#matrixproductstates

A Seoul National University-MIT physics collaboration produced a research in December 2025 that significantly changed our understanding of the early universe. Using Random Matrix Product States (RMPS), researchers Sunghoon Jung, Sungjung Kim, Jiwoo Park, and Seokhyeon Song have been able to study the universe's "initial state" when Einstein's General Relativity's smooth geometry disintegrates into a turbulent "quantum foam."

Breakthrough: Quantum Foam Mapping

This discovery centres on “gravitationally prepared states”. In quantum field theory, these states represent a closed world's quantum wave function. They are constructed by visualising a universe where gravity has boundary limitations but matter does not. These states preserve the whole history of gravitational events, which is vital for universe evolution.

Researching these states has been difficult due to their intricacy. Standard scientific methods struggle to explain “higher topologies” or “Wormhole Phase Transition,” complex spacetime geometries with many holes and bridges. Before this study, traditional semiclassical techniques were assumed to be unable to determine the contributions of these complex structures to the universe.

Understanding Gravity-Ready States

The research team solved these problems with Random Matrix Product States. The “tensor network” apparatus was designed for many-body quantum systems, including atom behaviour in crystals. The researchers added randomness to these matrices to represent quantum gravity's statistical behaviour.

The RMPS approach is unprecedentedly accurate. It can construct complex geometric configurations, including quantum entanglement "replica geometries," to all orders of approximation. With this precision, scientists may study how past gravitational history affects matter fields now.

Innovation: Random Matrix Product States Key findings include the “bra-ket wormhole phase transition” confirmation. “Bra” and “ket” are the two sides of a quantum probability computation. These two sides are connected gravitationally by a Wormhole Phase Transition.

As the universe's geometry alters fundamentally, this phase shift may be mathematically assured, the researchers found. This is guaranteed if the RMPS "transfer matrix" fits the spectral gapping property. This conclusion is essential because it provides a rigorous mathematical basis for understanding why and when wormholes dominate early cosmic physics, moving the issue from theoretical speculation to mathematics.

Bra-Ket Wormhole Phase Transition

The “off-shell” wormhole revelations were astounding. Classical physics calls configurations that follow equations of motion “on-shell” like a tossed ball. However, “off-shell” structures are quantum fluctuations and do not follow classical paths.

Because gravity models lack stable classical solutions, they often miss off-shell wormholes, whereas the RMPS model can include them. The researchers found that off-shell structures lead to nonzero long-distance correlations in gravitationally prepared states. This shows that a wormhole's quantum presence connects distant portions of the cosmos even if it isn't a fixed “bridge” and may leave quantitative evidence for researchers to locate.

Long-distance correlations and off-shell wormholes The researchers extended their model from two-dimensional to continuous space to study de Sitter gravitationally prepared states. This is relevant to our reality because de Sitter space provides the mathematical model for accelerated expansion, like cosmic inflation.

By applying matrix models to de Sitter space, the group created a new “toolkit” for studying quantum gravity events with non-perturbative effects that are too powerful or sophisticated for step-by-step approximation. This research sheds light on quantum phenomena and spatial geometry.

Cosmological Implications: de Sitter Space and Inflation

Information theory, condensed matter physics, and high-energy physics are the key scientific fields involved. The “holographic” view of the cosmos posits that spacetime emerges from quantum entanglement rather than being a basic “fabric.” The following are key study pillars:

An anti-de Sitter space conformal field theory-gravity duality: the AdS/CFT Correspondence.

The structure of spacetime can be understood using quantum entanglement and entropy.

Information Scrambling: The “butterfly effect” suggests that qubits, quantum events like black holes, can scramble information.

Future Research Roadmap

Even without a "Theory of Everything," this RMPS framework can guide future study. Future studies will focus on:

The Nature of Time: How previous gravitational history encodes the current quantum state.

Cosmic Inflation: Investigating whether long-distance correlations explain early universe matter distribution.

Quantum Error Correction: Comparing computational quantum coding to Wormhole Phase Transition mathematically.

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.

Matrix Product States

Matrix Product States: Unsung Heroes Solving quantum computing and classical simulation's Hardest Problems

Classical simulation methods are still needed to understand and advance quantum technologies. Matrix Product States (MPS) are a fundamental “workhorse” that solves many quantum mechanical issues with sophistication and efficiency. Recent advances show that they are vital for modelling quantum circuits and solving computational problems that were previously intractable, such as calculating symmetric group characters.

