This article needs additional citations for verification .(March 2011)
The time-evolving block decimation (TEBD) algorithm is a numerical scheme used to simulate one-dimensional quantum many-body systems, characterized by at most nearest-neighbour interactions. It is dubbed Time-evolving Block Decimation because it dynamically identifies the relevant low-dimensional Hilbert subspaces of an exponentially larger original Hilbert space. The algorithm, based on the Matrix Product States formalism, is highly efficient when the amount of entanglement in the system is limited, a requirement fulfilled by a large class of quantum many-body systems in one dimension.
This article is written like a personal reflection, personal essay, or argumentative essay that states a Wikipedia editor's personal feelings or presents an original argument about a topic.(October 2010)
Considering the inherent difficulties of simulating general quantum many-body systems, the exponential increase in parameters with the size of the system, and correspondingly, the high computational costs, one solution would be to look for numerical methods that deal with special cases, where one can profit from the physics of the system. The raw approach, by directly dealing with all the parameters used to fully characterize a quantum many-body system is seriously impeded by the lavishly exponential buildup with the system size of the amount of variables needed for simulation, which leads, in the best cases, to unreasonably long computational times and extended use of memory. To get around this problem a number of various methods have been developed and put into practice in the course of time, one of the most successful ones being the quantum Monte Carlo method (QMC). Also the density matrix renormalization group (DMRG) method, next to QMC, is a very reliable method, with an expanding community of users and an increasing number of applications to physical systems.
When the first quantum computer is plugged in and functioning, the perspectives for the field of computational physics will look rather promising, but until that day one has to restrict oneself to the mundane tools offered by classical computers. While experimental physicists are putting a lot of effort in trying to build the first quantum computer, theoretical physicists are searching, in the field of quantum information theory (QIT), for genuine quantum algorithms, appropriate for problems that would perform badly when trying to be solved on a classical computer, but pretty fast and successful on a quantum one. The search for such algorithms is still going, the best-known (and almost the only ones found) being the Shor's algorithm, for factoring large numbers, and Grover's search algorithm.
In the field of QIT one has to identify the primary resources necessary for genuine quantum computation. Such a resource may be responsible for the speedup gain in quantum versus classical, identifying them means also identifying systems that can be simulated in a reasonably efficient manner on a classical computer. Such a resource is quantum entanglement; hence, it is possible to establish a distinct lower bound for the entanglement needed for quantum computational speedups.
Guifré Vidal, then at the Institute for Quantum Information, Caltech, has recently proposed a scheme useful for simulating a certain category of quantumsystems. He asserts that "any quantum computation with pure states can be efficiently simulated with a classical computer provided the amount of entanglement involved is sufficiently restricted". This happens to be the case with generic Hamiltonians displaying local interactions, as for example, Hubbard-like Hamiltonians. The method exhibits a low-degree polynomial behavior in the increase of computational time with respect to the amount of entanglement present in the system. The algorithm is based on a scheme that exploits the fact that in these one-dimensional systems the eigenvalues of the reduced density matrix on a bipartite split of the system are exponentially decaying, thus allowing us to work in a re-sized space spanned by the eigenvectors corresponding to the eigenvalues we selected.
One can also estimate the amount of computational resources required for the simulation of a quantum system on a classical computer, knowing how the entanglement contained in the system scales with the size of the system. The classically (and quantum, as well) feasible simulations are those that involve systems only lightly entangled—the strongly entangled ones being, on the other hand, good candidates only for genuine quantum computations.
The numerical method is efficient in simulating real-time dynamics or calculations of ground states using imaginary-time evolution or isentropic interpolations between a target Hamiltonian and a Hamiltonian with an already-known ground state. The computational time scales linearly with the system size, hence many-particles systems in 1D can be investigated.
A useful feature of the TEBD algorithm is that it can be reliably employed for time evolution simulations of time-dependent Hamiltonians, describing systems that can be realized with cold atoms in optical lattices, or in systems far from equilibrium in quantum transport. From this point of view, TEBD had a certain ascendance over DMRG, a very powerful technique, but until recently not very well suited for simulating time-evolutions. With the Matrix Product States formalism being at the mathematical heart of DMRG, the TEBD scheme was adopted by the DMRG community, thus giving birth to the time dependent DMRG [ permanent dead link ], t-DMRG for short.
