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arXiv:1108.0531 · New Journal of Physics

A monomial matrix formalism to describe quantum many-body states

Maarten Van den Nest

Published Dec 1, 2011• 19 citations
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Abstract

We propose a framework to describe and simulate a class of many-body quantum states. We do so by considering joint eigenspaces of sets of monomial unitary matrices, called here ' M -spaces'; a unitary matrix is monomial if precisely one entry per row and column is nonzero. We show that M -spaces encompass various important state families, such as all Pauli stabilizer states and codes, the Affleck–Kennedy–Lieb–Tasaki (AKLT) model, Kitaev's (Abelian and non-Abelian) anyon models, group coset states, W states and the locally maximally entanglable states. We furthermore show how basic properties of M -spaces can be understood transparently by manipulating their monomial stabilizer groups. In particular, we derive a unified procedure to construct an eigenbasis of any M -space, yielding an explicit formula for each of the eigenstates. We also discuss the computational complexity of M -spaces and show that basic problems, such as estimating local expectation values, are NP-hard. Finally, we prove that a large subclass of M -spaces—containing, in particular, most of the aforementioned examples—can be simulated efficiently classically with a unified method.

DOI
10.1088/1367-2630/13/12/123004

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We propose a framework to describe and simulate a class of many-body quantum states. We do so by considering joint eigenspaces of sets of monomial unitary matrices, called here ' M -spaces'; a unitary matrix is monomial if precisely one entry per row and column is nonzero. We show that M -spaces encompass various important state families, such as all Pauli stabilizer states and codes, the Affleck–Kennedy–Lieb–Tasaki (AKLT) model, Kitaev's (Abelian and non-Abelian) anyon models, group coset states, W states and the locally maximally entanglable states. We furthermore show how basic properties of M -spaces can be understood transparently by manipulating their monomial stabilizer groups. In particular, we derive a unified procedure to construct an eigenbasis of any M -space, yielding an explicit formula for each of the eigenstates. We also discuss the computational complexity of M -spaces and show that basic problems, such as estimating local expectation values, are NP-hard. Finally, we prove that a large subclass of M -spaces—containing, in particular, most of the aforementioned examples—can be simulated efficiently classically with a unified method.
We propose a non-commutative extension of the Pauli stabilizer formalism. The aim is to describe a class of many-body quantum states which is richer than the standard Pauli stabilizer states. In our framework, stabilizer operators are tensor products of single-qubit operators drawn from the group 〈αI, X, S〉, where α = eiπ/4 and S = diag(1, i). We provide techniques to efficiently compute various properties related to bipartite entanglement, expectation values of local observables, preparation by means of quantum circuits, parent Hamiltonians, etc. We also highlight significant differences compared to the Pauli stabilizer formalism. In particular, we give examples of states in our formalism which cannot arise in the Pauli stabilizer formalism, such as topological models that support non-Abelian anyons.
We propose a non-commutative extension of the Pauli stabilizer for-malism. The aim is to describe a class of many-body quantum states which is richer than the standard Pauli stabilizer states. In our frame-work, stabilizer operators are tensor products of single-qubit operators drawn from the group 〈αI,X, S〉, where α = eipi/4 and S = diag(1, i). We provide techniques to efficiently compute various properties re-lated to bipartite entanglement, expectation values of local observ-ables, preparation by means of quantum circuits, parent Hamiltonians etc. We also highlight significant differences compared to the Pauli stabilizer formalism. In particular, we give examples of states in our formalism which cannot arise in the Pauli stabilizer formalism, such as topological models that support non-Abelian anyons. 1
The study of quantum computations that can be simulated efficiently classically is of interest for numerous reasons. On a fundamental level, such an investigation sheds light on the intrinsic computational power that is harnessed in quantum mechanics as compared to classical physics. More practically, understanding which quantum computations do not offer any speed-ups over classical computation provides insights in where (not) to look for novel quantum algorithmic primitives. On the other hand, classical simulation of many-body systems is a challenging task, as the dimension of the Hilbert space scales with the number of particles. Therefore, to understand the properties of the systems, suitable approximation methods need to be employed. The lectures will be divided into two parts. In the first part we discuss classical simulation of quantum computation from several perspectives. We review a number of well-known examples of classically simulatable quantum computations, such as the Gottesman-Knill theorem, matchgate simulation and tensor contraction methods. We further discuss simulation methods that are centred on classical sampling methods (‘weak simulation’), and illustrate how these techniques outperform methods that rely on the exact computation of measurement probabilities (‘strong simulation’). The second part focuses on "Entanglement and variational wavefunctions in quantum many body physics". We review the idea of entanglement in quantum many-body systems and how it helps us to understand the success of numerical renormalization group methods. In particular we will discuss a few variational wave-function based methods for simulating strongly correlated quantum systems, which include (1) matrix product states (2) multiscale entanglement renormalization ansatz (3) projected entangled pair states and (4) continuous matrix product states for quantum field theories.