Compression and Canonical Forms in Matrix Products

In essence, Matrix Product States represent quantum states in one dimension. Describe a complex n-qubit quantum state. Because the whole state vector grows exponentially, even moderate-scale systems cannot be directly managed. MPS converts the quantum state into rank-3 tensors or matrices to solve this. Known for their “matrix product” shape, the states allow significant compression.

This compression relies on SVD. SVD may split any matrix into diagonal singular value, right-unitary, and left-unitary portions. Because they carry minimal information, small, solitary values and their vectors can be removed to compress data. This concept applies to quantum states and is like compressing an image by maintaining the most important values.

The bond dimension, represented by hyper-parameter, controls MPS approximation and entanglement. Setting a finite introduces an approximation error, a frequent trade-off to avoid exponentially large computational even though an MPS can accurately represent any state if permitted to be exponentially large.

A key feature of MPS is its canonical form. When SVD is used to generate a Matrix Product States from a dense state vector from left to right, “left-orthonormal” tensors are produced. Contracting an MPS site from the left yields simply an identity matrix. This simplifies several calculations, including the state norm.

MPS can be “right-canonical” or “mixed canonical” with tensors to the left and right of a site. This mixed form is useful for determining local observable expectation values since it streamlines computation to a single central tensor. The powerful Density Matrix Renormalisation Group (DMRG) algorithm uses this structure to find Hamiltonian ground states.

Entanglement and area laws are also linked to MPS. The von Neumann entanglement entropy of a subsystem is directly encoded by the singular values of the bonds in a 1D system. The area law for 1D systems limits the entanglement entropy for finite bond dimensions to a constant.

PEPS have more intricate algorithms and higher computing costs, but they are better for 2D or 3D systems and meet area rules. Due to its simplicity and strength, widely available Matrix Product States code is often a better alternative, especially for systems where local legislation is not strictly relevant.

Using MPS to Simulate Quantum Circuits

Due to the noise and cost of actual quantum hardware, Matrix Product States are becoming crucial for classically emulating quantum processes. PennyLane, an open-source quantum computing and machine learning software framework, offers MPS-based tools like the Default Tensor device.

Local gates are easy to apply to a canonical MPS. Splitting the tensor with SVD after contracting the gate's matrix with MPS physical indices restores the MPS structure. However, non-local gates are harder.

There are two main methods:

The Swap-Unswap Method involves applying the gate, “un-swapping” the target qubits back, and “swapping out” sites till they are adjacent.

Creating an MPO for the gate allows the Matrix Product Operator (MPO) Method to work on all sites, including identity operations on intermediate qubits. Choosing the appropriate contraction path for MPOs can be NP-hard, however toggling between virtual and physical index contractions works for MPS.

Setting the bond dimension is critical for Matrix Product States simulations. When working with systems too large for precise comparisons, finite-size scaling (bond dimension scaling) is employed to ensure simulation results are legitimate. This involves increasing bond size in the simulation and seeing if the results converge to an extrapolated value, like zero noise extrapolation.

Finite-size scaling gives a quantitative method for picking a bond dimension, but a qualitative result from a cheaper, lower bond dimension simulation is sometimes acceptable. With a bond dimension of 32, a H6 molecule simulation yielded exact and qualitative results.

Symmetric Group Characters and Computational Hardness

MPS are used to simulate quantum circuits to solve theoretical physics and pure mathematics problems. A Matrix Product States method for calculating symmetric group FF characters has been developed. Computing irreducible characters is at least as difficult as counting the solutions to an NP-hard problem in the worst case. This makes it unlikely to find an algorithm whose runtime grows polynomially with n.

“Symmetric Group Characters via Matrix Product States” describes the novel method, which computes an MPS that encodes all irreducible characters of a permutation. This is based on Crichigno and Prakash's new mapping from quantum spin chain characteristics. Letters can be viewed as quantum state amplitudes on 2n fermionic modes.

This paper presents a poly(n)-size quantum circuit and a new classical MPS approach for these features that numerical benchmarks show can compete with top computer algebra systems.

Although the Matrix Product States' bond dimension might expand exponentially, this quantum circuit is significant since the preparation circuit's size is polynomial. This allows effective quantum methods for sampling challenges based on symmetric group properties, such as integrals over the unitary group, kernel functions in machine learning, and fractional quantum Hall effect Laughlin states. Representation theory and combinatorics require Kostka numbers, which can be calculated using the approach.

In conclusion