Around the same time, other groups have developed similar approaches in which quantum information plays a predominant role as, for example, in DMRG implementations for periodic boundary conditions , and for studying mixed-state dynamics in one-dimensional quantum lattice systems,.Those last approaches actually provide a formalism that is more general than the original TEBD approach, as it also allows to deal with evolutions with matrix product operators; this enables the simulation of nontrivial non-infinitesimal evolutions as opposed to the TEBD case, and is a crucial ingredient to deal with higher-dimensional analogues of matrix product states.
Consider a chain of N qubits, described by the function . The most natural way of describing would be using the local -dimensional basis :
where M is the on-site dimension.
The trick of TEBD is to re-write the coefficients :
This form, known as a Matrix product state, simplifies the calculations greatly.
To understand why, one can look at the Schmidt decomposition of a state, which uses singular value decomposition to express a state with limited entanglement more simply.
Consider the state of a bipartite system . Every such state can be represented in an appropriately chosen basis as:
where are formed with vectors that make an orthonormal basis in and, correspondingly, vectors , which form an orthonormal basis in , with the coefficients being real and positive, . This is called the Schmidt decomposition (SD) of a state. In general the summation goes up to . The Schmidt rank of a bipartite split is given by the number of non-zero Schmidt coefficients. If the Schmidt rank is one, the split is characterized by a product state. The vectors of the SD are determined up to a phase and the eigenvalues and the Schmidt rank are unique.
For example, the two-qubit state:
has the following SD:
On the other hand, the state:
is a product state:
At this point we know enough to try to see how we explicitly build the decomposition (let's call it D).
Consider the bipartite splitting . The SD has the coefficients and eigenvectors . By expanding the 's in the local basis, one can write:
The process can be decomposed in three steps, iterated for each bond (and, correspondingly, SD) in the chain:
Step 1: express the 's in a local basis for qubit 2:
The vectors are not necessarily normalized.
Step 2: write each vector in terms of the at most (Vidal's emphasis) Schmidt vectors and, correspondingly, coefficients :
Step 3: make the substitutions and obtain:
Repeating the steps 1 to 3, one can construct the whole decomposition of state D. The last 's are a special case, like the first ones, expressing the right-hand Schmidt vectors at the bond in terms of the local basis at the lattice place. As shown in, it is straightforward to obtain the Schmidt decomposition at bond, i.e. , from D.
The Schmidt eigenvalues, are given explicitly in D:
The Schmidt eigenvectors are simply:
Now, looking at D, instead of initial terms, there are . Apparently this is just a fancy way of rewriting the coefficients , but in fact there is more to it than that. Assuming that N is even, the Schmidt rank for a bipartite cut in the middle of the chain can have a maximal value of ; in this case we end up with at least coefficients, considering only the ones, slightly more than the initial ! The truth is that the decomposition D is useful when dealing with systems that exhibit a low degree of entanglement, which fortunately is the case with many 1D systems, where the Schmidt coefficients of the ground state decay in an exponential manner with :
Therefore, it is possible to take into account only some of the Schmidt coefficients (namely the largest ones), dropping the others and consequently normalizing again the state:
where is the number of kept Schmidt coefficients.
Let's get away from this abstract picture and refresh ourselves with a concrete example, to emphasize the advantage of making this decomposition. Consider for instance the case of 50 fermions in a ferromagnetic chain, for the sake of simplicity. A dimension of 12, let's say, for the would be a reasonable choice, keeping the discarded eigenvalues at % of the total, as shown by numerical studies, meaning roughly coefficients, as compared to the originally ones.
Even if the Schmidt eigenvalues don't have this exponential decay, but they show an algebraic decrease, we can still use D to describe our state . The number of coefficients to account for a faithful description of may be sensibly larger, but still within reach of eventual numerical simulations.
One can proceed now to investigate the behaviour of the decomposition D when acted upon with one-qubit gates (OQG) and two-qubit gates (TQG) acting on neighbouring qubits. Instead of updating all the coefficients , we will restrict ourselves to a number of operations that increase in as a polynomial of low degree, thus saving computational time.
