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arXiv:2608.03987

Realified tensor networks: quantum circuit simulation on real-valued matrix accelerators

Yusheng Zhao, Xiwei Pan, Enji Xiong, Chengkai Zhu, Jinguo Liu

quant-phSubmitted Aug 4, 2026• Announced Aug 5, 2026• 0 citations
v1 · Aug 4, 2026v2 · Aug 5, 2026
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Abstract

Tensor-network contraction simulates quantum circuits, but modern matrix accelerators (NPUs, TPUs) expose only real GEMM pipelines, so the complex networks of quantum simulation must be reconstructed in software. We resolve the mismatch by a realification rewrite that maps any complex tensor network to a real one. At each merge of two complex tensors, a rank-3 structure tensor realizes Gauss's three-multiplication (3M) formula; contractions with one or no complex operand need only two or one real products. We prove a tight cost law: overhead $1 + 2m + r$ in real multiplications, where $m$ and $r$ are the volume fractions of two- and one-complex-operand contractions, never exceeding $3\times$ relative to real contraction, with every intermediate at most doubled in size. On 67 circuits (random, Clifford+$T$, QAOA, VQE), the law holds across the real-to-complex range and complex-gate placement, not count, governs cost. Contraction orders transfer from the complex network with a relative arithmetic-cost gap below $5\times 10^{-4}$ on 66 of 67 circuits; the exception closes under a few steps of low-temperature simulated annealing. On an Ascend 910 NPU the rewrite beat both the four-real-GEMM baseline and a per-GEMM Gauss lowering on all twelve random circuits and on 52 of 55 structured cells (three cells slower by at most 12\%); the four-GEMM baseline was slower by a median $1.7\times$ (random) and $1.4\times$ (structured). Realification makes complex tensor-network contraction native to real-only matrix engines.

