In Shor's algorithm quantum subroutines occur with the structure $\mathcal{F}\mathcal{U}{\mathcal{F}}^{\ensuremath{-}1}$, where $\mathcal{F}$ is a unitary transform and $\mathcal{U}$ is performing a quantum computation. Examples are quantum adders and subunits of quantum modulo adders. In this paper we show, both analytically and numerically, that if, in analogy to spin echoes, $\mathcal{F}$ and ${\mathcal{F}}^{\ensuremath{-}1}$ can be implemented symmetrically when executing Shor's algorithm on actual, imperfect quantum hardware, such that $\mathcal{F}$ and ${\mathcal{F}}^{\ensuremath{-}1}$ have the same hardware errors, a symmetry boost in the fidelity of the combined $\mathcal{F}\mathcal{U}{\mathcal{F}}^{\ensuremath{-}1}$ quantum operation results when compared to the case in which the errors in $\mathcal{F}$ and ${\mathcal{F}}^{\ensuremath{-}1}$ are independently random. Running the complete gate-by-gate implemented Shor algorithm, we show that the symmetry-induced fidelity boost can be as large as a factor 4. While most of our analytical and numerical results concern the case of over- and under-rotation of controlled rotation gates, in the numerically accessible case of Shor's algorithm with a small number of qubits, we show explicitly that the symmetry boost is robust with respect to more general types of errors. While, expectedly, additional error types reduce the symmetry boost, we show explicitly, by implementing general off-diagonal $\mathrm{SU}(N)$ errors ( $N=2,4,8$ ), that the boost factor scales like a Lorentzian in $\ensuremath{\delta}/\ensuremath{\sigma}$, where $\ensuremath{\sigma}$ and $\ensuremath{\delta}$ are the error strengths of the diagonal over- and underrotation errors and the off-diagonal $\mathrm{SU}(N)$ errors, respectively. The Lorentzian shape also shows that, while the boost factor may become small with increasing $\ensuremath{\delta}$, it declines slowly (essentially like a power law) and is never completely erased. We also investigate the effect of diagonal nonunitary errors, which, in analogy to unitary errors, reduce but never erase the symmetry boost. Going beyond the case of small quantum processors, we present analytical scaling results that show that the symmetry boost persists in the practically interesting case of a large number of qubits. We illustrate this result explicitly for the case of Shor factoring of the semiprime RSA-1024, where, analytically, focusing on over- and underrotation errors, we obtain a boost factor of about 10. In addition, we provide a proof of the fidelity product formula, including its range of applicability.In Shor's algorithm quantum subroutines occur with the structure $\mathcal{F}\mathcal{U}{\mathcal{F}}^{\ensuremath{-}1}$, where $\mathcal{F}$ is a unitary transform and $\mathcal{U}$ is performing a quantum computation. Examples are quantum adders and subunits of quantum modulo adders. In this paper we show, both analytically and numerically, that if, in analogy to spin echoes, $\mathcal{F}$ and ${\mathcal{F}}^{\ensuremath{-}1}$ can be implemented symmetrically when executing Shor's algorithm on actual, imperfect quantum hardware, such that $\mathcal{F}$ and ${\mathcal{F}}^{\ensuremath{-}1}$ have the same hardware errors, a symmetry boost in the fidelity of the combined $\mathcal{F}\mathcal{U}{\mathcal{F}}^{\ensuremath{-}1}$ quantum operation results when compared to the case in which the errors in $\mathcal{F}$ and ${\mathcal{F}}^{\ensuremath{-}1}$ are independently random. Running the complete gate-by-gate implemented Shor algorithm, we show that the symmetry-induced fidelity boost can be as large as a factor 4. While most of our analytical and numerical results concern the case of over- and under-rotation of controlled rotation gates, in the numerically accessible case of Shor's algorithm with a small number of qubits, we show explicitly that the symmetry boost is robust with respect to more general types of errors. While, expectedly, additional error types reduce the symmetry boost, we show explicitly, by implementing general off-diagonal $\mathrm{SU}(N)$ errors ($N=2,4,8$), that the boost factor scales like a Lorentzian in $\ensuremath{\delta}/\ensuremath{\sigma}$, where $\ensuremath{\sigma}$ and $\ensuremath{\delta}$ are the error strengths of the diagonal over- and underrotation errors and the off-diagonal $\mathrm{SU}(N)$ errors, respectively. The Lorentzian shape also shows that, while the boost factor may become small with increasing $\ensuremath{\delta}$, it declines slowly (essentially like a power law) and is never completely erased. We also investigate the effect of diagonal nonunitary errors, which, in analogy to unitary errors, reduce but never erase the symmetry boost. Going beyond the case of small quantum processors, we present analytical scaling results that show that the symmetry boost persists in the practically interesting case of a large number of qubits. We illustrate this result explicitly for the case of Shor factoring of the semiprime RSA-1024, where, analytically, focusing on over- and underrotation errors, we obtain a boost factor of about 10. In addition, we provide a proof of the fidelity product formula, including its range of applicability.

