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AIP conference proceedings

Quantum Circuit Simulation Using the Density Matrix Renormalization Group

A. Kawaguchi

Published Jan 1, 2004• 0 citations
Publisher

Abstract

Simulating large quantum circuits with a classical computer requires enormous computational power, because the number of quantum states increases exponentially as a function of number of qubits. The density matrix renormalization group (DMRG) was introduced by White to study the properties of relatively large‐scale one‐dimensional quantum systems. This method corresponds to an efficient data compression for one‐dimensional quantum systems. In this study, we applied the DMRG method to quantum circuit simulation, and found that a simple quantum circuit based on Grover’s algorithm can be simulated with a classical computer.

DOI
10.1063/1.1834412

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Simulating large quantum circuits with a classical computer requires enormous computational power, because the number of quantum states increases exponentially as a function of number of qubits. The density matrix renormalization group (DMRG) was introduced by White to study the properties of relatively large‐scale one‐dimensional quantum systems. This method corresponds to an efficient data compression for one‐dimensional quantum systems. In this study, we applied the DMRG method to quantum circuit simulation, and found that a simple quantum circuit based on Grover’s algorithm can be simulated with a classical computer.
Quantum computing offers the potential for computational abilities that can go beyond classical machines. However, they are still limited by several challenges such as noise, decoherence, and gate errors. As a result, efficient classical simulation of quantum circuits is vital not only for validating and benchmarking quantum hardware but also for gaining deeper insights into the behavior of quantum algorithms. A promising framework for classical simulation is provided by tensor networks. Recently, the density-matrix renormalization group (DMRG) algorithm was developed for simulating quantum circuits using matrix product states (MPS). Although MPS are efficient for representing quantum states with one-dimensional correlation structures, the fixed linear geometry restricts the expressive power of the MPS. In this work, we extend the DMRG algorithm for simulating quantum circuits to tree tensor networks (TTNs). The framework employs a variational compression scheme that optimizes the TTN to approximate the evolved quantum state. To benchmark the method, we simulate random circuits and the quantum approximate optimization algorithm (QAOA) with various two-qubit gate connectivities. For the random circuits, we devise treelike gate layouts that are suitable for TTN and show that TTN requires less memory than MPS for the simulations. For the QAOA circuits, a naive TTN construction that exploits graph structure significantly improves the simulation fidelities. Our findings show that the DMRG algorithm with TTNs provides a promising framework for simulating quantum circuits, particularly when gate connectivities exhibit clustering or a hierarchical structure.
The density matrix renormalization group (DMRG) is widely acknowledged as a highly effective and accurate method for solving one-dimensional quantum many-body systems. However, the direct application of DMRG to the study of two-dimensional systems encounters challenges due to the limited entanglement encoded in the underlying wave-function Ansatz, known as the matrix product state. Conversely, Clifford circuits offer a promising avenue for simulating states with substantial entanglement, albeit confined to stabilizer states. In this work, we present the seamless integration of Clifford circuits within the DMRG algorithm, leveraging the advantages of both Clifford circuits and DMRG. This integration leads to a significant enhancement in simulation accuracy with small additional computational cost. Moreover, this framework is useful not only for its current application but also for its potential to be easily adapted to various other numerical approaches.
We develop a density-matrix renormalization group (DMRG) algorithm for the simulation of quantum circuits.This algorithm can be seen as the extension of the time-dependent DMRG from the usual situation of Hermitian Hamiltonian matrices to quantum circuits defined by unitary matrices.For small circuit depths, the technique is exact and equivalent to other matrix product state-based techniques.For larger depths, it becomes approximate in exchange for an exponential speed up in computational time.Like an actual quantum computer, the quality of the DMRG results is characterized by a finite fidelity.However, unlike a quantum computer, the fidelity depends strongly on the quantum circuit considered.For the most difficult possible circuit for this technique, the so-called "quantum supremacy" benchmark of Google LLC [Arute et al., Nature 574, 505 (2019)], we find that the DMRG algorithm can generate bit strings of the same quality as the seminal Google experiment on a single computing core.For a more structured circuit used for combinatorial optimization (quantum approximate optimization algorithm), we find a drastic improvement of the DMRG results with error rates dropping by a factor of 100 compared with random quantum circuits.Our results suggest that the current bottleneck of quantum computers is their fidelities rather than the number of qubits.
Abstract The physical properties of a quantum many-body system can, in principle, be determined by diagonalizing the respective Hamiltonian, but the dimensions of its matrix representation scale exponentially with the number of degrees of freedom. Hence, only small systems that are described through simple models can be tackled via exact diagonalization. To overcome this limitation, numerical methods based on the renormalization group paradigm that restrict the quantum many-body problem to a manageable subspace of the exponentially large full Hilbert space have been put forth. A striking example is the density-matrix renormalization group (DMRG), which has become the reference numerical method to obtain the low-energy properties of one-dimensional quantum systems with short-range interactions. Here, we provide a pedagogical introduction to DMRG, presenting both its original formulation and its modern tensor-network-based version. This colloquium sets itself apart from previous contributions in two ways. First, didactic code implementations are provided to bridge the gap between conceptual and practical understanding. Second, a concise and self-contained introduction to the tensor-network methods employed in the modern version of DMRG is given, thus allowing the reader to effortlessly cross the deep chasm between the two formulations of DMRG without having to explore the broad literature on tensor networks. We expect this pedagogical review to find wide readership among students and researchers who are taking their first steps in numerical simulations via DMRG. Graphic abstract
Over the last decade, the density-matrix renormalization group (DMRG) has emerged as the most powerful method for the simulation of strongly correlated one-dimensional (1D) quantum systems. Input from quantum information has allowed to trace the method's performance to the entanglement properties of quantum states, revealing why it works so well in 1D and not so well in 2D; it has allowed to devise algorithms for time-dependent quantum systems and, by clarifying the link between DMRG and Wilson's numerical renormalization group (NRG), for quantum impurity systems.