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arXiv:2309.15642 · Physical Review Research

Efficient tensor network simulation of IBM's largest quantum processors

Siddhartha Patra, Saeed S. Jahromi, Sukhbinder Singh, Román Orús

Published Mar 26, 2024• 63 citations
arXivHTMLPDF

Abstract

We show how quantum-inspired 2D tensor networks can be used to efficiently and accurately simulate the largest quantum processors from IBM, namely Eagle (127 qubits), Osprey (433 qubits), and Condor (1121 qubits). We simulate the dynamics of a complex quantum many-body system—specifically, the kicked Ising experiment considered recently by IBM in Y. Kim , —using graph-based projected entangled pair states (gPEPS), which was proposed by some of us in . Our results show that simple tensor updates are already sufficient to achieve very large unprecedented accuracy with remarkably low computational resources for this model. Apart from simulating the original experiment for 127 qubits, we also extend our results to 433 and 1121 qubits, and for evolution times around eight times longer, thus setting a benchmark for the newest IBM quantum machines. We also report accurate simulations for infinitely many qubits. Our results show that gPEPS are a natural tool to efficiently simulate quantum computers with an underlying lattice-based qubit connectivity, such as all quantum processors based on superconducting qubits. Published by the American Physical Society 2024

DOI
10.1103/physrevresearch.6.013326

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We show how quantum-inspired 2D tensor networks can be used to efficiently and accurately simulate the largest quantum processors from IBM, namely Eagle (127 qubits), Osprey (433 qubits), and Condor (1121 qubits). We simulate the dynamics of a complex quantum many-body system—specifically, the kicked Ising experiment considered recently by IBM in Y. Kim , —using graph-based projected entangled pair states (gPEPS), which was proposed by some of us in . Our results show that simple tensor updates are already sufficient to achieve very large unprecedented accuracy with remarkably low computational resources for this model. Apart from simulating the original experiment for 127 qubits, we also extend our results to 433 and 1121 qubits, and for evolution times around eight times longer, thus setting a benchmark for the newest IBM quantum machines. We also report accurate simulations for infinitely many qubits. Our results show that gPEPS are a natural tool to efficiently simulate quantum computers with an underlying lattice-based qubit connectivity, such as all quantum processors based on superconducting qubits. Published by the American Physical Society 2024
We report an accurate and efficient classical simulation of a kicked Ising quantum system on the heavy hexagon lattice. A simulation of this system was recently performed on a 127-qubit quantum processor using noise-mitigation techniques to enhance accuracy [Y. Kim ., Nature, 618, 500–5 (2023)]. Here we show that, by adopting a tensor network approach that reflects the geometry of the lattice and is approximately contracted using belief propagation, we can perform a classical simulation that is significantly more accurate and precise than the results obtained from the quantum processor and many other classical methods. We quantify the treelike correlations of the wave function in order to explain the accuracy of our belief propagation-based approach. We also show how our method allows us to perform simulations of the system to long times in the thermodynamic limit, corresponding to a quantum computer with an infinite number of qubits. Our tensor network approach has broader applications for simulating the dynamics of quantum systems with treelike correlations. Published by the American Physical Society 2024
We perform classical simulations of the 127-qubit kicked Ising model, which was recently emulated using a quantum circuit with error mitigation [Nature 618, 500 (2023)]. Our approach is based on the projected entangled pair operator (PEPO) in the Heisenberg picture. Its main feature is the ability to automatically identify the underlying low-rank and low-entanglement structures in the quantum circuit involving Clifford and near-Clifford gates. We assess our approach using the quantum circuit with 5+1 trotter steps which was previously considered beyond classical verification. We develop a Clifford expansion theory to compute exact expectation values and use them to evaluate algorithms. The results indicate that PEPO significantly outperforms existing methods, including the tensor network with belief propagation, the matrix product operator, and the Clifford perturbation theory, in both efficiency and accuracy. In particular, PEPO with bond dimension χ=2χ=2χ=2 already gives similar accuracy to the CPT with K=10K=10K=10 and MPO with bond dimension χ=1024χ=1024χ=1024. And PEPO with χ=184χ=184χ=184 provides exact results in 333 seconds using a single CPU. Furthermore, we apply our method to the circuit with 20 Trotter steps. We observe the monotonic and consistent convergence of the results with χχχ, allowing us to estimate the outcome with χ→∞χ\to\inftyχ→∞ through extrapolations. We then compare the extrapolated results to those achieved in quantum hardware and with existing tensor network methods. Additionally, we discuss the potential usefulness of our approach in simulating quantum circuits, especially in scenarios involving near-Clifford circuits and quantum approximate optimization algorithms. Our approach is the first use of PEPO in solving the time evolution problem, and our results suggest it could be a powerful tool for exploring the dynamical properties of quantum many-body systems.
