Logical Shadow Tomography with Gottesman-Kitaev-Preskill Codes for Continuous Variable Quantum Systems

Core Concepts

This paper introduces a novel shadow tomography protocol for continuous variable (CV) quantum systems using Gottesman-Kitaev-Preskill (GKP) codes, enabling efficient estimation of logical observables and reconstruction of logical states.

Abstract

• Bibliographic Information: Conrad, J., Eisert, J., & Flammia, S. T. (2024). Chasing shadows with Gottesman–Kitaev–Preskill codes. arXiv preprint arXiv:2411.00235v1.
• Research Objective: This paper aims to develop a shadow tomography protocol specifically designed for continuous variable quantum systems using GKP codes, addressing the challenge of infinite dimensionality in these systems.
• Methodology: The authors leverage the structure of GKP codes and their associated logical Clifford operations to design a twirling protocol for CV systems. They combine displacement operator twirling with Gaussian unitary twirling to effectively project physical channels onto logical depolarizing channels. This enables the application of classical shadow tomography techniques to estimate logical observables and reconstruct logical states.
• Key Findings: The paper demonstrates that by twirling a CV-POVM with logical GKP Clifford gates, the logical action of the POVM can be projected onto a depolarizing channel. This allows for the reconstruction of logical expectation values of operators relative to the chosen GKP codes. The authors provide specific examples using heterodyne and photon-parity measurements, highlighting the versatility of their approach. Notably, the heterodyne measurement protocol offers a method to approximate arbitrary physical states as convex combinations of Gaussian states, potentially aiding in simulating GKP error correction.
• Main Conclusions: The proposed GKP logical shadow tomography protocol provides an efficient and robust method for characterizing logical information encoded in CV quantum systems. This approach bridges the gap between discrete and continuous variable systems in the context of shadow tomography, offering new tools for analyzing and understanding GKP codes and their applications in quantum computation.
• Significance: This research significantly contributes to the field of quantum computing by introducing a practical method for characterizing and verifying logical information in CV systems, which are becoming increasingly relevant for fault-tolerant quantum computation.
• Limitations and Future Research: The sample complexity bounds derived in the paper, while rigorous, scale exponentially with system size. Future research could explore methods to improve these bounds and investigate the practical implementation of the proposed protocols in experimental settings. Additionally, exploring the application of this technique to other bosonic codes and more complex logical operations could be promising research directions.

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Chasing shadows with Gottesman-Kitaev-Preskill codes

by Jonathan Con... at arxiv.org 11-04-2024

https://arxiv.org/pdf/2411.00235.pdf

Deeper Inquiries

How does the performance of this GKP-based shadow tomography protocol compare to other tomography methods specifically designed for CV systems, considering both sample complexity and robustness to noise?

The GKP-based shadow tomography protocol presents a distinct approach compared to traditional CV tomography methods, each possessing strengths and weaknesses regarding sample complexity and noise resilience. Let's break down the comparison:
Sample Complexity:

GKP-based Shadow Tomography: This method suffers from exponential scaling of sample complexity with system size (number of modes, n, and local dimension, d). This arises from the need to twirl over the logical Clifford group, whose size grows super-exponentially. This scaling can be prohibitive for large systems.
Traditional CV Tomography: Methods like homodyne tomography and quantum state reconstruction using pattern functions typically exhibit polynomial scaling in the number of modes and the desired resolution in phase space. However, these methods often require a large number of measurement settings, which can be experimentally challenging.
Robustness to Noise:

GKP-based Shadow Tomography: This protocol inherits the inherent noise resilience of shadow tomography. The projection property of the Clifford twirl effectively mitigates certain types of noise, projecting noisy POVMs onto depolarizing channels with modified parameters that can be accounted for in post-processing. This makes the protocol suitable for NISQ environments.
Traditional CV Tomography: These methods can be sensitive to noise, particularly for states with large photon numbers or in the presence of experimental imperfections. Noise can lead to unphysical state reconstructions and biased estimates of observables.
Other Considerations:

GKP Code Overhead: The GKP-based approach requires encoding the quantum information into the GKP code, which introduces overhead in terms of the number of physical modes required.
Target Observables: GKP-based shadow tomography is well-suited for estimating logical observables, which are natural in the context of quantum error correction. Traditional methods might be more suitable for directly accessing physical properties of the CV state.
In summary: GKP-based shadow tomography offers noise resilience and the ability to estimate logical observables efficiently. However, the exponential scaling of sample complexity limits its applicability to smaller systems. Traditional CV tomography methods, while potentially more efficient for larger systems, can be more susceptible to noise. The choice of the most appropriate method depends on the specific application, the size of the system, and the desired trade-off between sample complexity and noise robustness.

