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
Permutation invariant qubit codes of distance \(d\) can protect against \(d-1\) deletion errors [3,4], i.e., erasures of subsystems at unknown locations.Gates
There is a measurement-free code-switching protocol between a qubit stabilizer code and a PI qubit code [8].Decoding
Schur-Weyl-transform based decoder [9]. Here, one first measures nested total angular momenta, i.e., that of the first qubit, the first and second, followed by the first, second, and third, etc. Then, for codes with spacing, one measures the projection of the angular momentum modulo the spacing. Recovery can be performed by applying geometric phase gates [10] or the quantum Schur transform.Cousins
- Qubit stabilizer code— There is a measurement-free code-switching protocol between a qubit stabilizer code and a PI qubit code [8].
- 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.
Member of code lists
Primary Hierarchy
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”, Quantum 8, 1552 (2024) arXiv:2307.14840 DOI
- [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]
- J. Yu, S. R. Muleady, Y.-X. Wang, N. Schine, A. V. Gorshkov, and A. M. Childs, “Efficient preparation of Dicke states”, (2024) arXiv:2411.03428
- [8]
- 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
- [9]
- Y. Ouyang and G. K. Brennen, “Finite-round quantum error correction on symmetric quantum sensors”, (2024) arXiv:2212.06285
- [10]
- 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