#tdvp

[artxssa]

I just know they would either hate or love each other HAHAHAHA

This actually only makes sense if you have seen that one elevator ad with Anna and Meryl😂

[govindhtech]

How TDVP Decodes Quantum Thermalization & Quantum Chaos

New Study Uses Time-Dependent Variational Principle TDVP to Find Periodic Orbits in Complex Quantum Worlds

Quantum many-body dynamics, which involve several particles, are notoriously difficult to characterise in quantum physics. Scale is the biggest issue: Hilbert space, the computing space, grows exponentially with system size. Physicists overcome computational crises using strong approximations.

One of the most successful is P. A. Dirac's Time-Dependent Variational Principle (TDVP). TDVP cleverly projects the complex, unitary quantum development onto a classical dynamical system, which is much simpler.

This projection greatly simplifies systems, allowing researchers to study them. TDVP maps dynamics onto the variational manifold of Matrix Product States (MPS) to create an effective classical system. MPS wave functions account for short-range entanglement, making them more expressive than basic product states. The bond dimension (χ) determines the amount of entanglement caught by the MPS. This gives the final classical system entanglement-related degrees of freedom.

This method connects classical chaos and quantum mechanics, making it interesting. This link allows physicists to study complex quantum phenomena like thermalisation and integrability using dynamical systems theory.

Chaos Skeleton: Periodic Orbits

According to classical chaos theory, periodic orbits organise. The “skeleton” of a system's phase space, they reveal its long-term behaviour, stability, and transport properties.

Despite its theoretical importance, finding these periodic orbits in TDVP-MPS dynamics' high-dimensional classical systems is difficult. The classical system with many pairs of canonical coordinates and momenta for a basic spin-1/2 chain is produced by TDVP even with minuscule bond dimensions.

Elena Petrova, Marko Ljubotina, and Maksym Serbyn developed a new method to locate and describe periodic orbits in the MPS variational manifold. The periodically kicking Ising model, used to study quantum thermalisation, was their focus.

Geometric Structure and Stable Orbits

The scientists found stable and unstable periodic orbits in the anticipated TDVP dynamics using their method. This suggests a mixed phase space in the TDVP approximation, a common feature of Hamiltonian dynamics.

TDVP dynamics resembled classical systems for the stable orbits found. The stable orbits were surrounded by Kolmogorov-Arnold-Moser (KAM) tori. We found that the dimensionality of these tori scales with the bond dimension squared (χ 2), which is directly related to the approximation's complexity. Computing Floquet multipliers, which measure the deformation produced by minor perturbations around an orbit over time, allowed the researchers to objectively classify orbit stability. The product of these multipliers is one because TDVP dynamics are symplectic, preserving phase volume.

See String Geometry Theory: The Future of Quantum Gravity.

Orbit Fate: Prethermal vs. Chaotic Regimes

The team's observation of these orbits as the quantum system transitioned from tranquil to maximal chaos revealed the truth.

We found that the low-leakage classical periodic orbits closely matched approximate eigenstates of the actual quantum evolution in the prethermal regime, a transitory stage where the system operates as if governed by an effective, almost conserved Hamiltonian. The orbit-finding approach generalises the Density Matrix Renormalisation Group (DMRG) method for ground state discovery in this low-leakage area, where TDVP dynamics correctly mimic quantum development.

When the model reached a maximally chaotic point and the coupling constant was increased, the orbits changed dramatically. Although the orbits continued, they got unstable and their Floquet exponent increased. They also entered MPS manifold high-entanglement zones.

This adjustment caused light leaking, a major shortcoming of the TDVP approximation. Leakage measures the precise unitary evolution's deviation from the fixed MPS manifold. In the chaotic domain, actual quantum dynamics often demand more entanglement than the fixed bond dimension (χ) can tolerate. Because of this, orbits in maximally entangled regions are unstable and leak more, reducing their "imprint," or link, on genuine quantum eigenstates.

A New Quantum Chaos Lens TDVP projection of quantum dynamics onto the MPS manifold creates a classical dynamical system whose properties can be rigorously studied utilising Quantum Chaos instruments.

This discovery opens exciting new research avenues. The method can be used to look for non-thermal eigenstates like quantum many-body scars in a systematic fashion. Periodic orbits with moderate entanglement growth (minimal leakage) have been linked to scars, rare states that prevent rapid thermalisation.

This clear link gives researchers a powerful tool for approximating quantum dynamics and revealing deep linkages between entanglement, chaos, and integrability in complex quantum systems. In conclusion, these classical skeletons of projected quantum motion shed light on why certain quantum systems behave chaotically while others remain regular.