Could the limitations of exponential scaling in sample complexity be mitigated by employing alternative coding schemes or leveraging specific properties of the target states or observables?

Yes, the exponential scaling limitation of GKP-based shadow tomography could potentially be mitigated by exploring alternative strategies:
Alternative Coding Schemes:

Lower Dimensional Codes: Employing GKP codes with lower local dimension d would directly reduce the size of the logical Clifford group, leading to improved sample complexity. However, this might come at the cost of reduced error correction capabilities.
Concatenated Codes: Concatenating GKP codes with other codes, such as qubit-based codes, could offer a trade-off between error correction performance and sample complexity. The outer code could be chosen to be more amenable to efficient shadow tomography.
Tailored Codes: Designing codes specifically optimized for shadow tomography, potentially by considering the structure of the Clifford group and its action on the code space, could lead to more efficient twirling protocols.
Leveraging State and Observable Properties:

Low-Rank Observables: If the target observables are known to be low-rank, the sample complexity can be significantly reduced, as shown in the original shadow tomography work [9].
State Preparation Assumptions: If the input states are known to belong to a specific class, such as Gaussian states or states with bounded photon number, tailored twirling protocols or alternative tomography methods might be more efficient.
Sparsity: If the logical state or the target observables exhibit sparsity in a suitable basis, compressed sensing techniques could be employed to reduce the sample complexity.
Other Strategies:

Approximate Twirling: Instead of performing a full twirl over the logical Clifford group, approximate twirling protocols using random walks or other sampling techniques could offer a trade-off between accuracy and sample complexity.
Adaptive Tomography: Adaptively choosing measurement settings based on prior information or intermediate measurement outcomes could potentially reduce the overall number of samples required.
In conclusion: While the exponential scaling presents a challenge, exploring alternative coding schemes, leveraging specific properties of the target states and observables, and employing advanced sampling and estimation techniques hold promise for mitigating this limitation and making GKP-based shadow tomography more practical for larger and more complex CV systems.

What are the potential implications of representing arbitrary physical states as convex combinations of Gaussian states, as achieved through the heterodyne measurement protocol, for broader applications in quantum information processing beyond error correction?

The ability to represent arbitrary physical states as convex combinations of Gaussian states, as achieved through the heterodyne measurement protocol in GKP-based shadow tomography, has intriguing implications for quantum information processing beyond error correction:
Efficient Classical Simulation:

Gaussian State Simulability: Gaussian states are known to be efficiently simulable classically [15]. This representation allows for efficient classical simulation of certain aspects of arbitrary quantum states, particularly those dominated by Gaussian features.
Resource Estimation: This decomposition could provide insights into the resources required to prepare or manipulate specific quantum states. The weights and characteristics of the constituent Gaussian states could serve as resource quantifiers.
State Engineering and Manipulation:

Gaussian State Engineering: The representation suggests a potential pathway for engineering arbitrary quantum states by appropriately combining and manipulating Gaussian states. This could be particularly relevant for optical platforms where Gaussian operations are readily available.
State Discrimination and Classification: The decomposition could facilitate the discrimination and classification of quantum states. By analyzing the weights and properties of the Gaussian components, one could potentially distinguish between different classes of states.
Quantum Communication and Cryptography:

Continuous-Variable Quantum Key Distribution: The representation might offer new approaches to continuous-variable quantum key distribution protocols, where Gaussian states are often employed.
Secret Sharing and Entanglement Distribution: The decomposition could be relevant for developing novel secret sharing and entanglement distribution protocols based on continuous-variable systems.
Quantum Metrology and Sensing:

Parameter Estimation: The representation could be beneficial for parameter estimation tasks in quantum metrology. By analyzing the response of the Gaussian components to the parameter of interest, one could potentially enhance the sensitivity of the measurement.
Beyond these specific applications, the representation of arbitrary states as convex combinations of Gaussian states provides a new lens through which to view and analyze quantum information processing tasks. It bridges the gap between the often-considered ideal case of Gaussian states and the more general and complex realm of arbitrary quantum states, potentially leading to new insights and practical tools for quantum technologies.