The OQGs are affecting only the qubit they are acting upon, the update of the state after a unitary operator at qubit k does not modify the Schmidt eigenvalues or vectors on the left, consequently the 's, or on the right, hence the 's. The only 's that will be updated are the 's (requiring only at most operations), as
The changes required to update the 's and the 's, following a unitary operation V on qubits k, k+1, concern only , and . They consist of a number of basic operations.
Following Vidal's original approach, can be regarded as belonging to only four subsystems:
The subspace J is spanned by the eigenvectors of the reduced density matrix :
In a similar way, the subspace K is spanned by the eigenvectors of the reduced density matrix:
The subspaces and belong to the qubits k and k + 1. Using this basis and the decomposition D, can be written as:
Using the same reasoning as for the OQG, the applying the TQG V to qubits k, k + 1 one needs only to update
We can write as:
To find out the new decomposition, the new 's at the bond k and their corresponding Schmidt eigenvectors must be computed and expressed in terms of the 's of the decomposition D. The reduced density matrix is therefore diagonalized:
The square roots of its eigenvalues are the new 's. Expressing the eigenvectors of the diagonalized matrix in the basis : the 's are obtained as well:
From the left-hand eigenvectors,
after expressing them in the basis , the 's are:
The dimension of the largest tensors in D is of the order ; when constructing the one makes the summation over , and for each , adding up to a total of operations. The same holds for the formation of the elements , or for computing the left-hand eigenvectors , a maximum of , respectively basic operations. In the case of qubits, , hence its role is not very relevant for the order of magnitude of the number of basic operations, but in the case when the on-site dimension is higher than two it has a rather decisive contribution.
The numerical simulation is targeting (possibly time-dependent) Hamiltonians of a system of particles arranged in a line, which are composed of arbitrary OQGs and TQGs:
It is useful to decompose as a sum of two possibly non-commuting terms, , where
Any two-body terms commute: , This is done to make the Suzuki–Trotter expansion (ST) of the exponential operator, named after Masuo Suzuki and Hale Trotter.
The Suzuki–Trotter expansion of the first order (ST1) represents a general way of writing exponential operators:
The correction term vanishes in the limit
For simulations of quantum dynamics it is useful to use operators that are unitary, conserving the norm (unlike power series expansions), and there's where the Trotter-Suzuki expansion comes in. In problems of quantum dynamics the unitarity of the operators in the ST expansion proves quite practical, since the error tends to concentrate in the overall phase, thus allowing us to faithfully compute expectation values and conserved quantities. Because the ST conserves the phase-space volume, it is also called a symplectic integrator.
The trick of the ST2 is to write the unitary operators as:
where . The number is called the Trotter number.
The operators , are easy to express, as:
since any two operators , (respectively, ,) commute for and an ST expansion of the first order keeps only the product of the exponentials, the approximation becoming, in this case, exact.
The time-evolution can be made according to
For each "time-step" , are applied successively to all odd sites, then to the even ones, and again to the odd ones; this is basically a sequence of TQG's, and it has been explained above how to update the decomposition when applying them.
Our goal is to make the time evolution of a state for a time T, towards the state using the n-particle Hamiltonian .
It is rather troublesome, if at all possible, to construct the decomposition for an arbitrary n-particle state, since this would mean one has to compute the Schmidt decomposition at each bond, to arrange the Schmidt eigenvalues in decreasing order and to choose the first and the appropriate Schmidt eigenvectors. Mind this would imply diagonalizing somewhat generous reduced density matrices, which, depending on the system one has to simulate, might be a task beyond our reach and patience. Instead, one can try to do the following:
i) construct the decomposition for a simple initial state, let us say, some product state , for which the decomposition is straightforward.
ii) relate to the ground state of a Hamiltonian by a sufficiently local transformation Q (one that can be expressed as a product of TQGs, for example)
iii) make an imaginary-time evolution towards the ground state of the Hamiltonian , according to:
or, alternatively, simulate an isentropic evolution using a time-dependent Hamiltonian, which interpolates between the Hamiltonian , which has the product state as its ground state, and the Hamiltonian ; the evolution must be done slow enough, such that the system is always in the ground state or, at least, very close to it.
iv)finally, make the time-evolution of the state towards using the Hamiltonian :
The errors in the simulation are resulting from the Suzuki–Trotter approximation and the involved truncation of the Hilbert space.
