Pair-cat codes: autonomous error-correction with low-order nonlinearity

Abstract

We introduce a driven-dissipative two-mode bosonic system whose reservoir causes simultaneous loss of two photons in each mode and whose steady states are superpositions of pair-coherent/Barut-Girardello coherent states. We show how quantum information encoded in a steady-state subspace of this system is exponentially immune to phase drifts (cavity dephasing) in both modes. Additionally, it is possible to protect information from arbitrary photon loss in either (but not simultaneously both) of the modes by continuously monitoring the difference between the expected photon numbers of the logical states. Despite employing more resources, the two-mode scheme enjoys two advantages over its one-mode cat-qubit counterpart with regards to implementation using current circuit QED technology. First, monitoring the photon number difference can be done without turning off the currently implementable dissipative stabilizing process. Second, a lower average photon number per mode is required to enjoy a level of protection at least as good as that of the cat-codes. We discuss circuit QED proposals to stabilize the code states, perform gates, and protect against photon loss via either active syndrome measurement or an autonomous procedure. We introduce quasiprobability distributions allowing us to represent two-mode states of fixed photon number difference in a twodimensional complex plane, instead of the full four-dimensional two-mode phase space. The twomode codes are generalized to multiple modes in an extension of the stabilizer formalism to nondiagonalizable stabilizers. The M -mode codes can protect against either arbitrary photon losses in up to M − 1 modes or arbitrary losses and gains in any one mode.

Quantum Computing with Rotation-Symmetric Bosonic Codes

“…As expected, these codes have similar performance for large average excitation number, n code [Eq. (11)], but can show significant differences for smallern code [84]. Remarkably, we find that the phase error-correction scheme approaches optimal for largē n code and small noise strength, and that the pretty good scheme is near optimal for almost alln code and small noise strengths.…”

“…Note that the cat codes we consider in this work are distinct from the two-mode cat codes in Ref. [84] and the single-mode cat codes in Refs. [4,99], where the codewords are manifestly nonorthogonal while still exhibiting discrete rotational symmetry.…”

Approximating the two-mode two-photon Rabi model

“…Each Hamiltonian block in this finer form acts on its corresponding subspace of fixed photon number difference ∆ = n−m. The Hamiltonian on the block is a tridiagonal matrix when written in the fixed-∆ Fock subspace {|n, n + ∆ } ∞ n=0 for ∆ ≥ 0 (with the entries swapped for ∆ < 0 [68]). The parity operator P earns its name by reducing to a diagonal operator proportional to (−1) n in this basis.…”

Generating Gottesman-Kitaev-Preskill qubit using a cross-Kerr interaction between a squeezed light and Fock states in optics

“…In a CV system, the bosonic code for encoding quantum information in CVs is essential to remove errors during quantum information processing. A variety of bosonic codes have been developed, e.g., the cat code [3][4][5][6] and the binomial code [7]. Among bosonic codes, the Gottesman-Kitaev-Preskill (GKP) qubit [8] is a promising way to encode a qubit in CVs, where qubit is encoded in the continuous Hilbert space of harmonic oscillator's position and momentum variables.…”