach purposefully removes elements that have little impact on the outcome while preserving just enough quantum behaviour to be accurate.

Prior until now, TWA was limited to isolated, highly idealised quantum systems, despite being a helpful shortcut. In these cases, no energy was gained or lost.

TWA expansion for "Messier Systems"

The team’s primary breakthrough, led by Dr. Jamir Marino, PhD, an assistant professor of physics at the UB College of Arts and Sciences, was to extend TWA to handle the “messier systems found in the real world”.

Real-world dynamics include particles that leak energy into their environment or are constantly pushed and pulled by outside forces. This phenomena is also known as dissipative spin dynamics.

Dr. Marino, the study's corresponding author, noted that other organisations had attempted to expand this application in the past. Even though it was recognised that some complex quantum systems might be successfully addressed using a semiclassical technique, Dr. Marino stressed that "the real challenge has been to make it accessible and easy to do." Dr. Marino conducted the initial research for this subject at Johannes Gutenberg University Mainz in Germany prior to joining UB this September.

Accessibility and Savings on Computation

The UB team's largest contribution, aside from the method's physical extension, was overcoming the method's immense complexity, which had previously deterred researchers. In the past, TWA physicists faced the difficult task of rederiving the complex, dense mathematics for every new quantum problem they tackled.

Dr Marino's team solved this accessibility issue by converting pages of "nearly impenetrable maths" into a straightforward conversion table. Using this template, a specific quantum problem is directly transformed into solveable equations. The result is a TWA template that is quite easy to use. According to Oksana Chelpanova, a co-author and postdoctoral researcher in Dr. Marino's lab at UB, "physicists can basically learn this method in one day, and by about the third day, they are running some of the most complex problems we present in the study," emphasising the rapid learning curve. With the help of this helpful framework, researchers can enter their problem and receive valuable results in a few hours.

The total benefits are significant and include much reduced processing costs and a much simpler formulation of the dynamical equations. The researchers believe that the new method could soon be the primary tool for studying these types of quantum dynamics on consumer-grade computers.

Supercomputers Are Only Allowed for the "Very Complex"

Spin-Adapted

Spin adapted representations increase simulation efficiency in quantum computing by directly using quantum systems' innate symmetries, especially the challenging non-Abelian SU(2) total-spin symmetry. The exponential growth of Hilbert space size with particle number is a fundamental difficulty in quantum system simulation. Despite classical algorithms' success in reducing computing costs using non-Abelian symmetries, quantum algorithms have struggled to include them.

Problems with Conventional Methods Total spin eigenstates may require an exponentially high number of coefficients to represent them in terms of the eigenbasis, the common computational basis. Most quantum computing methods use simpler Abelian symmetries. Previous research on non-Abelian symmetries focused on maintaining symmetry conservation with non-spin-adapted bases, which can be computationally difficult and may not avoid hardware noise or Trotterization from breaking symmetry. Explicit unitary basis modifications, like the quantum Schur transformation, are theoretically possible but impractical on most quantum hardware because they employ qubits, whose length and local dimensions rise with system scale.

New spin-adapted structure Researchers have developed a novel formalism for building quantum algorithms directly in a total-spin operator eigenbasis to circumvent these issues. This concept uses the symmetric group approach (SGA) to generate spin-adapted quantum Hamiltonians and associated unitarizes.

A truncation technique for total-spin eigenstates' internal degrees of freedom is innovative in this context. This method defines a hierarchy of spin-adapted subspace encodings onto quantum registers with increasing precision. In a site antiferromagnetic Heisenberg model, intermediate total spin values can be shortened to a maximum value, creating ever-larger subspaces that accurately describe the model's low-energy behaviour.

The step encoding, the other popular spin eigenstate encoding approach, is less suitable for quantum computing applications than height encoding, which retains quantum operator locality. However, step encoding often produces very non-local projector operators. The height encoding and height variable truncation provide sparse and local qubit Hamiltonians, making them excellent for quantum simulations on hardware with constrained connectivity.

