Permutation-invariant (PI) code[1]
Description
Block quantum code such that any permutation of the subsystems leaves any codeword invariant. In other words, the automorphism group of the code contains the symmetric group \(S_n\).
There is a notion of a Wigner functions for PI subspaces [2].
Qudit Dicke states and the discrete simplex mapping
For \(n\)-modular-qudit block codes with qudit dimension \(q\), an often used basis for the PI subspace consists of the qudit Dicke states.
Qudit Dicke states: A qudit Dicke state is an equal superposition of all qudit basis elements whose labels have the same composition, \begin{align} |D_{\mathbf{c}}\rangle=\frac{1}{\sqrt{\binom{n}{\mathbf{c}}}}\sum_{\substack{\mathbf{n}\in\mathbb{Z}_{q}^{n}\\ C(\mathbf{n})=\mathbf{c} } }|\mathbf{n}\rangle\,, \tag*{(1)}\end{align} where \(\binom{n}{\mathbf{c}}\) is the multinomial coefficient. Above, the composition \(C\) of a qudit basis label \(\mathbf{n}\) tabulates the number of each type of element present in the label. For example, the label \(\mathbf{n}=(0313)\) has composition \(C(\mathbf{n})=(1102)\), whose coordinates denote the number of zeroes (one), number of ones (one), number of twos (zero), and number of threes (two) present in the label. The \(q=2\) case reduces to the Dicke-state mapping.
Qudit Dicke states are in one-to-one correspondence with points on the discrete simplex \(\Delta_{q,n}\), which houses the totally symmetric irrep of \(SU(q)\) [3]. Since both constant-excitation Fock-state codes and single-spin codes defined on the totally symmetric \(SU(q)\)-irrep are in the same correspondence, the three types of codes and their distances are interconvertible [3].
Protection
Noise models can be categorized as those that cause the state to leave the maximally symmetric subspace and those that do not. PI codes of distance \(d\) can protect against \(d-1\) deletion errors [4–7], i.e., erasures of subsystems at unknown locations.
Other protection depends on the code family. The GNU PI family (parameterized by \(t\)) protects against arbitrary weight \(t\) qubit errors and approximately corrects spontaneous decay errors [8,9]. Other related codes protect against AD [10] while admitting a constant number of excitations.
Rate
There exists a family of PI codes for \(q=N\) with logical dimension \(K = o(2^N)\) and distance of order \(o(N/\log N)\) [3].Encoding
With quantum harmonic oscillators (superconducting charge qubits in a ultrastrong coupling regime) in \(O(N)\) as in [11]. Can be done in \(O(N^2)\) steps using quantum circuits [12], or using geometric phase gates in \(O(N)\) [13].Transversal Gates
Qudit Dicke states are in one-to-one correspondence with points on the discrete simplex \(\Delta_{q,n}\), which houses the totally symmetric irrep of \(SU(q)\) [3]. Any transversal gates of the form \(U^{\otimes N}\), with \(U \in SU(q)\), will implement logical operations in a subgroup of \(SU(q)\).Notes
PI codes can be constructed using real polynomials for high-dimensional qudit spaces [14].Qubit and qudit PI codes obtained from numerical optimization routines are useful for entanglement distillation [15; Appx. B.1].Qudit Dicke state preparation [16].Cousins
- Simplex integer-based code— Simplex integer-based codes can be partitioned into qudit PI codewords whose error-correction is guaranteed by the Tverberg theorem [3; Thm. VII.5].
- Constant-excitation (CE) code— Modular-qudit PI codes can be converted to constant-excitation Fock-state codes via the simplex mapping [3; Prop. V.2]. Any transversal gates are mapped to Gaussian gates on the Fock-state codes [3].
- Fock-state bosonic code— Modular-qudit PI codes can be converted to constant-excitation Fock-state codes via the simplex mapping [3; Prop. V.2]. Any transversal gates are mapped to Gaussian gates on the Fock-state codes [3].
