Trade-Offs on Number and Phase Shift Resilience in Bosonic Quantum Codes
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Abstract
Quantum codes typically rely on large numbers of degrees of freedom to achieve low error rates. However each additional degree of freedom introduces a new set of error mechanisms. Hence minimizing the degrees of freedom that a quantum code utilizes is helpful. One quantum error correction solution is to encode quantum information into one or more bosonic modes. We revisit rotation-invariant bosonic codes, which are supported on Fock states that are gapped by an integer <inline-formula> <tex-math notation="LaTeX">$g$ </tex-math></inline-formula> apart, and the gap <inline-formula> <tex-math notation="LaTeX">$g$ </tex-math></inline-formula> imparts number shift resilience to these codes. Intuitively, since phase operators and number shift operators do not commute, one expects a trade-off between resilience to number-shift and rotation errors. Here, we obtain results pertaining to the non-existence of approximate quantum error correcting <inline-formula> <tex-math notation="LaTeX">$g$ </tex-math></inline-formula>-gapped single-mode bosonic codes with respect to Gaussian dephasing errors. We show that by using arbitrarily many modes, <inline-formula> <tex-math notation="LaTeX">$g$ </tex-math></inline-formula>-gapped multi-mode codes can yield good approximate quantum error correction codes for any finite magnitude of Gaussian dephasing and amplitude damping errors.