arXiv:1601.07497Physical Review AMay 8, 20180 citations

Machine Learning to Simulate Quantum Computing System Errors from Physical Observations

J. H. Feng, Xingchen Zhang, Guan-Hao Feng +1 more

In the context of quantum computing, error correction remains a pivotal challenge, primarily due to imperfect gate operations and environmental interactions. This study introduces a machine learning-based method to simulate and analyze these errors. Utilizing a minimal scalable 2-Majorana-zero-mode (2-MZM) island model within a one-dimensional p-wave topological superconductor, this research employs a detailed Hamiltonian approach combined with a bosonic thermal bath interaction. The analysis is conducted using the Pauli master equation and Monte Carlo simulations. This work’s novel contribution lies in applying Boosted Decision Tree with Gradient boosting (BDTG) and Multi-Layer Perceptron (MLP) machine learning techniques. These methods, trained on Monte Carlo simulation data, showed proficiency in predicting the evolution of error probabilities in the quantum system. The results indicate a significant potential for machine learning to offer a more efficient alternative for simulating quantum computing errors, thereby contributing to developing more robust quantum computing systems.In the context of quantum computing, error correction remains a pivotal challenge, primarily due to imperfect gate operations and environmental interactions. This study introduces a machine learning-based method to simulate and analyze these errors. Utilizing a minimal scalable 2-Majorana-zero-mode (2-MZM) island model within a one-dimensional p-wave topological superconductor, this research employs a detailed Hamiltonian approach combined with a bosonic thermal bath interaction. The analysis is conducted using the Pauli master equation and Monte Carlo simulations. This work’s novel contribution lies in applying Boosted Decision Tree with Gradient boosting (BDTG) and Multi-Layer Perceptron (MLP) machine learning techniques. These methods, trained on Monte Carlo simulation data, showed proficiency in predicting the evolution of error probabilities in the quantum system. The results indicate a significant potential for machine learning to offer a more efficient alternative for simulating quantum computing errors, thereby contributing to developing more robust quantum computing systems.

UniverseApr 4, 20251 citation

Boosted Bell-state measurements for photonic quantum computation

Nico Hauser, Matthias J. Bayerbach, Simone Evaldo D’Aurelio +5 more

Abstract Fault-tolerant fusion-based photonic quantum computing (FBQC) greatly relies on entangling two-photon measurements, called fusions. These fusions can be realized using linear-optical projective Bell-state measurements (BSMs). These linear-optical BSMs are limited to a success probability of 50%, greatly reducing the performance of FBQC schemes. The performance of FBQC can be improved using boosting, thus achieving higher success probabilities by adding additional resources. Here, we realize a boosted BSM using a 4 × 4 multiport splitter and an additional entangled photon pair, allowing for a success probability of up to 75%. In our experiment, we obtain a success probability for our boosted BSM of (69.3 ± 0.3)%, clearly exceeding the 50% limit. We further demonstrate the significance of our boosted BSM for FBQC, showing a threefold increase in robustness to photon loss and a significant reduction of the logical error rates.Abstract Fault-tolerant fusion-based photonic quantum computing (FBQC) greatly relies on entangling two-photon measurements, called fusions. These fusions can be realized using linear-optical projective Bell-state measurements (BSMs). These linear-optical BSMs are limited to a success probability of 50%, greatly reducing the performance of FBQC schemes. The performance of FBQC can be improved using boosting, thus achieving higher success probabilities by adding additional resources. Here, we realize a boosted BSM using a 4 × 4 multiport splitter and an additional entangled photon pair, allowing for a success probability of up to 75%. In our experiment, we obtain a success probability for our boosted BSM of (69.3 ± 0.3)%, clearly exceeding the 50% limit. We further demonstrate the significance of our boosted BSM for FBQC, showing a threefold increase in robustness to photon loss and a significant reduction of the logical error rates.

arXiv:2410.16380npj Quantum InformationMar 8, 202514 citations