Approximate simulation of complex quantum circuits using sparse tensors

TL;DR

The paper tackles the challenge of classically simulating complex quantum circuits to delineate the boundary of quantum advantage and to aid hardware/software development. It introduces TruSTS, a sparse-tensor based approach that represents the quantum state with a fixed-length sparse state $\ket{\phi}$ and applies gates via a bitwise contraction that groups basis states into independent subspaces using a mask $m_g$, followed by truncation back to $k$ terms. A key contribution is the analytic relationship between the truncation parameter $k$, the normalization factor $\gamma$, and the resulting fidelity $f$, with $f$ approximately equal to $\gamma^2$, plus bounds like $\bar{f}_{min}=1/2^N$ and a Porter-Thomas-based upper limit for random circuits. Empirical results show the runtime scales linearly with the number of kept terms $k$ and is nearly independent of the number of qubits $N$ for $k \ll 2^N$, while comparisons with MPS highlight regime-dependent tradeoffs, indicating TruSTS as a complementary sparsity-driven tool to traditional tensor-network methods.

Abstract

The study of quantum circuit simulation using classical computers is a key research topic that helps define the boundary of verifiable quantum advantage, solve quantum many-body problems, and inform development of quantum hardware and software. Tensor networks have become forefront mathematical tools for these tasks. Here we introduce a method to approximately simulate quantum circuits using sparsely-populated tensors. We describe a sparse tensor data structure that can represent quantum states with no underlying symmetry, and outline algorithms to efficiently contract and truncate these tensors. We show that the data structure and contraction algorithm are efficient, leading to expected runtime scalings versus qubit number and circuit depth. Our results motivate future research in optimization of sparse tensor networks for quantum simulation.

Figures (6)

• Figure 1: Diagram of the data structure and bitwise contraction algorithm for a sample state with $N = 5$ qubits, and $k = 8$. The contraction is performed on lists of coordinates and data (shown in (a)) by sorting and grouping based on a bitmask (b). After sorting, the gate may be applied independently to each group of coordinates (c).
• Figure 2: Representation of the TruSTS algorithm for $N=3$ qubits and $k = 2$, with sparse tensor terms represented as Feynman paths. The blue paths represent the terms in the exact circuit output shaded with an opacity corresponding to the probability amplitude, while terms kept in the sparse tensor truncation are shown in orange. The exact and approximate final state are listed numerically below.
• Figure 3: Example of the fully coupled random architecture with 3 layers shown.
• Figure 4: Final state fidelity vs the truncation fraction (a), and renormalization parameter $\gamma$ (b) for different layer numbers. In both (a) and (b), the square dotted gray horizontal line indicates the soft lower limit of fidelity $\bar{f}_\textrm{min}$ derived in Eq. \ref{['eq:fmin']}, and circles represent top-$k$ truncation while triangles represent random-$k$ truncation. In (a), the solid lines represent the upper limit for the given layer number, and the dashed black line represents the upper limit for a circuit reaching a Porter-Thomas distribution as derived in Eq. \ref{['eq:tpresult']}. In both (a) and (b), the diagonal dash-dotted black line has a slope of 1 and represents the expected fidelity derived in Eq. \ref{['eq:fidelityvsgamma']}.
• Figure 5: Runtime per gate vs $k$ (a), and qubit number (b). In (b) the green triangles represent the runtime per gate when running the exact circuit.