- Single-spin code— Modular-qudit PI codes can be converted to spin codes defined on the completely symmetric irrep of \(SU(q)\) via the simplex mapping [3; Prop. IV.2]. Any transversal gates are mapped to \(SU(q)\) gates on the spin codes [3].
- Editing code— PI codes of distance \(d\) can protect against \(d-1\) (quantum) deletion errors.
Primary Hierarchy
References
- [1]
- H. Pollatsek and M. B. Ruskai, “Permutationally Invariant Codes for Quantum Error Correction”, (2004) arXiv:quant-ph/0304153
- [2]
- M. Calixto and J. Guerrero, “Wigner quasi-probability distribution for symmetric multi-quDit systems and their generalized heat kernel”, (2025) arXiv:2507.14866
- [3]
- A. Aydin, V. V. Albert, and A. Barg, “Quantum error correction beyond \(SU(2)\): spin, bosonic, and permutation-invariant codes from convex geometry”, (2025) arXiv:2509.20545
- [4]
- M. Hagiwara and A. Nakayama, “A Four-Qubits Code that is a Quantum Deletion Error-Correcting Code with the Optimal Length”, (2020) arXiv:2001.08405
- [5]
- A. Nakayama and M. Hagiwara, “Single Quantum Deletion Error-Correcting Codes”, (2020) arXiv:2004.00814
- [6]
- Y. Ouyang, “Permutation-invariant quantum coding for quantum deletion channels”, 2021 IEEE International Symposium on Information Theory (ISIT) 1499 (2021) arXiv:2102.02494 DOI
- [7]
- T. Shibayama and M. Hagiwara, “Permutation-Invariant Quantum Codes for Deletion Errors”, 2021 IEEE International Symposium on Information Theory (ISIT) 1493 (2021) arXiv:2102.03015 DOI
- [8]
- Y. Ouyang, “Permutation-invariant quantum codes”, Physical Review A 90, (2014) arXiv:1302.3247 DOI
- [9]
- Y. Ouyang and J. Fitzsimons, “Permutation-invariant codes encoding more than one qubit”, Physical Review A 93, (2016) DOI
- [10]
- Y. Ouyang and R. Chao, “Permutation-Invariant Constant-Excitation Quantum Codes for Amplitude Damping”, IEEE Transactions on Information Theory 66, 2921 (2020) arXiv:1809.09801 DOI
- [11]
- C. Wu, Y. Wang, C. Guo, Y. Ouyang, G. Wang, and X.-L. Feng, “Initializing a permutation-invariant quantum error-correction code”, Physical Review A 99, (2019) DOI
- [12]
- A. Bärtschi and S. Eidenbenz, “Deterministic Preparation of Dicke States”, Lecture Notes in Computer Science 126 (2019) arXiv:1904.07358 DOI
- [13]
- M. T. Johnsson, N. R. Mukty, D. Burgarth, T. Volz, and G. K. Brennen, “Geometric Pathway to Scalable Quantum Sensing”, Physical Review Letters 125, (2020) arXiv:1908.01120 DOI
- [14]
- Y. Ouyang, “Permutation-invariant qudit codes from polynomials”, Linear Algebra and its Applications 532, 43 (2017) DOI
- [15]
- B. Desef and M. B. Plenio, “Optimizing quantum codes with an application to the loss channel with partial erasure information”, Quantum 6, 667 (2022) arXiv:2105.13233 DOI
- [16]
- N. B. Kerzner, F. Galeazzi, and R. I. Nepomechie, “Simple ways of preparing qudit Dicke states”, (2025) arXiv:2507.13308
Page edit log
- Victor V. Albert (2022-07-26) — most recent
- Victor V. Albert (2021-12-16)
- Benjamin Quiring (2021-12-16)
Cite as:
“Permutation-invariant (PI) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/permutation_invariant