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Efficient simulation of quantum circuits has become indispensable with the rapid development of quantum hardware. The primary simulation methods are based on state vectors and tensor networks. As the number of qubits and quantum gates grows larger in current quantum devices, traditional state-vector-based quantum circuit simulation methods prove inadequate due to the overwhelming size of the Hilbert space and extensive entanglement. Consequently, brutal force tensor network simulation algorithms become the only viable solution in such scenarios. The two main challenges faced in tensor network simulation algorithms are optimal contraction path finding and efficient execution on modern computing devices, with the latter determining the actual efficiency. In this study, we investigate the optimization of such tensor network simulations on modern GPUs and propose general optimization strategies from two aspects: computational efficiency and accuracy. First, we propose to transform critical Einstein summation operations into GEMM operations, leveraging the specific features of tensor network simulations to amplify the efficiency of GPUs. Second, by analyzing the data characteristics of quantum circuits, we employ extended precision to ensure the accuracy of simulation results and mixed precision to fully exploit the potential of GPUs, resulting in faster and more precise simulations. Our numerical experiments demonstrate that our approach can achieve a 3.96× reduction in verification time for random quantum circuit samples in the 18-cycle case of Sycamore, with sustained performance exceeding 21 TFLOPS on one A100. This method can be easily extended to the 20-cycle case, maintaining the same performance, accelerating by 12.5× compared to the state-of-the-art CPU-based results and 4.48–6.78× compared to the state-of-the-art GPU-based results reported in the literature. Furthermore, the strategies presented in this article hold general applicability for accelerating problems that can be tackled by contracting tensor networks such as graphical models and combinatorial optimizations.
Classical simulation of quantum circuits is an essential tool in quantum information science, but its applicability is constrained by the exponential growth of the Hilbert space and the entanglement structure of quantum states. In this work, we introduce Matrix Product Evolution (MPE), a tensor-train representation of quantum circuits constructed along the circuit depth rather than along the qubit index. Within this formulation, the simulation of a quantum circuit is modeled as the contraction of multiple MPE tensors. We develop an efficient contraction strategy based on a zip-up procedure to carry out this contraction in practice. We investigate the numerical behavior of this MPE-based contraction framework through simulations of random quantum circuits and the time evolution of a quantum many-body state. Our results characterize the growth of temporal bond dimensions, clarify how post-selection modifies the contraction cost and approximation accuracy, and identify regimes in which depth-oriented tensor-network contractions provide a useful complement to standard MPS-based simulation approaches.
Tensors are a natural generalization of matrices, and tensor networks are a natural generalization of matrix products. Despite the simple definition of tensor networks, they are versatile enough to represent many different kinds of "products" that arise in various theoretical and practical problems. In particular, the powerful computational model of quantum computation can be defined almost entirely in terms of matrix products and tensor products, both of which are special cases of tensor networks. As such, (classical) algorithms for evaluating tensor networks have profound importance in the study of quantum computation. In this thesis, we design and implement a parallel algorithm for tensor network contraction. In addition to finding efficient contraction orders for a tensor network, we also dynamically slice it into multiple sub-tasks with lower space and time costs, in order to evaluate the tensor network in parallel. We refer to such an evaluation strategy as a contraction scheme for the tensor network. In addition, we introduce a local optimization procedure that improves the efficiency of the contraction schemes we find. We also investigate the applications of our parallel tensor network contraction algorithm in quantum computation. The most ready application is the simulation of random quantum supremacy circuits, where we benchmark our algorithm to demonstrate its advantage over other similar tensor network based simulators. Other applications we found include evaluating the energy function of a Quantum Approximate Optimization Algorithm (QAOA), and simulating surface codes under a realistic error model with crosstalk.
Simulation of quantum systems is challenging due to the exponential size of the state space. Tensor networks provide a systematically improvable approximation for quantum states. 2D tensor networks such as Projected Entangled Pair States (PEPS) are well-suited for key classes of physical systems and quantum circuits. However, direct contraction of PEPS networks has exponential cost, while approximate algorithms require computations with large tensors. We propose new scalable algorithms and software abstractions for PEPS-based methods, accelerating the bottleneck operation of contraction and refactorization of a tensor subnetwork. We employ randomized SVD with an implicit matrix to reduce cost and memory footprint asymptotically. Further, we develop a distributed-memory PEPS library and study accuracy and efficiency of alternative algorithms for PEPS contraction and evolution on the Stampede2 supercomputer. We also simulate a popular near-term quantum algorithm, the Variational Quantum Eigensolver (VQE), and benchmark Imaginary Time Evolution (ITE), which compute ground states of Hamiltonians.
Most research in quantum computing today is performed against simulations of quantum computers rather than true quantum computers. Simulating a quantum computer entails implementing all of the unitary operators corresponding to the quantum gates as tensors. For high numbers of qubits, performing tensor multiplications for these simulations becomes quite expensive, since NNN-qubit gates correspond to 2N2^{N}2N-dimensional tensors. One way to accelerate such a simulation is to use field programmable gate array (FPGA) hardware to efficiently compute the matrix multiplications. Though FPGAs can efficiently perform tensor multiplications, they are memory bound, having relatively small block random access memory. One way to potentially reduce the memory footprint of a quantum computing system is to represent it as a tensor network; tensor networks are a formalism for representing compositions of tensors wherein economical tensor contractions are readily identified. Thus we explore tensor networks as a means to reducing the memory footprint of quantum computing systems and broadly accelerating simulations of such systems.
We classically simulate the IBM doped Clifford random circuit sampling experiment, comprising 707070 qubits, 707070 entangling layers, and 468468468 inserted TTT gates, as reported in the first version of arXiv:2607.25941. A deterministic temporal-boundary tensor network contraction approach is specifically designed to tackle such open-boundary one-dimensional brickwork circuits with operator-Schmidt-rank-222 entangling gates. For an nnn-qubit circuit of depth ddd, the resulting unsliced path evaluates an exact amplitude with contraction width ⌈d/2⌉\lceil d/2\rceil⌈d/2⌉; Ratcatcher calculations certify that no smaller width is possible for the tested instances. Because one-qubit gates are absorbed without changing the network topology, the width and dense scheduled contraction cost are independent of their values and of the number and placement of TTT gates. For this initial instance, its largest intermediate tensor contains 2352^{35}235 complex64 entries (256 times smaller than IBM's estimation), corresponding to a tensor payload of 256256256 GiB, and is distributed across eight GPUs within a node. Using 32 nodes, with eight NVIDIA H100 GPUs per node, we completed all 2051 amplitude batches corresponding to IBM's published output bitstrings in 37.3 minutes. The resulting probabilities yield a log-XEB estimate of 0.350340.350340.35034 with a 95% interval of [0.29763,0.40305][0.29763,0.40305][0.29763,0.40305]. Under the Porter--Thomas and scrambled-noise assumptions, this is numerically compatible with IBM's fidelity lower bound; separately, fidelity-weighted resource accounting projects a 583-contraction workload with a 10.6-minute makespan on the same 32 nodes. More broadly, the approach provides a practical diagnostic for experimental outputs and a quantitative tool for designing future doped Clifford sampling experiments.