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arXiv:2302.10436 · npj Quantum Information

Error-mitigated quantum simulation of interacting fermions with trapped ions

Wentao Chen, Shuaining Zhang, Jialiang Zhang, Xiaolu Su, Yao Lu, Kuan Zhang, Mu Qiao, Ying Li, Jing-Ning Zhang, Kihwan Kim

Published Dec 7, 2023• 7 citations
arXivHTMLPDF

Abstract

Abstract Quantum error mitigation has been extensively explored to increase the accuracy of the quantum circuits in noisy-intermediate-scale-quantum (NISQ) computation, where quantum error correction requiring additional quantum resources is not adopted. Among various error-mitigation schemes, probabilistic error cancellation (PEC) has been proposed as a general and systematic protocol that can be applied to numerous hardware platforms and quantum algorithms. However, PEC has only been tested in two-qubit systems and a superconducting multi-qubit system by learning a sparse error model. Here, we benchmark PEC using up to four trapped-ion qubits. For the benchmark, we simulate the dynamics of interacting fermions with or without spins by applying multiple Trotter steps. By tomographically reconstructing the error model and incorporating other mitigation methods such as positive probability and symmetry constraints, we are able to increase the fidelity of simulation and faithfully observe the dynamics of the Fermi–Hubbard model, including the different behavior of charge and spin of fermions. Our demonstrations can be an essential step for further extending systematic error-mitigation schemes toward practical quantum advantages.