Key benefits and advantages:

This approach generates a hierarchy of spin-adapted Hamiltonians for the antiferromagnetic Heisenberg model whose ground-state energy and wave function rapidly converge to their exact equivalents, even with very small truncation thresholds. Ground state energy estimates for a 16-site Heisenberg chain converged quickly. Hardware suitability: Truncated Hamiltonians can be encoded into sparse and local qubit Hamiltonians, making them suitable for quantum simulations on hardware with limited connectivity. Avoids Impractical Changes: This method achieves results comparable to those of complex unitary basis modifications like the quantum Schur transformation without explicitly implementing them on most qubit-based quantum devices. For smaller truncated subspaces, the approach may reduce qubit count and improve noise resilience. Using the lowest non-trivial truncation subspace, a ground state approximation for a 16-site chain only needs 3 qubits for the singlet state (down from 9) or 7 for the triplet state (down from 14 for the barStrunc le 3/2 subspace). This reduced qubit count could increase noise robustness in hardware quantum simulations of low-energy states. Reducing Sampling Overhead: Quantum algorithms can be formulated in total-spin eigenbases, so the generated wave functions can be sampled there instead of the projected spin eigenbasis. Significant compression of the ground-state wave function has been shown, which should reduce the sampling overhead needed for observable quantum state estimation. See also National Quantum Computing Centre Gets NPL Ion Trap.

Application and Demo:

The researchers solved the one-dimensional antiferromagnetic Heisenberg Hamiltonian using their method. A resource-efficient adiabatic state-preparation technique was used to create shallow quantum-circuit approximations for the ground states of the Heisenberg Hamiltonian in various symmetry sectors. This includes singlet and triplet variations. Even with slight truncation values, numerical simulations showed that these approximation time-evolution unitaries approximate low-energy beginning states' temporal dynamics. The adiabatic schedules using simple linear ramp functions have very high final instantaneous fidelities.

Future Outlook:

Future research will use real quantum hardware to extend this protocol to non-integrable spin Hamiltonians and electronic structure Hamiltonians to test its scalability. Using spin-adapted operator-specific scalable error mitigation approaches, the researchers seek to enable utility-scale quantum simulations. Compressing the ground-state wave function in the spin-adapted basis should speed up quantum simulations by reducing sampling overhead during observable estimation. More compact wave functions could improve methods like sample-based quantum diagonalisation (SQD).

Scientists improved particle physics simulation by utilising quantum computers to simulate and watch “string breaking” in real time. Traditional computers couldn't replicate this complicated process, in which subatomic particles like quarks are joined by "strings" of force fields that release energy when they break.

The groundbreaking findings are the latest step towards using quantum computers for simulations that surpass conventional machines. These quantum simulations are “incredibly encouraging,” says LBNL scientist Christian Bauer. He said “string breaking is a very important process that is not yet fully understood from first principles”. Classical computers can calculate the ultimate effects of particle collisions involving string generation or breaking, but not the intermediate dynamics.

Two Simulation Methods Revealed

Two worldwide academic-business research teams conducted the experiments. Two teams worked at theGoogle Quantum AI Lab in Santa Barbara, California, and Cambridge startup QuEra Computing. These groups discovered string breaking employing “diametrically opposite quantum-simulation philosophies”:

Analogue Quantum Simulation (QuEra Computing):

This team included Harvard, Innsbruck, and QuEra Computing researchers using QuEra's Aquila computer.

The data was encoded in rubidium atoms kept in place by optical “tweezers” in a 2D honeycomb or kagome-geometry pattern.

The electric field at a given position in space was reflected by each atom's qubit, which could be stimulated or relaxed.

This analogue quantum simulation relied on arranging the atoms so that their electrostatic forces mimicked the electric field. This arrangement allowed the system to continuously attain lower energy levels.

This technology allowed the first observation of string breaking in a programmable two-dimensional quantum simulator. The “tabletop analogue of quark confinement,” a key property of QCD, was achieved.

◦ Daniel González-Cuadra, co-author of the QuEra paper and theoretical physicist and assistant professor at the Institute for Theoretical Physics (IFT) in Madrid, said neutral-atom devices can now solve theoretical difficulties. He said “seeing string breaking in a controlled 2D environment marks a critical step towards using quantum simulators to explore high-energy physics”.

Alexei Bylinskii, QuEra's VP of Quantum Computing Services, said this alliance “underscores the value of open, programmable neutral-atom hardware for fundamental research.” Research in condensed-matter, high-energy, and quantum-information science is enhanced by flexible access to Aquila's multi-qubit capabilities.

