Description
Block quantum code defined on two-dimensional subsystems such that any permutation of the subsystems leaves any codeword invariant.
Dicke states: For \(n\)-qubit block codes, an often used basis for the \(n+1\)-dimensional PI subspace consists of the Dicke states \(|D^n_w\rangle\) -- normalized PI states of \(w\) excitations, i.e., a normalized sum over all binary-string basis elements with \(w\) ones and \(n - w\) zeroes. For example, the single-excitation Dicke state on three qubits is \begin{align} |D_{1}^{3}\rangle=\frac{1}{\sqrt{3}}\left(|001\rangle+|010\rangle+|100\rangle\right)~. \tag*{(1)}\end{align} The \(n+1\)-dimensional PI space can be thought of as a standalone spin-\(n/2\) quantum system, yielding a way to convert between permutation-invatiant qubit codes and \(SU(2)\) spin codes. A single-spin code for the \(SU(2)\) group correcting spherical tensors can be mapped into a PI qubit code with an analogous distance [1][2; Thm. 1].
Protection
Encoding
Gates
Decoding
Parents
Children
- Post-selected PI code
- Binary dihedral PI code
- Combinatorial PI code
- Qudit GNU PI code
- \(((n,1,2))\) Bravyi-Lee-Li-Yoshida PI code
- Clifford spin code — Clifford codes for spins housing representations of \(SU(2)\) yield PI qubit codes with non-trivial distance when the single spin-\(n/2\) is treated as the permutationally invariant subspace of \(n\) qubits via the Dicke-state mapping. The subgroup of gates of a Clifford spin code is implemented transversally via this mapping [1].
Cousins
- Qubit stabilizer code — There is a measurement-free code-switching protocol between a qubit stabilizer code and a PI qubit code [7].
- Eigenstate thermalization hypothesis (ETH) code — Several instances of ETH codes contain PI qubit codewords.
- Single-spin code — Single-spin codes are subspaces of a single large spin, which can be either standalone or correspond to the PI subspace of a set of spins via the Dicke state mapping.
References
- [1]
- S. Omanakuttan and J. A. Gross, “Multispin Clifford codes for angular momentum errors in spin systems”, Physical Review A 108, (2023) arXiv:2304.08611 DOI
- [2]
- E. Kubischta and I. Teixeira, “Permutation-Invariant Quantum Codes with Transversal Generalized Phase Gates”, (2024) arXiv:2310.17652
- [3]
- 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
- [4]
- A. Nakayama and M. Hagiwara, “Single Quantum Deletion Error-Correcting Codes”, (2020) arXiv:2004.00814
- [5]
- H. Buhrman, M. Folkertsma, B. Loff, and N. M. P. Neumann, “State preparation by shallow circuits using feed forward”, (2024) arXiv:2307.14840
- [6]
- L. Piroli, G. Styliaris, and J. I. Cirac, “Approximating Many-Body Quantum States with Quantum Circuits and Measurements”, Physical Review Letters 133, (2024) arXiv:2403.07604 DOI
- [7]
- Y. Ouyang, Y. Jing, and G. K. Brennen, “Measurement-free code-switching for low overhead quantum computation using permutation invariant codes”, (2024) arXiv:2411.13142
- [8]
- Y. Ouyang and G. K. Brennen, “Finite-round quantum error correction on symmetric quantum sensors”, (2024) arXiv:2212.06285
- [9]
- X. Wang and P. Zanardi, “Simulation of many-body interactions by conditional geometric phases”, Physical Review A 65, (2002) arXiv:quant-ph/0111017 DOI
Page edit log
- Victor V. Albert (2023-04-18) — most recent
Cite as:
“PI qubit code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/qubit_permutation_invariant