Designing quantum error correction codes for practical spin qudit

TL;DR

This work analyzes quantum error correction for spin-qudit memories in solid-state systems under realistic Hamiltonians that include hyperfine and quadrupole interactions. It shows that a naive spin-7/2 code-word fails Knill-Laflamme criteria at finite fields, but a spin-9/2 code-word tailored to the full Hamiltonian can satisfy error-correction criteria at accessible fields; it also explores multi-qudit encoding to mitigate electric-field noise. The authors provide explicit code-words and encoding/decoding sequences for Sb- and Bi-donor systems and propose a three-spin-7/2 scheme as a practical alternative, discussing experimental feasibility and required fidelities. Overall, the paper advances fault-tolerant quantum memory design by combining Hamiltonian-aware code construction with multi-qudit protection strategies for robust operation beyond the NISQ era.

Abstract

The implementation of practical error correction protocols is essential for deployment of quantum information technologies. Ways of exploiting high-spin nuclei, which have multi-level quantum resources, have attracted interest in this context because they offer additional Hilbert space dimensions in a spatially compact and theoretically efficient structure. We present a quantitative analysis of the performance of a spin-qudit-based error-correctable quantum memory, with reference to the actual Hamiltonians of several potential candidate systems. First, the ideal code-word implemented on a spin-7/2 nucleus, which provides first order Pauli-$X$, $Y$ and $Z$ error correction, has intrinsic infidelity due to mixed eigenstates under realistic conditions. We confirm that expansion to a spin-9/2 system with tailored code-words can compensate this infidelity. Second, we claim that electric field fluctuations -- which are inevitable in real systems -- should also be considered as a noise source, and we illustrate an encoding/decoding scheme for a multi-spin-qudit-based error correction code that can simultaneously compensate for both electric and magnetic field perturbations. Such strategies are important as we move beyond the current noisy-intermediate quantum era, and fidelities above two or three nines becomes crucial for implementation of quantum technologies.

Figures (16)

• Figure 1: (a) A energy level diagram for the spin states of the Si:Sb system. (b) the nuclear spin transition frequencies within the $m_s = -1/2$ manifold. (c) the shift of nuclear transition frequencies from their values at a field of 1 T as a function of magnetic field.
• Figure 2: The expectation value for $\langle0_L\rvert I_Z\lvert0_L\rangle$ and $\langle1_L\rvert I_Z\lvert1_L\rangle$ for original 7/2 code-words, with practical environment of Si:Sb.
• Figure 3: The lines show set of $\epsilon_1$ and $\epsilon_2$, which satisfying zero expectation value for $\langle0_L\rvert I_Z\lvert0_L\rangle - \langle1_L\rvert I_Z\lvert1_L\rangle$ (blue) , $\langle0_L\rvert I^{\dagger}_X I_X \lvert1_L\rangle$ and (red) $\langle0_L\rvert I^{\dagger}_X I_Y \lvert1_L\rangle$ (black).
• Figure 4: The lines show set of $\epsilon_1$ and $\epsilon_2$, which satisfying zero expectation value for $\langle0_L\rvert I_Z\lvert0_L\rangle - \langle1_L\rvert I_Z\lvert1_L\rangle$ (black) , $\langle0_L\rvert I^{\dagger}_X I_X \lvert0_L\rangle - \langle1_L\rvert I^{\dagger}_X I_X \lvert1_L\rangle$ (red) , with ideal 9/2 code-word and additional $\epsilon_1$ and $\epsilon_2$ parameters.
• Figure 5: (a) $\epsilon_1 , \epsilon_2$ for the partially tailored 7/2 code-words, which satisfies $\langle0_L\rvert I_Z\lvert0_L\rangle - \langle1_L\rvert I_Z\lvert1_L\rangle =0$ and $\langle0_L\rvert I^{\dagger}_X I_X\lvert1_L\rangle = 0$ conditions. (b) The expectation values for KL criteria, for partially tailored code-words and original 7/2 code-words.