In the case of a Trotter approximation of order, the error is of order . Taking into account steps, the error after the time T is:
The unapproximated state is:
where is the state kept after the Trotter expansion and accounts for the part that is neglected when doing the expansion.
The total error scales with time as:
The Trotter error is independent of the dimension of the chain.
Considering the errors arising from the truncation of the Hilbert space comprised in the decomposition D, they are twofold.
First, as we have seen above, the smallest contributions to the Schmidt spectrum are left away, the state being faithfully represented up to:
where is the sum of all the discarded eigenvalues of the reduced density matrix, at the bond . The state is, at a given bond , described by the Schmidt decomposition:
is the state kept after the truncation and
is the state formed by the eigenfunctions corresponding to the smallest, irrelevant Schmidt coefficients, which are neglected. Now, because they are spanned by vectors corresponding to orthogonal spaces. Using the same argument as for the Trotter expansion, the error after the truncation is:
After moving to the next bond, the state is, similarly:
The error, after the second truncation, is:
and so on, as we move from bond to bond.
The second error source enfolded in the decomposition is more subtle and requires a little bit of calculation.
As we calculated before, the normalization constant after making the truncation at bond is:
Now let us go to the bond and calculate the norm of the right-hand Schmidt vectors ; taking into account the full Schmidt dimension, the norm is:
Taking into account the truncated space, the norm is:
Taking the difference, , we get:
Hence, when constructing the reduced density matrix, the trace of the matrix is multiplied by the factor:
The total truncation error, considering both sources, is upper bounded by:
When using the Trotter expansion, we do not move from bond to bond, but between bonds of same parity; moreover, for the ST2, we make a sweep of the even ones and two for the odd. But nevertheless, the calculation presented above still holds. The error is evaluated by successively multiplying with the normalization constant, each time we build the reduced density matrix and select its relevant eigenvalues.
One thing that can save a lot of computational time without loss of accuracy is to use a different Schmidt dimension for each bond instead of a fixed one for all bonds, keeping only the necessary amount of relevant coefficients, as usual. For example, taking the first bond, in the case of qubits, the Schmidt dimension is just two. Hence, at the first bond, instead of futilely diagonalizing, let us say, 10 by 10 or 20 by 20 matrices, we can just restrict ourselves to ordinary 2 by 2 ones, thus making the algorithm generally faster. What we can do instead is set a threshold for the eigenvalues of the SD, keeping only those that are above the threshold.
TEBD also offers the possibility of straightforward parallelization due to the factorization of the exponential time-evolution operator using the Suzuki–Trotter expansion. A parallel-TEBD has the same mathematics as its non-parallelized counterpart, the only difference is in the numerical implementation.
In quantum field theory, the Dirac spinor is the spinor that describes all known fundamental particles that are fermions, with the possible exception of neutrinos. It appears in the plane-wave solution to the Dirac equation, and is a certain combination of two Weyl spinors, specifically, a bispinor that transforms "spinorially" under the action of the Lorentz group.
In the general theory of relativity, the Einstein field equations relate the geometry of spacetime to the distribution of matter within it.
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a principal connection on the frame bundle – see affine connection. In the special case of a manifold isometrically embedded into a higher-dimensional Euclidean space, the covariant derivative can be viewed as the orthogonal projection of the Euclidean directional derivative onto the manifold's tangent space. In this case the Euclidean derivative is broken into two parts, the extrinsic normal component and the intrinsic covariant derivative component.
In physics, the S-matrix or scattering matrix relates the initial state and the final state of a physical system undergoing a scattering process. It is used in quantum mechanics, scattering theory and quantum field theory (QFT).
In quantum mechanics, the Hellmann–Feynman theorem relates the derivative of the total energy with respect to a parameter, to the expectation value of the derivative of the Hamiltonian with respect to that same parameter. According to the theorem, once the spatial distribution of the electrons has been determined by solving the Schrödinger equation, all the forces in the system can be calculated using classical electrostatics.
In particle physics, neutral particle oscillation is the transmutation of a particle with zero electric charge into another neutral particle due to a change of a non-zero internal quantum number, via an interaction that does not conserve that quantum number. Neutral particle oscillations were first investigated in 1954 by Murray Gell-mann and Abraham Pais.
