Fault-Tolerant Encoding of Logical Qudits in Spin Systems

TL;DR

A quantitative comparison shows that the logical qudit encoding proposed here offers an exponential resource advantage over multi-level mappings from logical qubits, and therefore it is believed this strategy can pave the way for realizing logical qudit encodings in finite multi-level physical systems.

Abstract

The universal quantum computer will enable the simulation of arbitrary quantum states governed by arbitrary Hamiltonians. In this context, it is essential to equip future quantum processors with fault-tolerant logical qudits, since qudits naturally align with the simulation of multi-level physical systems. In this study, we present a general framework and working examples of fault-tolerant logical qudit encoding using spin systems, which are among the most coherent and robust finite multi-level physical platforms. The d-dimensional logical qudit encoding with distance-3 (or 5) codewords can be designed within a 12d (or 40d)-dimensional Hilbert space, and the design can be further generalized to 2t+1-distance codes and to encodings exploiting multiple physical qudits. A quantitative comparison shows that the logical qudit encoding proposed here offers an exponential resource advantage over multi-level mappings from logical qubits, and therefore we believe this strategy can pave the way for realizing logical qudit encodings in finite multi-level physical systems.

Figures (5)

• Figure 1: (a) Brief schematic of qutrit Z-error correction code. (see main text for details) (b) The qutrit fidelity as a function of time with distance-1 (black), distance-3 (red), and distance-5 (blue) QEC code. Inset: Gain from fault-tolerant encoding, for distance-3 (red) and distance-5 (blue) encoding.
• Figure 2: Required Hilbert space for logical qudit encoding in this work (blue) and convensional mapping into multiple logical qubits (orange), according qudit dimension $d$ (a) and code-word distance (b).
• Figure 3: Qutrit fidelity as a function of time without (black) and with (red) error correction. The effects of decoherence during the decoding sequence and imperfect gate rotation angle errors are also simulated, shown as green and purple solid lines, and a blue dotted line (see main text for details).
• Figure A1: The gates for encoding pulses of qutrit Z-error correction code. Unitary rotations $U_{\theta i}$ are around $y$ axis, with $cos(\theta_i) = \sqrt{1/2}, \sqrt{3/10}, \sqrt{3/7}, \sqrt{7/20}, \sqrt{7/13}$ for $i = 1,2,3,4,5$, respectively.
• Figure A2: The gates for decoding pulses of qutrit Z-error correction code. Unitary rotations $U_{\theta i}$ are around $y$ axis, with $cos(\theta_i) = \sqrt{1/2}, \sqrt{2/5}, \sqrt{3/10}$ for $i = 1,2,3$, respectively.