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arXiv:2210.03048 · ACM Transactions on Quantum Computing

Scalable Experimental Bounds for Entangled Quantum State Fidelities

Shamminuj Aktar, Andreas Bärtschi, Abdel‐Hameed A. Badawy, Stephan Eidenbenz

Published Nov 11, 2024• 1 citation
arXivHTMLPDF

Abstract

Estimating the state preparation fidelity of highly entangled states on noisy intermediate-scale quantum (NISQ) devices is important for benchmarking and application considerations. Unfortunately, exact fidelity measurements quickly become prohibitively expensive, as they scale exponentially as O (3 N for N -qubit states, using full state tomography with measurements in all Pauli bases combinations. However, Somma et al.established that the complexity could be drastically reduced when looking at fidelity lower bounds for states that exhibit symmetries, such as Dicke states and GHZ states. These bounds must still be tight enough for larger states to provide reasonable estimations on NISQ devices. For the first time and more than 15 years after the theoretical introduction, we report meaningful lower bounds for the state preparation fidelity of all Dicke states up to N =10 and all GHZ states up to N =20 on Quantinuum H1 ion-trap systems using efficient implementations of recently proposed scalable circuits for these states. Our achieved lower bounds match or exceed previously reported exact fidelities on superconducting systems for much smaller states. Furthermore, we provide evidence that for large Dicke states \(\left|\smash{D_{N/2}^{N}} \right\rangle\) , we may resort to a GHZ-based approximate state preparation to achieve better fidelity. This work provides a path forward to benchmarking entanglement as NISQ devices improve in size and quality.