In theoretical physics, the Wess–Zumino model has become the first known example of an interacting four-dimensional quantum field theory with linearly realised supersymmetry. In 1974, Julius Wess and Bruno Zumino studied, using modern terminology, dynamics of a single chiral superfield whose cubic superpotential leads to a renormalizable theory.
In quantum field theory and statistical mechanics, loop integrals are the integrals which appear when evaluating the Feynman diagrams with one or more loops by integrating over the internal momenta. These integrals are used to determine counterterms, which in turn allow evaluation of the beta function, which encodes the dependence of coupling for an interaction on an energy scale .
The Newman–Penrose (NP) formalism is a set of notation developed by Ezra T. Newman and Roger Penrose for general relativity (GR). Their notation is an effort to treat general relativity in terms of spinor notation, which introduces complex forms of the usual variables used in GR. The NP formalism is itself a special case of the tetrad formalism, where the tensors of the theory are projected onto a complete vector basis at each point in spacetime. Usually this vector basis is chosen to reflect some symmetry of the spacetime, leading to simplified expressions for physical observables. In the case of the NP formalism, the vector basis chosen is a null tetrad: a set of four null vectors—two real, and a complex-conjugate pair. The two real members asymptotically point radially inward and radially outward, and the formalism is well adapted to treatment of the propagation of radiation in curved spacetime. The Weyl scalars, derived from the Weyl tensor, are often used. In particular, it can be shown that one of these scalars— in the appropriate frame—encodes the outgoing gravitational radiation of an asymptotically flat system.
In the Newman–Penrose (NP) formalism of general relativity, Weyl scalars refer to a set of five complex scalars which encode the ten independent components of the Weyl tensor of a four-dimensional spacetime.
The theoretical and experimental justification for the Schrödinger equation motivates the discovery of the Schrödinger equation, the equation that describes the dynamics of nonrelativistic particles. The motivation uses photons, which are relativistic particles with dynamics described by Maxwell's equations, as an analogue for all types of particles.
In mathematics, the Schur orthogonality relations, which were proven by Issai Schur through Schur's lemma, express a central fact about representations of finite groups. They admit a generalization to the case of compact groups in general, and in particular compact Lie groups, such as the rotation group SO(3).
In 1927, a year after the publication of the Schrödinger equation, Hartree formulated what are now known as the Hartree equations for atoms, using the concept of self-consistency that Lindsay had introduced in his study of many electron systems in the context of Bohr theory. Hartree assumed that the nucleus together with the electrons formed a spherically symmetric field. The charge distribution of each electron was the solution of the Schrödinger equation for an electron in a potential , derived from the field. Self-consistency required that the final field, computed from the solutions, was self-consistent with the initial field, and he thus called his method the self-consistent field method.
In general relativity, a point mass deflects a light ray with impact parameter by an angle approximately equal to
Learning with errors (LWE) is the computational problem of inferring a linear -ary function over a finite ring from given samples some of which may be erroneous. The LWE problem is conjectured to be hard to solve, and thus to be useful in cryptography.
An LC circuit can be quantized using the same methods as for the quantum harmonic oscillator. An LC circuit is a variety of resonant circuit, and consists of an inductor, represented by the letter L, and a capacitor, represented by the letter C. When connected together, an electric current can alternate between them at the circuit's resonant frequency:
Overcompleteness is a concept from linear algebra that is widely used in mathematics, computer science, engineering, and statistics. It was introduced by R. J. Duffin and A. C. Schaeffer in 1952.
Coherent states have been introduced in a physical context, first as quasi-classical states in quantum mechanics, then as the backbone of quantum optics and they are described in that spirit in the article Coherent states. However, they have generated a huge variety of generalizations, which have led to a tremendous amount of literature in mathematical physics. In this article, we sketch the main directions of research on this line. For further details, we refer to several existing surveys.
In the ADM formulation of general relativity one splits spacetime into spatial slices and time, the basic variables are taken to be the induced metric, , on the spatial slice, and its conjugate momentum variable related to the extrinsic curvature, ,. These are the metric canonical coordinates.
In theoretical physics, more specifically in quantum field theory and supersymmetry, supersymmetric Yang–Mills, also known as super Yang–Mills and abbreviated to SYM, is a supersymmetric generalization of Yang–Mills theory, which is a gauge theory that plays an important part in the mathematical formulation of forces in particle physics.