Permutation-invariant codes: a numerical study and qudit constructions

TL;DR

It is conjecture that qubit PI codes correcting up to $d-1$ deletion errors have block length, which implies an upper bound $d \leq \sqrt{12n-3}/3$ on their code distance, and that PR codes can saturate this bound.

Abstract

We investigate Permutation-Invariant (PI) quantum error-correcting codes encoding a logical qudit of dimension $\mathrm{d}_\mathrm{L}$ in PI states using physical qudits of dimension $\mathrm{d}_\mathrm{P}$. We extend the Knill--Laflamme (KL) conditions for $d-1$ deletion errors from qubits to qudits and investigate numerically both qubit ($\mathrm{d}_\mathrm{L} = \mathrm{d}_\mathrm{P} = 2$) and qudit ($\mathrm{d}_\mathrm{L} > 2$ or $\mathrm{d}_\mathrm{P} > 2$) PI codes. We analyze the scaling of the block length $n$ in terms of the code distance $d$, and compare to existing families of PI codes due to Ouyang, Aydin--Alekseyev--Barg (AAB) and Pollatsek--Ruskai (PR). Our three main findings are: (i) We conjecture that qubit PI codes correcting up to $d-1$ deletion errors have block length $n(d) \geq (3d^2 + 1) / 4$, which implies an upper bound $d \leq \sqrt{12n-3}/3$ on their code distance, and that PR codes can saturate this bound. (ii) For qudit PI codes encoding a single qudit we numerically observe that increasing $\mathrm{d}_\mathrm{P}$ results in $n$ monotonically decreasing and approaching the quantum Singleton bound $n(d) \geq 2d-1$. (iii) We propose a semi-analytic extension of the qubit AAB construction to qudits that finds explicit solutions by solving a linear program. Our results therefore provide key insights into lower bounds on the block length scaling of both qubit and qudit PI codes, and demonstrate the benefit of increased physical local dimension in the context of PI codes.

Figures (26)

• Figure 1: Numerical optimization of the cost function of \ref{['eq:f_cost']} with (a) codewords with complex coefficients and (b) codewords restricted to the PR family pollatsek2004permutationally. For each $n$ and $t$ the optimization is run up to $1000$ times (solid points), with each repetition using a maximum of $1.2\times10^6$ optimization steps. Vertical dashed lines show, for each $t$, the smallest $n$ for which a solution exists, i.e. when $f_{\rm cost} < 10^{-18}$ (black dashed line). This is the minimal block length, $n_{\rm min}(t)$. We find $n_{\rm min}(t=1)=7$, $n_{\rm min}(t=2)=19$ and $n_{\rm min}(t=3)=37$ for both codeword families. Due to the reduced computational resources required, we are able to obtain $n_{\rm min}(t=4)=61$ and $n_{\rm min}(t=5)=91$ for the PR code family.
• Figure 2: Block length $n(t)$ versus error weight $t$ for minimal PI codes with complex coefficients (blue dots) and for minimal PR PI codes pollatsek2004permutationally (blue crosses) determined numerically at $n_{\rm min} = 7,19,37,61,91$ for $t=1,2,3,4,5$, c.f. \ref{['fig:NumericsOptimization']}. The quadratic fit (solid blue line) to $n_{\rm min}(t) = 3t^2 + 3t+1$ demonstrates perfect agreement, which forms the basis of \ref{['con:min_PI_qubit_scaling']}. We also plot the block length scaling of the AAB aydin2024family (orange line) and Ouyang ouyang2014permutation (green line) code families.
• Figure 3: Codeword coefficients $\alpha_i$ for $\ket{c_0}$ (solid bars) and $\beta_i$ for $\ket{c_1}$ (hollow bars) for one example real minimal qubit PI code for block length $n = 7$ and error weight $t = 1$. The absolute value of the codeword coefficients have mirror symmetry with respect to reversing the composition labels, as emphasized by the bar coloring. We also observe that every second coefficient has the opposite sign compared to its mirror pair, starting at $\alpha_0$ (for $\ket{c_0}$) and $\beta_1$ (for $\ket{c_1}$). These mirror and phase-flip symmetries are defined in \ref{['eqn:mirror_phase_sym']}.
• Figure 4: Codeword coefficients $\alpha_i$ for $\ket{c_0}$ (a) and $\beta_i$$\ket{c_1}$ (b) for real minimal qubit PI codes with block length $n = 7$ and error weight $t = 1$. We fix $M = 2$ coefficients in the first codeword, i.e. $\alpha_0 = \alpha_1 = 0$. We numerically minimize the cost function of \ref{['eq:f_cost']} using the remaining $14$ codeword coefficients. Repeating the minimization $1000$ times from random initial choices for the non-fixed codeword coefficients, we converge to the same eight solutions shown here. A valid solution exists for both positive and negative signs of each of the $\alpha_2$, $\alpha_7$, $\beta_0$ and $\beta_5$, giving a total of eight solutions.
• Figure 5: Table of all solution space projections on coordinate planes defined by the $26$ possible pairs of $\ket{c_0}$ codeword coefficient fixings, $\alpha_i,\alpha_j$ for $0 \leq i < j \leq 7$, found numerically for real minimal qubit PI codes with $(n,t)=(7,1)$. We do not enforce the phase-flip nor mirror symmetry. The grid search is performed only in the top left quadrant with $0 \leq \alpha_i,\alpha_j \leq 0.85$ and with the constraint $\alpha_i^2 + \alpha_j^2 \leq 1$. The remaining three quadrants are constructed using the symmetries observed in \ref{['sec:codewordsymmetries']}. The solution corresponding to the AAB code family of Ref. aydin2024family is indicated by the red star.

Theorems & Definitions (39)

• Definition 2.1
• Definition 2.2
• Definition 2.3
• Theorem 2.1
• proof
• Definition 2.4
• Theorem 2.2
• proof
• Corollary 2.1
• Definition 2.5