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arXiv:2207.05612 · PRX Quantum

Density-Matrix Renormalization Group Algorithm for Simulating Quantum Circuits with a Finite Fidelity

Thomas Ayral, Thibaud Louvet, Yiqing Zhou, Cyprien Lambert, Edwin Miles Stoudenmire, Xavier Waintal

Published Apr 10, 2023• 66 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/prxquantum.4.020304

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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.
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.
We optimize matrix-product state-based algorithms for simulating quantum circuits with finite fidelity, specifically the time-evolving block decimation (TEBD) and the density-matrix renormalization group (DMRG) algorithms, by exploiting the irregular arrangement of entangling operations in circuits. We introduce a variation of the standard TEBD algorithm, we termed “cluster-TEBD,” which dynamically arranges qubits into entanglement clusters, enabling the exact contraction of multiple circuit layers in a single time step. Moreover, we enhance the DMRG algorithm by introducing an adaptive protocol, which analyzes the entanglement distribution within each circuit section to be contracted, dynamically adjusting the qubit grouping at each iteration. We analyze the performances of these enhanced algorithms in simulating both stabilizer and nonstabilizer random-structured quantum circuits, with up to 1000 qubits and 100 layers of Clifford and non-Clifford gates, and in simulating Shor's quantum algorithm with up to hundreds of thousands of layers. Our findings show that, even with reasonable computational resources per task, cluster-based approaches can significantly speed up simulations of large-sized quantum circuits and improve the fidelity of the final states.
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.
In this paper recent substantial progress in applying the density-matrix renormalization-group (DMRG) to the simulation of the time-evolution of strongly correlated quantum systems in one dimension is reviewed. Various approaches to generating a time-evolution numerically are considered. The key focus of this review is on current strategies to circumvent the limitations of the quite small subspace well approximated by DMRG, by either enlarging or changing it as time evolves. All these approaches can be extended to the simulation of mixed, i.e. finite-temperature states. While this paper is phrased almost entirely in standard DMRG language, I finish by considering the alternative formulation of time-evolutions in the language of matrix product states which is less well-known but conceptually more powerful.
The numerical study of anyonic systems is known to be highly challenging due to their nonbosonic, nonfermionic particle-exchange statistics, and with the exception of certain models for which analytical solutions exist, very little is known about their collective behavior as a result. Meanwhile, the density matrix renormalization group (DMRG) algorithm is an exceptionally powerful numerical technique for calculating the ground state of a low-dimensional lattice Hamiltonian, and has been applied to the study of bosonic, fermionic, and group-symmetric systems. The recent development of a tensor network formulation for anyonic systems opened up the possibility of studying these systems using algorithms such as DMRG, though this has proved challenging in terms of both programming complexity and computational cost. This paper presents the implementation of DMRG for finite anyonic systems, including a detailed scheme for the implementation of anyonic tensors with optimal scaling of computational cost. The anyonic DMRG algorithm is demonstrated by calculating the ground-state energy of the ``golden chain,'' which has become the benchmark system for the numerical study of anyons and is shown to produce results comparable to those of the anyonic time-evolving block decimation algorithm and superior to the variationally optimized anyonic multiscale renormalization ansatz, at far lesser computational cost.
As demonstrated in the density matrix renormalization group (DMRG) method, approximating many-body wave function of electrons using a matrix product state (MPS) is a promising way to solve electronic structure problems. The expressibility of an MPS is determined by the size of the matrices or, in other words, the bond dimension, which unfortunately may be required to be very large in quantum chemistry simulations. In this study, we propose to calculate the ground state energies of molecular systems by variationally optimizing quantum circuit MPS (QCMPS) with a relatively small number of qubits. It is demonstrated that with carefully chosen circuit structure and orbital localization scheme, QCMPS can reach a similar accuracy as that achieved in DMRG with an exponentially large bond dimension. QCMPS simulation of a linear hydrogen molecular chain with 50 orbitals can reach the chemical accuracy using only 6 qubits at a moderate circuit depth. These results suggest that QCMPS is a promising wave function ansatz in the variational quantum eigensolver algorithm for molecular systems.