#singularvaluedecomposition

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