Professor Peter Zoller, a senior author at IQOQI and the University of Innsbruck and “founding father of modern quantum simulation,” said “Gauge theories govern much of modern physics.” By showing non-abelian gauge fields and topological matter in two dimensions where strings can bend and fluctuate, the basis is established for studying them.

The experiment featured dynamic quenches using local detuning ‘kicks’ to watch strings snap and re-form in real time, revealing resonance peaks signalling many-body tunnelling processes; programmable geometry, where atoms were placed on hexagonal lattice links to enforce Gauss's-law constraints via Rydberg blockade; and tuneable string tension by varying laser detuning and interaction radius. This work stretched one-dimensional demonstrations to two spatial dimensions, when theoretical and numerical techniques near saturation.

Google Quantum AI Lab (Digital Quantum Simulation) utilised the Sycamore processor.

The chip's superconducting loop states encoded the 2D quantum field, unlike the analogue method.

This “digital” quantum simulator delicately controls the quantum field's evolution “by hand” using discrete manipulations.

Frank Pollmann, a physicist from the Technical University of Munich (TUM) in Garching, Germany, who led the Google experiment, said both teams placed strings in the field that “effectively acted like rubber bands connecting two particles.” Researchers adjusted settings to make strings stiff, wobbly, or breakable. Pollmann sometimes said, “The whole string just dissolves: the particles become deconfined.”

Importance and Future

These experiments are necessary to employ quantum computers for simulations beyond regular machines. The results demonstrate the scalability of neutral-atom platforms like Aquila for simulating complex quantum field theories and set a benchmark for quantum simulation by pushing classical computational capabilities in real-time gauge-theory dynamics. This confirms the growing importance of quantum hardware for scientific study.

Simulating strings in a 2D electric field can be useful in material physics, but high-energy interactions like those in particle colliders, which require the stronger nuclear force, are difficult to replicate. Monika Aidelsburger, a physicist at Munich's Max Planck Institute of Quantum Optics, says these more complex simulations have “no clear path at this point how to get there”.

She added that quantum simulation has advanced “really amazing and very fast” overall. Because ‘qudits’ quantum systems with more than two quantum states may produce more accurate representations of a quantum field and enhance simulation power, researchers are considering using them. Christian Bauer and LBNL colleague Anthony Ciavarella were among the first to model the strong nuclear force with a quantum computer last year.

This research will boost particle physics and demonstrate quantum computing's scientific discovery potential.

Financial Support and Recognition

Quantum-classical mechanics as an alternative to quantum mechanics in molecular and chemical physics

In quantum mechanics, the theory of quantum transitions is grounded on the convergence of a series of time-dependent perturbation theory. In nuclear and atomic physics, this series converges because the dynamics of quantum transitions (quantum jumps) are absent by definition.

Global Particle Physics Excellence Awards

website url: physicistparticle.com/

Nomination link: https://physicistparticle.com/award-nomination/?ecategory=Awards&rcategory=Awardee

Что-то я забросил свой блог... А так то я болел и не до этого было :-(  Но мы продолжаем

Сегодня хотелось бы рассказать о своем ресерче, а точнее о разделе физики, в котором мы проводим свои исследования. И это, квантовые симуляторы, а если быть более точным, то создание ультрахолодных молекул(!) в оптических ловушках. Но сегодня о квантовых симуляторах. Я попытаюсь ответить на вопрос, зачем они нужны и какой от них вообще толк?

Ну начнем с того, что скажем, что квантовый симулятор, это не квантовый компьютер и никакого отношения к quantum computing не имеет. А суть вот в чем. Известно, что любую даже относительно простенькую квантовую модель исследовать либо сложно, либо практически нереально. Допустим мы имеем систему из 50 частиц со спином \( \uparrow \) и \( \downarrow \) и хотим решить для этой системы уравнение Шредингера

\(  i \hbar \frac{d}{dt} \phi = H \phi \)

,то есть найти \( \phi(t) \) ( \(  \phi(t) = exp (-i \hbar H t ) \psi(0)   \) ). И так вот, чтобы только записать число всевозможных конфигураций нам понадобится \( 2^{50} \) или \(  10^{15} \) бит информации или около 500 Тб, что очень много. Не говоря уже о других операциях, которые тоже потребуют памяти. Итого, просто технически исследовать даже такую простенькую квантовую модель на компьютере нереально. Но исследовать то хочется. Так что же делать? Есть довольно лаконичный выход из этой ситуации, который реализуется посредством совершенно нового подхода. Этот подход и именуются квантовой симуляцией. В чем же фишка?

