The simulation of quantum systems is a task for which quantum computers are believed to give an exponential speed up as compared with classical ones. While ground states of one-dimensional systems can be efficiently approximated using matrix product states (MPS), their time evolution can encode quantum computations, so that simulating the latter should be hard classically. However, one might believe that for systems with high enough symmetry, and thus insufficient parameters to encode a quantum computation, efficient classical simulation is possible. We discuss supporting evidence to the contrary: we provide a rigorous proof of the observation that a time-independent local Hamiltonian can yield a linear increase of the entropy when acting on a product state in a translational invariant framework. This criterion has to be met by any classical simulation method, which in particular implies that every global approximation of the evolution requires exponential resources for any MPS-based method.
The simulation of quantum systems is a task for which quantum computers are believed to give an exponential speed up as compared with classical ones. While ground states of one-dimensional systems can be efficiently approximated using matrix product states (MPS), their time evolution can encode quantum computations, so that simulating the latter should be hard classically. However, one might believe that for systems with high enough symmetry, and thus insufficient parameters to encode a quantum computation, efficient classical simulation is possible. We discuss supporting evidence to the contrary: we provide a rigorous proof of the observation that a time-independent local Hamiltonian can yield a linear increase of the entropy when acting on a product state in a translational invariant framework. This criterion has to be met by any classical simulation method, which in particular implies that every global approximation of the evolution requires exponential resources for any MPS-based method.
We investigate the relation between the scaling of block entropies and the efficient simulability by matrix product states (MPSs) and clarify the connection both for von Neumann and Rényi entropies. Most notably, even states obeying a strict area law for the von Neumann entropy are not necessarily approximable by MPSs. We apply these results to illustrate that quantum computers might outperform classical computers in simulating the time evolution of quantum systems, even for completely translational invariant systems subject to a time-independent Hamiltonian.
Classical simulation of quantum many-body systems is in general a challenging problem for the simple reason that the dimension of the Hilbert space grows exponentially with the system size. In particular, merely encoding a generic quantum many-body state requires an exponential number of bits. However, condensed matter physicists are mostly interested in local Hamiltonians and especially their ground states, which are highly non-generic. Thus, we might hope that at least some physical systems allow efficient classical simulation.Starting with one-dimensional (1D) quantum systems (i.e., the simplest nontrivial case), the first basic question is: Which classes of states have efficient classical representations? It turns out that this question is quantitatively related to the amount of entanglement in the state, for states with ``little entanglement'' are well approximated by matrix product states (a data structure that can be manipulated efficiently on a classical computer). At a technical level, the mathematical notion for ``little entanglement'' is area law, which has been proved for unique ground states in 1D gapped systems. We establish an area law for constant-fold degenerate ground states in 1D gapped systems and thus explain the effectiveness of matrix-product-state methods in (e.g.) symmetry breaking phases. This result might not be intuitively trivial as degenerate ground states in gapped systems can be long-range correlated.Suppose an efficient classical representation exists. How can one find it efficiently? The density matrix renormalization group is the leading numerical method for computing ground states in 1D quantum systems. However, it is a heuristic algorithm and the possibility that it may fail in some cases cannot be completely ruled out. Recently, a provably efficient variant of the density matrix renormalization group has been developed for frustration-free 1D gapped systems. We generalize this algorithm to all (i.e., possibly frustrated) 1D gapped systems. Note that the ground-state energy of 1D gapless Hamiltonians is computationally intractable even in the presence of translational invariance.It is tempting to extend methods and tools in 1D to two and higher dimensions (2+D), e.g., matrix product states are generalized to tensor network states. Since an area law for entanglement (if formulated properly) implies efficient matrix product state representations in 1D, an interesting question is whether a similar implication holds in 2+D. Roughly speaking, we show that an area law for entanglement (in any reasonable formulation) does not always imply efficient tensor network representations of the ground states of 2+D local Hamiltonians even in the presence of translational invariance. It should be emphasized that this result does not contradict with the common sense that in practice quantum states with more entanglement usually require more space to be stored classically; rather, it demonstrates that the relationship between entanglement and efficient classical representations is still far from being well understood.Excited eigenstates participate in the dynamics of quantum systems and are particularly relevant to the phenomenon of many-body localization (absence of transport at finite temperature in strongly correlated systems). We study the entanglement of excited eigenstates in random spin chains and expect that its singularities coincide with dynamical quantum phase transitions. This expectation is confirmed in the disordered quantum Ising chain using both analytical and numerical methods.Finally, we study the problem of generating ground states (possibly with topological order) in 1D gapped systems using quantum circuits. This is an interesting problem both in theory and in practice. It not only characterizes the essential difference between the entanglement patterns that give rise to trivial and nontrivial topological order, but also quantifies the difficulty of preparing quantum states with a quantum computer (in experiments).
The dynamics of one-dimensional quantum many-body systems is often numerically simulated with matrix-product states (MPSs). The computational complexity of MPS methods is known to be related to the growth of entropies of reduced density matrices for bipartitions of the chain. While for closed systems the entropy relevant for the complexity is uniquely defined by the entanglement entropy, for open systems it depends on the choice of the representation. Here, we systematically compare the growth of different entropies relevant to the complexity of matrix-product representations in open-system simulations. We simulate an XXZ spin-1/2 chain in the presence of spontaneous emission and absorption and dephasing. We compare simulations using a representation of the full density matrix as a matrix-product density operator (MPDO) with a quantum trajectory unraveling, where each trajectory is itself represented by an MPS (QT + MPS). We show that the bipartite entropy in the MPDO description (operator entanglement, OE) generally scales more favorably with time than the entropy in QT + MPS (trajectory entanglement, TE): (i) For spontaneous emission and absorption the OE vanishes while the TE grows and reaches a constant value for large dissipative rates and sufficiently long times; (ii) for dephasing the OE exhibits only logarithmic growth while the TE grows polynomially. Although QT + MPS requires a smaller local state space, the more favorable entropy growth can thus make MPDO simulations fundamentally more efficient than QT + MPS. Furthermore, MPDO simulations allow for easier exploitation of higher-order Trotter decompositions and translational invariance, allowing for larger time steps and system sizes.