DOI
10.1038/s41534-023-00784-8

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Abstract Quantum error mitigation has been extensively explored to increase the accuracy of the quantum circuits in noisy-intermediate-scale-quantum (NISQ) computation, where quantum error correction requiring additional quantum resources is not adopted. Among various error-mitigation schemes, probabilistic error cancellation (PEC) has been proposed as a general and systematic protocol that can be applied to numerous hardware platforms and quantum algorithms. However, PEC has only been tested in two-qubit systems and a superconducting multi-qubit system by learning a sparse error model. Here, we benchmark PEC using up to four trapped-ion qubits. For the benchmark, we simulate the dynamics of interacting fermions with or without spins by applying multiple Trotter steps. By tomographically reconstructing the error model and incorporating other mitigation methods such as positive probability and symmetry constraints, we are able to increase the fidelity of simulation and faithfully observe the dynamics of the Fermi–Hubbard model, including the different behavior of charge and spin of fermions. Our demonstrations can be an essential step for further extending systematic error-mitigation schemes toward practical quantum advantages.
Quantum error mitigation (QEM) has emerged as a powerful tool for the extraction of useful quantum information from quantum devices. Here, we introduce the Subspace Noise Tailoring (SNT) algorithm, which efficiently combines the cheap cost of Symmetry Verification (SV) and low bias of Probabilistic Error Cancellation (PEC) QEM techniques. We study the performance of our method by simulating the Trotterized time evolution of the spin-1/2 Fermi-Hubbard model (FHM) using a variety of local fermion-to-qubit encodings, which define a computational subspace through a set of stabilizers, the measurement of which can be used to post-select noisy quantum data. We study different combinations of QEM and encodings and uncover a rich state diagram of optimal combinations, depending on the hardware performance, system size and available shot budget. We then demonstrate how SNT extends the reach of current noisy quantum computers in terms of the number of fermionic lattice sites and the number of Trotter steps, and quantify the required hardware performance beyond which a noisy device may compete with current state-of-the-art classical computational methods.
We show that combining quantum error detection (QED) with probabilistic error cancellation (PEC) gives more accurate and lower-variance estimates than PEC alone, provided that the symmetry measurements required for QED are carefully chosen. Because noisy symmetry measurements can negate the benefits of the PEC+QED approach, we cast the selection of measurement configurations as a classical optimization problem that systematically suppresses the impact of noise. Applying optimized PEC+QED to GHZ-state output distributions and to simulating the time-dynamics of a generalized superfast encoded Fermi-Hubbard model, we find consistent improvements over PEC. For GHZ states, the optimization over symmetry measurement configurations is essential for achieving an advantage. For the Fermi-Hubbard model, PEC+QED improves observable estimation on a 2×22 \times 22×2 lattice and for larger systems the mitigation overheads can be reduced by measuring only subsets of stabilizers. Our results demonstrate the importance of circuit-specific tailoring of QEM techniques and that fault-tolerant design principles may already provide value for near-term devices.
Abstract Various quantum applications can be reduced to estimating expectation values, which are inevitably deviated by operational and environmental errors. Although errors can be tackled by quantum error correction, the overheads are far from being affordable for near-term technologies. To alleviate the detrimental effects of errors on the estimation of expectation values, quantum error mitigation techniques have been proposed, which require no additional qubit resources. Here we benchmark the performance of a quantum error mitigation technique based on probabilistic error cancellation in a trapped-ion system. Our results clearly show that effective gate fidelities exceed physical fidelities, i.e., we surpass the break-even point of eliminating gate errors, by programming quantum circuits. The error rates are effectively reduced from (1.10 ± 0.12) × 10−3 to (1.44 ± 5.28) × 10−5 and from (0.99 ± 0.06) × 10−2 to (0.96 ± 0.10) × 10−3 for single- and two-qubit gates, respectively. Our demonstration opens up the possibility of implementing high-fidelity computations on a near-term noisy quantum device.
The Fermi-Hubbard model (FHM) is a simple yet rich model of strongly interacting electrons with complex dynamics and a variety of emerging quantum phases. These properties make it a compelling target for digital quantum simulation. Trotterization-based quantum simulations have shown promise, but implementations on current hardware are limited by noise, necessitating error mitigation techniques like circuit optimization and post-selection. A mapping of the FHM to a Z2 LGT was recently proposed that restricts the dynamics to a subspace protected by additional symmetries, and its ability for post-selection error mitigation was verified through noisy classical simulations. In this work, we propose and demonstrate a suite of algorithm-hardware co-design strategies on a trapped-ion quantum computer, targeting two key aspects of NISQ-era quantum simulation: circuit compilation and error mitigation. In particular, a novel combination of iteratively preconditioned gradient descent (IPG) and subsystem von Neumann Entropy compression reduces the 2-qubit gate count of FHM quantum simulation by 35%, consequently doubling the number of simulatable Trotter steps when used in tandem with error mitigation based on conserved symmetries, debiasing and sharpening techniques. Our work demonstrates the value of algorithm-hardware co-design to operate digital quantum simulators at the threshold of maximum circuit depths allowed by current hardware, and is broadly generalizable to strongly correlated systems in quantum chemistry and materials science.
Abstract The practical application of Noisy Intermediate-Scale Quantum (NISQ) hardware is fundamentally limited by gate-level noise, which corrupts computational results. This research addresses this challenge through the development and validation of a modular software framework for Quantum Error Mitigation (QEM). We implement and compare two leading QEM strategies: Zero-Noise Extrapolation (ZNE) and Probabilistic Error Cancellation (PEC). The framework is built using the Qiskit library and is validated using a realistic, hardware-calibrated noisy simulator that mimics the error characteristics of a real IBM Quantum device. Our methodology focuses on a robust implementation of ZNE via unitary folding and a proof-of-concept demonstration of the PEC algorithm. The framework's effectiveness is demonstrated on a simulated noisy backend where, for a 3-qubit Greenberger-Horne-Zeilinger (GHZ) state, the baseline state fidelity was increased from 93.5% to 97.3% via ZNE, corresponding to a 59% reduction in the state's infidelity. This work provides a practical demonstration of how software-based error mitigation can enhance the reliability of quantum computations on current noisy hardware.
Error mitigation is essential for unlocking the full potential of quantum algorithms and accelerating the timeline toward quantum advantage. As quantum hardware progresses to push the boundaries of classical simulation, efficient and robust error mitigation methods are becoming increasingly important for producing accurate and reliable outputs. However, existing error-mitigation approaches face a fundamental tradeoff between practical performance and reliability: heuristic methods such as zero-noise extrapolation (ZNE) enjoy faster runtime but lack accuracy guarantees, while rigorous techniques such as probabilistic error cancellation (PEC) provide unbiased estimates at prohibitive computational cost. We introduce a characterization-based, rigorously-grounded quantum error mitigation and error suppression framework (QESEM) that resolves this tradeoff by leveraging the accuracy guarantees of quasi-probabilistic mitigation with dramatically reduced overhead. We explain the innovative methods underlying QESEM and demonstrate its capabilities in the largest utility-scale error mitigation experiment based on an unbiased method. This experiment simulates the kicked transverse field Ising model with far-from-Clifford parameters on an IBM Heron device. We further validate QESEM's versatility across arbitrary quantum circuits and devices through high-accuracy error-mitigated molecular VQE circuits executed on IBM Heron and IonQ trapped-ion devices. Compared with multiple variants of the widely used zero-noise extrapolation method, QESEM consistently achieves higher accuracy while avoiding the prohibitive runtime overhead associated with PEC. These results mark a significant step forward in accuracy and reliability for running quantum circuits on current devices across diverse applications. Finally, we provide projections of QESEM's performance on near-term devices toward quantum advantage.