Classical simulations of quantum circuits play a vital role in the development of quantum computers and in taking the temperature of the field. Here, we classically simulate various physically-motivated circuits using 2D tensor network ansätze for the many-body wavefunction, which match the geometry of the underlying quantum processor. We employ a generalized boundary Matrix Product State contraction algorithm to controllably generate samples from the resultant tensor network states. Our approach systematically converges both the quality of the final state and the samples drawn from it to the true distribution defined by the circuit, with GPU hardware providing significant speedups over CPU hardware. Using these methods, we simulate the largest local unitary Jastrow ansatz circuit from recent IBM experiments to numerical precision. We also study a domain-wall quench in a 2D discrete-time Heisenberg model on large heavy-hex and rotated square lattices, reflecting IBM’s and Google’s latest processors, respectively. We observe a rapid buildup of complex loop correlations on the Google Willow geometry, which significantly impacts local properties of the system. Meanwhile, loop correlations build up extremely slowly on heavy-hex processors, with almost negligible impact even at large circuit depths. Our results underscore the role processor geometry plays in classical simulability.
We present a general graph-based projected entangled-pair state (gPEPS) algorithm to approximate ground states of nearest-neighbor local Hamiltonians on any lattice or graph of infinite size. By introducing the structural matrix, which codifies the details of tensor networks on any graphs in any dimension ddd, we are able to produce a code that can be essentially launched to simulate any lattice. We further introduce an optimized algorithm to compute simple tensor updates as well as expectation values and correlators with a mean-field-like effective environments. Though not being variational, this strategy allows to cope with PEPS of very large bond dimension (e.g., D=100D=100D=100) and produces remarkably accurate results in the thermodynamic limit in many situations, and specially when the correlation length is small and the connectivity of the lattice is large. We prove the validity of our approach by benchmarking the algorithm against known results for several models, i.e., the antiferromagnetic Heisenberg model on a chain, star and cubic lattices, the hardcore Bose-Hubbard model on square lattice, the ferromagnetic Heisenberg model in a field on the pyrochlore lattice, as well as the three-state quantum Potts model in field on the kagome lattice and the spin-1 bilinear-biquadratic Heisenberg model on the triangular lattice. We further demonstrate the performance of gPEPS by studying the quantum phase transition of the 2d2d2d quantum Ising model in transverse magnetic field on the square lattice, and the phase diagram of the Kitaev-Heisenberg model on the hyperhoneycomb lattice. Our results are in excellent agreement with previous studies.
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.
High-performance numerical methods are essential not only for advancing quantum many-body physics but also for enabling integration with emerging quantum computing platforms. We present a scalable, general-purpose parallel algorithm for quantum simulations based on -represented quantum algebra (ORQA). This framework applies to arbitrary spin systems and naturally integrates with quantum circuit simulation in the Heisenberg picture, making it particularly relevant to recent large-scale experiments on superconducting qubit processors [Y. Kim ]. As a benchmark, we simulate the kicked Ising model on a 127-qubit heavy-hexagon lattice, tracking the time evolution of local magnetization using up to 1 × 10 12 Pauli strings. Executed on the supercomputer Fugaku, our simulations exhibit strong scaling up to 2 17 parallel processes with near-linear communication overhead. These results establish ORQA as a practical and high-performance tool for quantum many-body dynamics and highlight its potential for integration into hybrid quantum-classical computational frameworks, complementing recent advances in tensor network and surrogate simulation techniques.