Incorporating conservation laws explicitly into matrix product states (MPSs) has proven to make numerical simulations of quantum many-body systems much less resource consuming. We will discuss here to what extent this concept can be used in simulation where the dynamically evolving entities are matrix product operators (MPOs). Quite counter-intuitively the expectation of gaining in speed by sacrificing information about all but a single symmetry sector is not in all cases fulfilled. It turns out that often the entanglement imposed by the global constraint of fixed particle number is the limiting factor.
Incorporating conservation laws explicitly into matrix product states (MPSs) has proven to make numerical simulations of quantum many-body systems much less resource consuming. We will discuss here to what extent this concept can be used in simulation where the dynamically evolving entities are matrix product operators (MPOs). Quite counter-intuitively the expectation of gaining in speed by sacrificing information about all but a single symmetry sector is not in all cases fulfilled. It turns out that often the entanglement imposed by the global constraint of fixed particle number is the limiting factor.
The eigenvectors of the particle number operator in second quantization are characterized by the block sparsity of their matrix product state representations. This is shown to generalize to other classes of operators. Imposing block sparsity yields a scheme for conserving the particle number that is commonly used in applications in physics. Operations on such block structures, their rank truncation, and implications for numerical algorithms are discussed. Explicit and rank-reduced matrix product operator representations of one- and two-particle operators are constructed that operate only on the non-zero blocks of matrix product states.
Matrix product states (MPSs) and matrix product operators (MPOs) have been proven to be a powerful tool to study quantum many-body systems but are restricted to moderately entangled states as the number of parameters scales exponentially with the entanglement entropy. While MPSs can efficiently find ground states of one-dimensional systems, their capacities are limited when simulating their dynamics, where the entanglement can increase ballistically with time. On the other hand, quantum devices appear as a natural platform to encode and perform the time evolution of correlated many-body states. However, accessing the regime of long-time dynamics is hampered by quantum noise. In this paper we use the best of worlds: the short-time dynamics is efficiently performed by MPSs, compiled into short-depth quantum circuits, and performed further in time on a quantum computer thanks to efficient MPO-optimized quantum circuits. We quantify the capacities of this hybrid classical-quantum scheme in terms of fidelities taking into account a noise model. We show that using classical knowledge in the form of tensor networks provides a way to better use limited quantum resources and lowers the noise requirements to reach a practical quantum advantage. Finally, we successfully demonstrate our approach with an experimental realization of the technique. Combined with efficient circuit transpilation we simulate a ten-qubit system on an actual quantum device over a longer time scale than low-bond-dimension MPSs and purely quantum Trotter evolution.
Abstract The eigenvectors of the particle number operator in second quantization are characterized by the block sparsity of their matrix product state representations. This is shown to generalize to other classes of operators. Imposing block sparsity yields a scheme for conserving the particle number that is commonly used in applications in physics. Operations on such block structures, their rank truncation, and implications for numerical algorithms are discussed. Explicit and rank-reduced matrix product operator representations of one- and two-particle operators are constructed that operate only on the non-zero blocks of matrix product states.
Tensor networks [1] and in particular matrix-product states [2] have proven extremely useful for the investigation of quantum many-body states, such as, e.g., ground states of local many-body Hamiltonians. They provide an efficient description of these states, avoiding the exponential increase of Hilbert space with particle number at the cost of limiting the entanglement. In particular for matrix-product states, the relation between area laws of Rényi entanglement entropies and efficient simulability of quantum states is nowadays well understood [3]. However, it is not always clear a priori whether or not the relevant states of a given quantum system (e.g., its eigenstates) fulfil such an area law and can thus efficiently be represented by matrix-product states. By investigating specific states of relevance for many-body quantum dynamics, in particular symmetric states and Fock states, we here aim to derive criteria beyond the area law to assess for which quantum systems a description by matrix-product states or more general tensor networks is beneficial. [1] S. Montangero, Introduction to Tensor Networks (Springer, Cham, 2018) [2] J. I. Cirac, D. Pérez-García, N. Schuch, and F. Verstraete, Rev. Mod. Phys 93, 045003 (2021) [3] N. Schuch, M. W. Wolf, F. Verstraete, and J. I. Cirac, Phys. Rev. Lett. 100, 030504 (2008)
The Gutzwiller variational wavefunction (GVW) is commonly employed to capture correlation effects in condensed matter systems such as ferromagnets, ultracold bosonic gases, correlated superconductors, etc. By noticing that the grand-canonical and number-conserving Gutzwiller Ansätze are in fact the zero-order approximation of an expansion in the truncation parameter of a Matrix Product State (MPS), we argue that MPSs, and the algorithms used to operate on them, are not only flexible computational tools but also a unifying theoretical framework that can be used to generalize and improve on the GVW. In fact, we show that a number-conserving GVW is less efficient in capturing the ground state of a quantum system than a more general MPS which can be optimized with comparable computational resources. Moreover, we suggest a corrected time-dependent density matrix renormalization group algorithm that ensures the conservation of the expectation value of the number of particles when a GVW or a MPS are not explicitly number-conserving. The GVW dynamics obtained with our algorithm compares very well with the exact one in 1D. Most importantly, the algorithm works in any dimension for a GVW. We thus expect it to be of great value in the study of the dynamics of correlated quantum systems.
Tensor networks like matrix product states (MPSs) and matrix product operators (MPOs) are powerful tools for representing exponentially large states and operators, with applications in quantum many-body physics, machine learning, numerical analysis, and other areas. In these applications, computing a compressed representation of the MPO-MPS product is a fundamental computational primitive. For this operation, this paper introduces a new single-pass, randomized algorithm, called successive randomized compression (SRC), that improves on existing approaches in speed or in accuracy. The performance of the new algorithm is evaluated on synthetic problems and unitary time evolution problems for quantum spin systems.