Фишка в следующем. Допустим у нас есть система, которую и численно исследовать трудно и экспериментально. То есть всё против нас :-) Так вот, давайте на секундочку забудем про эту систему и постараемся воссоздать схожую, но которую можно было бы контролировать, то есть менять параметры системы и самое главное(!), получать данные непосредственно из измерений. Математически, соответствие между системами выражается в эквивалентности их гамильтонианов. 

\( H_{sys} \longleftrightarrow H_{sim} \)

И уже исследуя эту систему (квантовый симулятор), мы сможем предсказать и проанализировать поведение прежней системки. Чтобы, лучше понять о чем идет речь, приведу хрестоматийный пример. 

Все мы слышали о высокотемпературной сверхпроводимости, которую открыли в середине 80ых 20 века. После этого открытия произошел бум в области высокотемпературной сверхпроводимости и ученым даже удалось выделять целый класс материалов, у которых наблюдается высокое \( T_{c} \). Этот класс именуется купратами и вся загвоздка заключается в том, что соединения есть, а понять почему и чем вызвана сверхпроводимость не получается. То бишь физика данного действва не ясна до сих пор.Теоретически все чики-брики для низкотемпературной сверхпроводимости (БКШ теория), а вот для высокотемпературной никакой толковой теории создать не удалось.Почему? Да потому что купраты относятся к  классу сильно коррелированных систем, где волновую функцию нельзя взять и разбить на мини-составляющие, что делает любое исследование довольно сложным. Однако(!), есть подозрения на то, что сильно коррелированные системы (то есть системы у которых электроны взаимодействуют и воздействуют друг на друга посредством кулоновского притяжения) могут описываться с помощью модели Хаббарда. Что это такое? 

\( H = -J \sum\limits_{i, j} (c_{i, \uparrow}^\dagger c_{j, \uparrow} + c_{i, \downarrow}^{\dagger} c_{j, \downarrow} +h.c.) + U \sum\limits_{i} n_{i, \uparrow} n_{i, \downarrow} \)

То есть имеется два члена - \( J \) и \( U \), где один характеризует перескоки с одной соты в другую, а второй член описывает взаимодействие между частицами в соте. Казалось бы какая простенькая модель, а хрена с два! Решить аналитически в общем случае нельзя, разве что в очень частных случаях, да и численно не все так просто. Короче, модель Хаббарда, как и еще одна излюбленная физиками модель Изинга довольно красивенько и миленько выглядит, но если покопаться, то понимаешь, что этой лютый треш и лучше туда не соваться. Прямо как некоторые девушки ;-) Ну да ладно, есть модель Хаббарда и именно эта модель может объяснить и приоткрыть завесу тайны над высокотемпературной сверхпроводимостью. Так по крайней мере заявляют специалисты в области сверхпроводимости, которые последние лет 10-15 особого прогресса не достигли в своей области. И так вот, что же делать? 

Физическая система есть - купраты. Модель тоже - модель Хаббарда. А вот продвинуться не можем. В этот момент и вступают квантовые симуляторы. А давайте попробуем создать системку с точки зрения квантового симулятора, который также будет описываться моделью Хаббарда, но уже имея полный доступ к его параметрам ( \( J \) и  \( U \) , в реальной системе эти величины жестко фиксированы )и мы сможем менять их как нам заблагорассудится и сможем проводить измерения, получая необходимые физические величины непосредственно из эксперимента. А давайте. Но как? 

А с помощью ультрахолодных атомов и оптических решеток. То есть заменим наш кристалл на систему из атомов, которые предварительно помещены в оптическую решетку.

По сути будем иметь то, что на картинке. И вот такая система абсолютно эквивалентна кристаллу. Я даже больше скажу. Она лучше :-) И вот почему

Оптическая решетка свободна от всяких дефектов

Член туннелирования (\( J \) легко управляем и мы его можем менять, впрочем как и параметр взаимодействия \( U \) 

Величина сот у решетки порядка 500-1000 nm

Временные характеристики у данной системы довольно низки, на уровне \( 10^{-3} \) секунд, что позволяет нас довольно детально исследовать все что там происходит

и еще много всего 

Сразу возникают вопросы.

