[Paper Review] A density-matrix renormalization group algorithm for simulating quantum circuits with a finite fidelity

Thomas Ayral, Thibaud Louvet+4 more|arXiv (Cornell University)|Jul 12, 2022

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper introduces a density-matrix renormalization group (DMRG) algorithm tailored for simulating quantum circuits with finite fidelity, extending DMRG from hermitian Hamiltonians to unitary quantum circuits. It achieves exponential speedup over exact methods for deep circuits, matching Google's quantum supremacy experiment fidelity on a single core and reducing error rates by 100× for QAOA circuits, demonstrating that fidelity—not qubit count—is the primary bottleneck in current quantum computing.

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 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 (MPS) 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 Inc. , 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 or QAOA), 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.

Motivation & Objective

• To develop a DMRG-based method for simulating quantum circuits with finite fidelity, extending the technique beyond hermitian Hamiltonians to unitary evolution.
• To benchmark the algorithm on three circuit types: Google's quantum supremacy circuit, a random circuit, and a QAOA circuit for combinatorial optimization.
• To assess whether computational cost or fidelity is the dominant limitation in simulating near-term quantum devices.
• To demonstrate that structured circuits like QAOA benefit significantly more from DMRG than random circuits, revealing fidelity dependence on circuit structure.

Proposed method

• Adapt the time-dependent DMRG algorithm to handle unitary evolution matrices instead of hermitian Hamiltonians, enabling simulation of quantum circuits.
• Use matrix product state (MPS) representations to compress the quantum state, truncating to a bond dimension χ for computational efficiency.
• Apply singular value decomposition (SVD) to the MPS to identify and retain the most significant entanglement modes, with singular values scaling as Sμ ∝ 2−N/4g(μ/2N/2).
• Use the scaling law g(x) = 2cos(½A⁻¹(πx)) to model singular value distribution and predict fidelity decay under truncation.
• Implement Metropolis sampling to generate bitstrings distributed according to |Ψx|², enabling efficient sampling of measurement outcomes.
• Use cross-entropy benchmarking to quantify fidelity and compare results with actual quantum hardware performance.

Experimental results

Research questions

• RQ1 Can DMRG be generalized to simulate quantum circuits governed by unitary evolution, rather than Hamiltonian time evolution?
• RQ2 How does the fidelity of the DMRG simulation depend on the structure of the quantum circuit, particularly for random vs. structured circuits?
• RQ3 To what extent can DMRG achieve exponential speedup over exact simulation while maintaining high fidelity for deep quantum circuits?
• RQ4 Why does the DMRG algorithm perform significantly better on QAOA circuits than on random quantum supremacy circuits, and what does this imply about the role of circuit structure?
• RQ5 Is the fidelity of the DMRG simulation comparable to that of actual quantum hardware, and can it reproduce key experimental benchmarks like Google's supremacy experiment?

Key findings

• The DMRG algorithm reproduces the bitstring statistics of Google's quantum supremacy circuit with fidelity comparable to the original experiment, achieving this on a single computing core.
• For the QAOA circuit, the DMRG method reduces error rates by a factor of 100 compared to random circuits, indicating strong dependence of fidelity on circuit structure.
• The fidelity of the DMRG simulation scales as F(χ) ≈ 4χ / 2^{N/2} in the chaotic limit, which is four times better than the worst-case uniform singular value distribution.
• The singular value spectrum follows a non-uniform scaling law Sμ ∝ 2−N/4g(μ/2N/2), with g(x) = 2cos(½A⁻¹(πx)), explaining the improved performance for structured circuits.
• Metropolis sampling enables efficient bitstring generation with ~70% average acceptance ratio, requiring only ~1.5 amplitudes per accepted bitstring.
• The algorithm's fidelity plateaus at log(2)/D for large N, indicating a fundamental limit tied to circuit depth D rather than qubit count.