Permutation-invariant qudit codes from polynomials (1604.07925v4)

Abstract: A permutation-invariant quantum code on $N$ qudits is any subspace stabilized by the matrix representation of the symmetric group $S_N$ as permutation matrices that permute the underlying $N$ subsystems. When each subsystem is a complex Euclidean space of dimension $q \ge 2$, any permutation-invariant code is a subspace of the symmetric subspace of $(\mathbb C^q)^N.$ We give an algebraic construction of new families of of $d$-dimensional permutation-invariant codes on at least $(2t+1)^2(d-1)$ qudits that can also correct $t$ errors for $d \ge 2$. The construction of our codes relies on a real polynomial with multiple roots at the roots of unity, and a sequence of $q-1$ real polynomials that satisfy some combinatorial constraints. When $N > (2t+1)^2(d-1)$, we prove constructively that an uncountable number of such codes exist.

• The paper presents an algebraic framework using polynomials to construct new families of permutation-invariant qudit codes designed for correcting errors in N-qudit systems.
• The construction relies on specific polynomial root conditions and sequence properties to meet the combinatorial distance requirements necessary for correcting t errors.
• The paper demonstrates the existence of uncountably many such codes and provides examples, showing the method generalizes previous techniques and broadens the scope of quantum error correction.

Overview of "Permutation-invariant qudit codes from polynomials"

This paper, authored by Yingkai Ouyang, presents an algebraic approach for constructing new families of permutation-invariant quantum codes. These codes are designed for systems of $N$ qudits, where each qudit represents a $q$-level system, and they are invariant under permutations of the subsystems. Particularly, the paper focuses on permutation-invariant codes that can correct a specified number of errors, $t$.

Permutation-invariant quantum codes (PIQCs) are a subclass of quantum error-correcting codes (QECCs), which are crucial for safeguarding quantum information against decoherence and operational errors. PIQCs are robust against symmetric noise, which is common in many physical systems. Unlike conventional QECCs, which are often linked to finite field theories and binary systems, PIQCs deal with the symmetric subspace of $( \mathbb{C}^q )^{\otimes N}$ and utilize non-additive code structures.

Key Contributions

1. Construction Framework: The core contribution of the paper lies in an algebraic framework for constructing PIQCs using polynomials. The construction is based on a polynomial $f(x)$ with specific roots conditions and a sequence of polynomials $p_1(z), \ldots, p_q(z)$ that satisfy combinatorial distance requirements. This setup ensures that codes can correct up to $t$ errors.
2. Polynomial Roots and Combination: The construction demands that $f(x)$ has a root at $x = 1$ with multiplicity $2t + 1$, ensuring robustness in error correction. The polynomials $p_i(z)$ partition the integer $N$ and are arranged to meet a distance criterion, which is key for their error-correction capabilities.
3. Existence of Uncountable Codes: The paper convincingly argues, using algebraic techniques, that there exists an uncountable number of such codes for a given configuration, provided $N$ exceeds a threshold based on the desired error correction level and dimensionality.
4. Examples and Construction: Several explicit examples of PIQCs are demonstrated, including codes that encode different $d$-level systems while correcting errors. For instance, codes for qubits that correct single errors are showcased, validating the theoretical framework with concrete instances.
5. Comparison and Implications: A comparison with previous PIQC constructions, like those relying on binomial distributions, illustrates how the proposed approach generalizes earlier methods. The paper also discusses how the construction relates to previously known permutation-invariant codes.

Implications and Future Directions

The construction of permutation-invariant qudit codes proposed in the paper has significant theoretical and practical implications for quantum error correction. The algebraic method enables the design of PIQCs that are not restricted to binary or finite field systems, broadening the scope of QECCs to encompass more complex quantum systems found in practical quantum computing and information tasks.

Future research can explore:

• Optimization of PIQCs:

Further work could focus on optimizing these codes for hardware implementations, perhaps targeting specific quantum architectures or error models beyond symmetric noise.

• Combination with Other Techniques:

Exploring combinations of the described PIQC constructions with other quantum redundancy and error-mitigation techniques may yield novel approaches with enhanced error tolerance.

• Generalization to Multi-qubit Systems:

Extending the ideas to multi-qubit systems while maintaining permutation-invariance could lead to a richer understanding of QECCs in diverse quantum information protocols.

Ultimately, this paper advances the field by providing a scalable and flexible framework for developing powerful quantum codes, enhancing the robustness of quantum information systems in the presence of realistic noise.