DOI
10.1145/3700885

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Estimating the state preparation fidelity of highly entangled states on noisy intermediate-scale quantum (NISQ) devices is important for benchmarking and application considerations. Unfortunately, exact fidelity measurements quickly become prohibitively expensive, as they scale exponentially as O (3 N for N -qubit states, using full state tomography with measurements in all Pauli bases combinations. However, Somma et al.established that the complexity could be drastically reduced when looking at fidelity lower bounds for states that exhibit symmetries, such as Dicke states and GHZ states. These bounds must still be tight enough for larger states to provide reasonable estimations on NISQ devices. For the first time and more than 15 years after the theoretical introduction, we report meaningful lower bounds for the state preparation fidelity of all Dicke states up to N =10 and all GHZ states up to N =20 on Quantinuum H1 ion-trap systems using efficient implementations of recently proposed scalable circuits for these states. Our achieved lower bounds match or exceed previously reported exact fidelities on superconducting systems for much smaller states. Furthermore, we provide evidence that for large Dicke states (\left|\smash{D_{N/2}^{N}} \right\rangle) , we may resort to a GHZ-based approximate state preparation to achieve better fidelity. This work provides a path forward to benchmarking entanglement as NISQ devices improve in size and quality.
Estimating the state preparation fidelity of highly entangled states on noisy intermediate-scale quantum (NISQ) devices is an important task for benchmarking and application considerations. Unfortunately, exact fidelity measurements quickly become prohibitively expensive, as they scale exponentially as O(3N) for N-qubit states, using full state tomography with measurements in all Pauli bases combinations. However, Somma et al. [20] established that the complexity could be drastically reduced when looking at fidelity lower bounds for states that exhibit symmetries, such as Dicke States and GHZ States. For larger states, these bounds still need to be tight enough to provide reasonable estimations on NISQ devices.
Estimating the fidelity of state preparation in multiqubit systems is generally a time-consuming task. Nevertheless, this complexity can be reduced if the desired state can be characterized by certain symmetries measurable with the corresponding experimental setup. In this paper we give simple expressions to estimate the fidelity of multiqubit state preparation for rotational-invariant, stabilizer, and generalized coherent states. We specifically discuss the GHZ-type and W-type states, and obtain efficiently measurable lower bounds for the fidelity. We use these techniques to estimate the fidelity of a quantum simulation of an Ising-like interaction model using two trapped ions. These results are directly applicable to experiments using fidelity-based entanglement witnesses, such as quantum simulations and quantum computation.
We define scalable benchmarks for two important quantum subroutines, namely the quantum Fourier transform (QFT) and the multi-controlled-Z gate. Our benchmarks are based on applying the subroutines to a set of states that are sufficient for lower-bounding the process fidelity from the set of measured output state fidelities. By randomly sampling from the set of input states, the lower boundonthe process fidelity can be estimated. We implement the QFT and multi-controlled-Z circuits with up to 20 and 12 qubits, respectively, on the Quantinuum H1-1 quantum computer. Our circuits make use of parameterized-angle entangling gates, which for these circuits reduces the number of two-qubit gates significantly.
Reliable characterization of quantum states and quantum processes is central to the development of scalable quantum computing. Standard characterization techniques such as full quantum state tomography become rapidly impractical as system size increases. In this work we present two software-oriented fidelity estimation protocols that avoid full tomographic reconstruction while remaining experimentally relevant for noisy intermediate-scale quantum devices. The first protocol is direct fidelity estimation (DFE), where the fidelity between a target state and an experimentally prepared state is estimated from importance-sampled Pauli measurements. The second protocol is a survival-probability-based method that reconstructs fidelity from experimentally accessible moments of a noisy evolution and its approximate inverse, and is intrinsically robust against state-preparation-and-measurement (SPAM) errors. We discuss the theoretical basis of both approaches, analyze their noise sensitivity, and present numerical evidence on representative multiqubit states under depolarizing, dephasing, and amplitude-damping channels. For DFE we show that attenuation-aware correction improves robustness against measurement noise. For the survival-probability approach we show that an L2L_2L2​-optimized expansion substantially outperforms naive Taylor truncation in high-noise regimes.
As quantum technology advances and the size of quantum computers grow, it becomes increasingly important to understand the extent of quality in the devices. As large-scale entanglement is a quantum resource crucial for achieving quantum advantage, the challenge in its generation makes it a valuable benchmark for measuring the performance of universal quantum devices. In this paper, we study entanglement in Greenberger-Horne-Zeilinger (GHZ) and graph states prepared on the range of IBM Quantum devices. We generate GHZ states and investigate their coherence times with respect to state size and dynamical decoupling techniques. A GHZ fidelity of 0.519 ± 0.014 is measured on a 32-qubit GHZ state, certifying its genuine multipartite entanglement (GME). We show a substantial improvement in GHZ decoherence rates for a seven-qubit GHZ state after implementing dynamical decoupling, and observe a linear trend in the decoherence rate of α = ( 7.13 N + 5.54 ) × 10 − 3 µ s − 1 for up to N = 15 qubits, confirming the absence of superdecoherence. Additionally, we prepare and characterize fully bipartite entangled native-graph states on 22 superconducting quantum devices with qubit counts as high as 414 qubits, all active qubits of the 433-qubit IBM Osprey device. Analysis of the decay of two-qubit entanglement within the prepared states shows suppression of coherent noise signals with the implementation of dynamical decoupling techniques. Additionally, we observe that the entanglement in some qubit pairs oscillates over time, which is likely caused by residual ZZ-interactions. Characterizing entanglement in native-graph states, along with detecting entanglement oscillations, can be an effective approach to low-level device benchmarking that encapsulates two-qubit error rates along with additional sources of noise, with possible applications to quantum circuit compilation. We develop several tools to automate the preparation and entanglement characterization of GHZ and graph states. In particular, a method to characterize graph state bipartite entanglement using just 36 circuits, constant with respect to the number of qubits. Published by the American Physical Society 2024
Dicke states are multipartite entangled states in which a fixed number of quantum excitations is coherently shared among many qubits. Originally introduced in the context of cooperative emission and superradiance, they are now important resources for quantum sensing, networking, and collective quantum phenomena. Preparing prescribed Dicke states with high fidelity, however, remains challenging, particularly as the system size and excitation number increase. Here we present an open dataset and accompanying code for preparing Dicke states using a collision-based quantum protocol. The dataset covers systems from five to fourteen qubits over a broad range of excitation numbers and records how the best-found noiseless preparation fidelity changes with circuit depth. It also provides circuit-resource estimates and experimental measurements for selected states on the 54-qubit IQM Emerald superconducting processor. The accompanying code reproduces the processed data and validation checks, providing a reusable benchmark for studying the trade-off between state-preparation fidelity, circuit cost, and hardware noise.