In contrast to classical many-particle systems for which the number of degrees of freedom grows linearly with the system size, it grows exponentially for quantum many-particle systems. On the one hand, the resulting complexity of the quantum systems has a high potential for technological advances if it can be exploited directly. For example, in recent years, the fundamentals for the realization of quantum computers and their possible applications have been investigated intensely. On the other hand, the high complexity, especially in strongly correlated regimes, often makes the analytical or numerical analysis of underlying physical phenomena very difficult. This is why some systems of high technological relevance, like the high-temperature superconductors, are still unsatisfactorily understood. A main topic of this thesis is the development of efficient numerical methods for the simulation of strongly correlated quantum lattice models. For one-dimensional systems, the density-matrix renormalization-group (DMRG) is such a very successful method. The physical states of interest are approximated within a certain class of ansatz states. These ansatz states are designed in a way that the number of degrees of freedom are prevented from growing exponentially. With the DMRG method, ground states of interacting one-dimensional systems can often be determined with very high precision. This observation indicates already that ground states actually do not exhaust the available exponential number of degrees of freedom but can be described efficiently with much fewer parameters. The ansatz states used in DMRG are the so-called matrix product states. A very useful criterion for the construction of corresponding appropriate ansatz states for higher-dimensional systems is the quantum nonlocality as for example quantified by the entanglement entropy. The ansatz states should be able to reproduce the scaling properties of the quantum nonlocality in the physical states of interest with as few parameters as possible. The first part of the thesis, therefore, provides analytical and numerical analysis of the scaling of quantum nonlocality with the system size or time in different, physically relevant scenarios. For example, the scaling of Renyi entropies and their dependence on boundary conditions is derived within the 1+1-dimensional conformal field theory. Conjectures and analytical indications concerning the properties of entanglement entropy in critical fermionic and bosonic systems are confirmed numerically with high precision. For integrable models in the thermodynamic limit, general preconditions are derived under which subsystems converge to steady states. These steady states are non-thermal and retain information about the initial state. It is shown that the entanglement entropy in such steady states is extensive. For short times, the entanglement entropy grows typically linearly with time, causing an exponential increase in computation costs for the DMRG method. Consequently, time-dependent DMRG simulations usually have no direct access to the long time regime. The second part of the thesis focuses on the development and improvement of the above-mentioned numerical techniques. The time-dependent DMRG is complemented with an extrapolation technique for the evaluated observables. In this way, the problem of the entropy increase can be circumvented, allowing for a precise determination of spectral functions. The method is demonstrated using the example of the Heisenberg antiferromagnet and results are compared to Bethe-Ansatz data for T=0 and quantum Monte Carlo data for T>0. For the simulation of higher-dimensional systems, projected-entangled pair-states (PEPS) and the multiscale entanglement renormalization ansatz (MERA) turn out to be appropriate wavefunction classes. Initially, they could only be used for spin systems. Within this thesis, variants of those state classes are being suggested for the fermionic case. It is shown how the known algorithms for spin systems can be translated into corresponding algorithms for the fermionic systems. The occurring computation costs increase only by a marginal overhead. First numerical benchmarking tests proceeded successfully. Thus, new techniques are available which can for example be used to examine the two-dimensional Hubbard model. This model is a candidate for the description of high-temperature superconductivity.
Simulating quantum many-body systems (QMBS) is one of the long-standing, highly non-trivial challenges in condensed matter physics and quantum information due to the exponentially growing size of the system's Hilbert space. To date, tensor networks have been an essential tool for studying such quantum systems, owing to their ability to efficiently capture the entanglement properties of the systems they represent. One of the well-known tensor network architectures, namely matrix product states (MPS), is the standard method for simulating one-dimensional QMBS. Here, we propose a simple, yet efficient, method to augment the already available MPS algorithms to simulate the dynamics of time-dependent Hamiltonians with better accuracy and a faster convergence rate, giving a second-order convergence compared to the first-order convergence of the standard method. We apply our proposed method to simulate the dynamics of a chain of single spins associated with nitrogen-vacancy color centers in diamonds, which has potential applications for practical and scalable quantum technologies, and find that our method improves the average error for a system of few NV centers by a factor of about 1000 for moderate step sizes. Our work paves the way for efficient simulation of QMBS under the influence of time-dependent Hamiltonians.
We study the computational complexity of simulating the time-dependent expectation value of a local operator in a one-dimensional quantum system by using temporal matrix product states(MPSs) and argue that it is intimately related to that of encoding temporal transition matrices and their partial traces. We show that we can upper-bound the rank of these reduced transition matrices by one of the Heisenberg evolution of local operators, thus making aconnection between two apparently different quantities, the temporal entanglement and the local operator entanglement(OE). As a result, whenever the local OE grows slower than linearly in time, we show that computing time-dependent expectation values of local operators using temporal MPSs is likely advantageous with respect to computing the same quantities using standard MPS techniques. Published by the American Physical Society 2024