[Paper Review] Quantum error mitigation for rotation symmetric bosonic codes with symmetry expansion

Suguru Endo, Yasunari Suzuki+5 more|arXiv (Cornell University)|Nov 11, 2022

Quantum Computing Algorithms and Architecture4 citations

TL;DR

This paper introduces a generalized symmetry expansion (SE) method for quantum error mitigation in rotation symmetric bosonic codes (RSBCs), enabling virtual state preparation and noise suppression using only two controlled-rotation gates and an ancilla qubit. The method dramatically reduces photon loss errors, achieving high-fidelity logical states with trace distances close to ideal, especially as the code's rotation order $M$ increases.

ABSTRACT

The rotation symmetric bosonic code (RSBC) is a unified framework of practical bosonic codes that have rotation symmetries, such as cat codes and binomial codes. While cat codes achieve the break-even point in which the coherence time of the encoded qubits exceeds that of unencoded qubits, with binomial codes nearly approaching that point, the state preparation fidelity needs to be still improved for practical quantum computing. Concerning this problem, we investigate the framework of symmetry expansion, a class of quantum error mitigation that virtually projects the state onto the noise-free symmetric subspace by exploiting the system's intrinsic symmetries and post-processing of measurement outcomes. Although symmetry expansion has been limited to error mitigation of quantum states immediately before measurement, we successfully generalize symmetry expansion for state preparation. To implement our method, we use an ancilla qubit and only two controlled-rotation gates via dispersive interactions between the bosonic code states and the ancilla qubit. Interestingly, this method also allows us to virtually prepare the RSBC states only from easy-to-prepare states, e.g., coherent states. We also discuss that the conventional symmetry expansion protocol can be applied to improve the computation fidelity when the symmetries of rotation bosonic codes are unavailable due to low measurement fidelity. By giving comprehensive analytical and numerical arguments regarding the trace distance between the error-mitigated state and the ideal state and the sampling cost of quantum error mitigation, we show that symmetry expansion dramatically suppresses the effect of photon loss. Our novel error mitigation method will significantly enhance computation accuracy in the near-term bosonic quantum computing paradigm.

Motivation & Objective

• Address the challenge of low state preparation fidelity in near-term bosonic quantum computing, particularly for rotation symmetric bosonic codes (RSBCs) like cat and binomial codes.
• Overcome limitations of existing error mitigation techniques that are restricted to post-measurement correction, extending them to pre-measurement and state preparation contexts.
• Develop a practical, low-overhead method to virtually prepare high-fidelity RSBC states from easily preparable input states such as coherent states.
• Demonstrate that symmetry expansion can suppress photon loss errors effectively, improving logical state fidelity and enabling near-term quantum advantage.
• Analyze the sampling cost and trace distance performance to establish theoretical and numerical bounds on error mitigation efficacy.

Proposed method

• Generalize symmetry expansion (SE) to work during state preparation by leveraging system symmetries and post-processing of measurement outcomes.
• Implement the method using a single ancilla qubit and only two controlled-rotation gates via dispersive interactions between the ancilla and the bosonic code.
• Use the symmetry projector $\hat{\mathcal{P}}_{2M}^{(0)}$ to virtually project the noisy state onto the noise-free symmetric subspace, effectively mitigating photon loss.
• Apply the formalism to both cat codes and binomial codes, showing that the method preserves logical expectation values and improves state fidelity.
• Derive analytical expressions for projection probability and trace distance under photon loss noise, validated numerically.
• Extend the framework to other symmetries, such as number-phase symmetries, by considering truncated projectors $\hat{\mathcal{P}}_X^{(L)}$ for phase noise mitigation.

Experimental results

Research questions

• RQ1 Can symmetry expansion be generalized beyond post-measurement error mitigation to include state preparation in bosonic codes?
• RQ2 How effective is the proposed SE method in suppressing photon loss errors in rotation symmetric bosonic codes?
• RQ3 What is the scaling of the sampling cost and trace distance with respect to the code's rotation order $M$ and average photon number?
• RQ4 Can the method virtually prepare high-fidelity RSBC states from simple input states like coherent states?
• RQ5 Can symmetry expansion mitigate non-rotation-symmetric noise, such as phase noise, when combined with alternative projectors like $\hat{\mathcal{P}}_X^{(L)}$?

Key findings

• The symmetry expansion method reduces the trace distance between the error-mitigated state and the ideal state significantly, with analytical results closely matching exact numerical solutions for cat codes.
• For cat codes, the error mitigation performance improves with increasing $M$, and the trace distance decreases as $M$ grows, indicating better fidelity.
• The sampling cost scales as $M^2$ for the rotation order $M$, which is favorable for near-term implementation with moderate $M$.
• The method enables virtual preparation of RSBC states from coherent states with high effective fidelity, bypassing the need for complex pulse sequences.
• Truncated projectors $\hat{\mathcal{P}}_X^{(L)}$ effectively suppress phase noise, which is not mitigated by rotation symmetry-based projectors, demonstrating complementary utility.
• Numerical results confirm that increasing the truncation level $L$ enhances the error mitigation effect for phase noise, with Wigner function reconstructions showing improved state purity.