In this paper we compute the entanglement, as quantified by negativity, between two blocks of length LA{L}_{A}LA​ and LB{L}_{B}LB​, separated by LLL sites in the one-dimensional spin-1 Affleck-Kennedy-Lieb-Tasaki (AKLT) model. We took the model with two different boundary conditions. We consider the case of NNN spin-1 particles in the bulk and one spin-1/2 particle at each boundary, which constitute a unique ground state, and the case of just spin 1, even at the end of the chain, where the degeneracy of the ground state is 4. In both scenarios we made a partition consisting of two blocks AAA and BBB, containing LA{L}_{A}LA​ and LB{L}_{B}LB​ sites, respectively. The separation of these two blocks is LLL. In both cases we explicitly obtain the reduced density matrix of the blocks AAA and BBB. We prove that the negativity in the first case vanishes identically for $L\ensuremath{\geqslant}1$, while in the second scenario it may approach a constant value N=1/2N=1/2N=1/2 for each degenerate eigenstate depending on the way one constructs these eigenstates. However, as there is some freedom in constructing these eigenstates, vanishing entanglement is also possible in the latter case. Additionally, we also compute the entanglement between noncomplementary blocks in the case of periodic boundary conditions for the spin-1 AKLT model for which there is a unique ground state. Even in this case, we find that the negativity of separated blocks of spins is zero.
Affleck-Kennedy-Lieb-Tasaki (AKLT) states are an important class of many-body quantum states that are useful in quantum information processing, including measurement-based quantum computation in particular. Here we propose a general approach for constructing efficient verification protocols for AKLT states on arbitrary graphs with local spin measurements. Our verification protocols build on bond verification protocols and matching covers (including edge coloring) of the underlying graphs, which have a simple geometric and graphic picture. We also provide rigorous performance guarantee that is required for practical applications. With our approach, most AKLT states of wide interest, including those defined on one- and two-dimensional lattices, can be verified with a constant sample cost, which is independent of the system size and is dramatically more efficient than all previous approaches. As an illustration, we construct concrete verification protocols for AKLT states on various lattices and on arbitrary graphs up to five vertices.
This paper explores the potential of utilizing entangled multi-body states for enhanced quantum simulation. Traditional quantum simulation methods face limitations when simulating complex quantum systems, primarily due to their inability to effectively represent multi-particle correlations. We propose a novel approach leveraging the inherent properties of entangled states to achieve a more accurate and efficient simulation of these systems. Specifically, we examine the construction of quantum computers incorporating multi-body entangled states and their application to simulating quantum field theories, a domain where complex many-body interactions are prevalent. The core idea centers on exploiting the correlations encoded within entangled states to reduce the computational complexity associated with simulating high-dimensional quantum systems. We present a theoretical framework outlining the key challenges and potential strategies for realizing this approach, focusing on the design of appropriate quantum circuits and the optimization of entanglement protocols. The ultimate goal is to demonstrate a pathway towards simulating intricate quantum phenomena that are currently intractable using conventional simulation techniques. The simulation method is based on mapping the problem to a quantum computer. The quantum computer performs the simulation based on the state of the qubits. The final result is the answer to the original problem. The key mathematical tool is the quantum circuit. The quantum circuit is a series of quantum gates applied to qubits. The quantum gates are unitary transformations. The unitary transformations are represented by matrices. The matrix is a square matrix. The matrix is a linear transformation. The linear transformation is a transformation that preserves linear relationships. The eigenvalues of the matrix are the solutions to the equation det(A - λI) = 0, where A is the matrix and λ is the eigenvalue and I is the identity matrix. The eigenvectors of the matrix are the vectors that are transformed to themselves by the matrix. The eigenvectors are the solutions to the equation (A - λI)v = 0, where A is the matrix, λ is the eigenvalue and v is the eigenvector. The eigenvalues and eigenvectors are important concepts in linear algebra.