Как создать ультрахолодные атомы?

Как создать оптическую решетку?

Ну ультрахолодные атомы создаются на раз два и сейчас это довольно обкатанная процедура. Хотя если вы хотите поиграться с более сложными системами, вроде Ферми-Ферми молекул в основном состоянии, то есть соединениями, которые состоят из ферми атомов, как LiK (эта молекула интересна тем, что у нее высокое диполь-диполь взаимодействие, которое может быть полезно), то тут уже не все так просто.  А вообще используются типичные методы охлаждения с помощью магнито-оптических ловушек и прочих штучек-дрючек. Вообще хотелось бы более детально рассмотреть данный вопрос с точки зрения, как теории, так и эксперимента. Как будет время, попробую запилить материал. Что же касается оптических решеток, то тут просто посылаются навстречу друг друга два лазерных луча и они образуют стоячую волну, которую и можно характеризовать как решетку. И уже потом на эту решетку, “насаживаются” охлажденные атомы.

И так вот, есть ультрахолодные атомы в оптических решетках и они ведут себя как кристаллы. Ну и исследование таких уже (новых) систем и представляет собой раздел, квантовую симуляцию. Итого, всю процедуру можно разбить на 4 шага.

Соответствие между реальной физической моделью и той, которую мы хотим воссоздать с помощью ультрахолодных атомов (так это не обязательное условие, но о других методах симуляций я пока что ничего не знаю).

Подготовка начальной системы

Изменение параметров квантовой системы путем экспериментального вмешательства (поменял интенсивность, включил магнитное поле и т.д.).

Ну и напоследок, проведение измерений и извлечение реально наблюдаемых величин.

Хочу еще раз опять повториться, что основная фишка во всех этих действиях в том, что при исследовании новых систем, мы измеряем напрямую квантовые величины. Это очень круто.

Ну и это я так, дал так сказать обзорчик на область. Тон в этой области задают немцы, а именно группа Иммануэля Блоха в Германии (институт Макса Планка). Он является так сказать батей данного направления. А одна из самых первых статей в этой области Quantum Phase Transition from a Super uid to a Mott Insulator in a Gas of Ultracold Atoms (M. Greiner, O. Mandel, T. Esslinger, T.W. Hansch, and I. Bloch Nature 415, 39-44, 2002 ) сейчас имеет порядка 4900 цитирований! Это прямо-таки Нобелевский размах. Ну и помимо Блоха главные игроки в эксперименте, это

Greiner (MIT)

Zwierlein (MIT)

Spielmann (JQI)

Esslinger (ETH Zurich)

Jun Ye (Colorado Boudler)

и другие

Кстати что касается той статьи. Что же они нашли такого крутого? А вот что. Оказывается, при исследовании модели Хаббарда возникают два крайних состоянии - это сверхтекучее состояние (\( J  \gg \ U\)) и так называемое моттовское состояние (\( J \ll U \)). И так вот, Грейнер, Блох и компания смогли пронаблюдать за тем, как частицы из сверхтекучего состояния попадают в моттовское, меняя глубину оптической решетки. По сути они увидели фазовый переход в реальном времени, прям как переход вода-лед :-) Менять же глубину можно тупо уменьшая/увеличивая интенсивность нашего лазера. Хочу правда оговориться, что вся эта картина не имеет никакого отношения к высокотемпературной сверхпроводимости, так как там модель Ферми-Хаббард, а в исследуемой немцами модели, Бозе-Хаббард. И напоследок, вот главная картинка  с их эксперимента, где изображено распределение частиц по импульсам в зависимости от глубины потенциальной решетки. Более смазанные распределения относятся к моттовскому состоянию, а более пикообразные к сверхтекучему.  

P.S. За основу этого репортика был взят PhD thesis парня из группы Маркуса Грейнера, Widagdo Setiawan, который просто изумительно описал в своем введении мотивацию и то что из себя представляет quantum simulation. Кстати, сейчас он работает квонт трейдером в KCG.