Designing quantum error correction codes for practical spin qudit systems

The implementation of practical error correction protocols is essential for deployment of quantum information technologies. Ways of exploiting high-spin nuclei, which have multilevel 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-<a:math xmlns:a="http://www.w3.org/1998/Math/MathML"><a:mi>X</a:mi><a:mo>,</a:mo><a:mo> </a:mo><a:mi>Y</a:mi></a:math> and <b:math xmlns:b="http://www.w3.org/1998/Math/MathML"><b:mi>Z</b:mi></b:math> 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 for 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 and 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 become crucial for the implementation of quantum technologies.