[govindhtech]

Quantum Many Body Dynamics Simulation Via CQD Algorithm

Quantum Many Body Dynamics

The New Quantum Many Body Dynamics Simulation Hybrid Algorithm ‘Classically Corrected Quantum Dynamics CQD’ Overcomes Scale and Noise Limitations

Understanding microscopic physical processes involves the ability to predict the evolution of complex quantum systems across time. Simulating quantum many-body dynamics remains a fundamental and difficult physics problem. The exponential growth of Hilbert space makes simulating arbitrary quantum systems above a certain size difficult for traditional approaches, even as computational tools are improving. Quantum computing may solve the problem since the resources needed to mimic quantum time evolution rise polynomially with particle count. The Noisy Intermediate-Scale Quantum (NISQ) period is especially difficult for these devices to use because to noise and scalability issues. Simulations that use Trotterization of the time evolution unitary often have large Trotter errors due to limited coherence time and the number of implementable Trotter steps. Non-local Hamiltonians may require extra SWAP gates on restricted-connectivity hardware, increasing circuit depth. Starting Classically Corrected Quantum Dynamics Gian Gentinetta, Friederike Metz, and Giuseppe Carleo from EPFL developed a hybrid approach that carefully mixes quantum and classical computation to handle these significant practical difficulties. Classically Corrected Quantum Dynamics (CQD) streamlines the quantum process to generate surprising results even on noisy hardware. The core of CQD is outsourcing computing to a classical model. Trotterization based on a reduced Hamiltonian evolves the quantum computer's initial state, focussing on terms that are difficult to duplicate conventionally but well implemented on hardware. Classical models adjust the simulation by compensating for approximations or adding terms left out of the quantum circuit. CQD optimisation is a significant invention. The classical component of the wavefunction is parameterised using time-dependent parameters that are periodically optimised over time evolution using an enlarged version of the Time-Dependent Variational Principle (TDVP). The quantum circuit component has no variational parameters, which is crucial. This eliminates the requirement for computationally demanding quantum hardware operations like calculating out complicated gradients or overlaps (as needed by variational quantum algorithms using the parameter shift rule). Simply sampling configurations from the time-evolved quantum circuit in multiple bases may compute all derivatives and terms, including the quantum geometry tensor and forces, classically.

Three Strong Applications Showcased The researchers demonstrated CQD's adaptability with three demanding applications: Correcting Trotter Errors: Due to limited coherence time, digital quantum simulations often require large Trotter steps, which can lead to substantial errors. The CQD framework uses a classical Jastrow ansatz to fix these faults in the transverse-field Ising model (TFIM), resulting in better realism than simulations using only the classical ansatz or the Trotterized quantum circuit. This classical correction softened the fluctuating, non-smooth fidelities from piecewise constant Trotter evolution. The hybrid CQD ansatz gave accurate predictions, confirming the need for the quantum component in addition to classical error mitigation, while the classical ansatz alone was unable to capture long-range correlations for longer periods. Hardware-Efficient Time Evolution: Non-local Hamiltonians require additional SWAP gates, which deepens the circuit, because near-term quantum hardware often has limited connectivity. CQD allows researchers to confine the quantum circuit to a hardware-efficient approximation that only contains terms that follow the device's topology. The classical ansatz adjusts for missing non-local terms. For a two-dimensional TFIM simulation with weaker next-nearest-neighbor couplings and strong nearest-neighbor couplings, the CQD ansatz had the highest fidelity and could correct for missing Hamiltonian terms, avoiding SWAP gates and lowering circuit depth. CQD extends system size by adding degrees of freedom simply in the classical model, allowing simulations to exceed current devices' qubit counts. This works best with a weakly connected classical bath and a strongly correlated quantum component. The conventional simulation immediately degraded, whereas the CQD ansatz modelled a partitioned TFIM chain with good realism over time. Outlook: More Expressivity and Stability The findings show that adding a quantum circuit considerably enhances the expressivity of simpler classical ansätze like the Jastrow ansatz. This higher expressivity allows accurate simulations over long timescales, whereas classical models could not. The inversion of the quantum geometric tensor causes instabilities when optimising complex ansätze, however employing simpler classical ansätze with fewer variational elements decreases these instabilities. The CQD framework may simulate approximation dynamics using quantum hardware in any system with a known effective Hamiltonian acting on the quantum partition. Future study may include scaling assessments on noisy quantum hardware and tensor network or neural network quantum states. The technique could improve simulations of complex, physically intriguing systems like quantum impurity models or molecular systems with active and inactive orbitals.

[chrisskw]

Am I tired? Yes Did I go to bed at 5am and have an 11am flight organised? Yes Did a strange gentlemen glare at me and @pup.hero while taking this photo of @pup_russ and @pup_rolo asleep at this terminal? Yes Did I turn 28 today? Yes. This is the face of 4 people. The other 2 may still be drunk, the other 2 may have gotten sleep the night before. All I know is that I got read for filth for announcing my birthday. The rest of the Irish contingent ignored our antics at the gate🤗😇😘 Happy to be back in dublin so I can catch up on sleep. #folsomeurope was superb. Our #TDVP pack has grown by 1 and we are ready for our next adventure. Negative Craic because work tomorrow... #twinknoises #homewardbound #sleepdeprived #berlin #birthdaypost #amIaDaddyYet #zaddy #adultish #pushinmytailplug (at Berlin Schönefeld) https://www.instagram.com/p/B2ekkaFoqqA/?igshid=1ldg6dza9bwb

[chrisskw]

We have landed. Berlin we are in you. Cum and entertain us #TDVP #pahi #hapi (at Scandic Berlin Kurfürstendamm) https://www.instagram.com/p/B2TvC-MoI3u/?igshid=12nmrxql0lakx

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The devil wears prada-Still Fly #cover #guitar #clean #tdvp

[pizzagirll]