"Binomial code" fault-tolerant logical qubits encoded in bosonic modes of superconducting cavities

Figure

Description

The bosonic modes of long-lifetime machined aluminum cavities may be universally controlled when coupled to a nonlinear oscillator, such as a transmon. This allows for the preparation and manipulation of exotic quantum states, some of which have desirable properties for quantum error detection and correction. One such scheme is a family of codes referred to as “binomial codes”, which use finite superpositions of Fock states with definite generalized photon number parity to detect photon loss, the dominant error channel in the cavity.

Motivation

A superconducting qubit with long lifetime, a known dominant error channel, and a readily-implementable error correction scheme is desirable for quantum information processing.

References

• https://journals.aps.org/prx/abstract/10.1103/PhysRevX.6.031006
• https://www.nature.com/articles/s41467-017-00045-1
• https://rsl.yale.edu/sites/default/files/files/RSL_Theses/Reinhold-Thesis%20(1).pdf

Linked Papers

• michael-2016-binomial-codes

Related Entries

• cat-codes
• gkp-codes

Seed Metadata

• date_published: 2016-07-14

Physics

Logical qubit encoded in weighted superpositions of Fock states of a harmonic oscillator (superconducting cavity), with binomial coefficients. The simplest code protecting against single photon loss ($L=1$):

$|0_L⟩\propto |0⟩+|4⟩,|1_L⟩=|2⟩$

General code words use Fock states spaced by $S+1$ with binomial weights, exactly correcting up to $L$ photon losses, $G$ gains, and $D$ dephasing events. Error detection via photon number parity measurement; correction via conditional displacements controlled by an ancilla transmon.

Key Metrics

Metric	Value	Notes	Fidelity reference
Logical error rate	Below break-even demonstrated	Ofek et al. 2016	—
Cavity $T_1$	>1 ms	3D aluminum cavity	—
QEC cycle time	~1–5 μs	Parity measurement + feedback	—
Fock space size	$N\le 10$	For $L=1$ code	—
Ancilla (transmon) $T_1$	50–200 μs	Limits QEC performance	—
Operating temperature	10–20 mK	Dilution refrigerator	—

Related Qubits

• cat-codes — related bosonic code (coherent state basis)
• gkp-codes — related bosonic code (grid states)
• circuit-qed — hardware platform