Bosonic quantum Fourier codes

While 2-level systems, aka qubits, are a natural choice to perform a logical quantum computation, the situation is less clear at the physical level. Encoding information in higher-dimensional physical systems can indeed provide a first level of redundancy and error correction that simplifies the overall fault-tolerant architecture. A challenge then is to ensure universal control over the encoded qubits. Here, we explore an approach where information is encoded in an irreducible representation of a finite subgroup of $U(2)$ through an inverse quantum Fourier transform. We illustrate this idea by applying it to the real Pauli group $\langle X, Z\rangle$ in the bosonic setting. The resulting two-mode Fourier cat code displays good error correction properties and admits an experimentally-friendly universal gate set that we discuss in detail.

💡 Research Summary

The paper introduces a novel bosonic quantum error‑correcting code, called the “Fourier cat code,” which exploits the inverse quantum Fourier transform (QFT) of a finite subgroup G ⊂ U(2) to encode two qubits into two bosonic modes. The authors start from a general framework: given a finite group G with unitary representation λ, one can define orthonormal “group states” |g⟩ and an encoding map E that applies the inverse QFT F_G† to the basis |λ,ℓ,m⟩. This map produces a logical register L and a multiplicity (gauge) register M. Because the physical representation π(g) acts exactly as the left regular representation L(g) on the group states, applying π(g) implements the logical operation λ(g) on L while leaving M unchanged.

To obtain a concrete, experimentally relevant code, the authors specialize to the real Pauli group G = ⟨X,Z⟩, the smallest non‑abelian subgroup of U(2). Its defining (2‑dimensional) representation λ has λ(X)=σ_x and λ(Z)=σ_z. They consider two bosonic modes and a passive Gaussian representation π where π(X) is the mode‑swap (SWAP) and π(Z)=(-1)^{\hat n_2} applies a π‑phase to the second mode. Choosing the coherent‑state seed |α⟩|iα⟩ with α = pπ/2 ensures that the four coherent states |gα⟩ (g∈G) are linearly independent and that their Gram matrix Γ is diagonalized by the group Fourier transform. Consequently, the four encoded states |d_{ℓ,m}⟩ are simple products of single‑mode cat states: - |d_{0,0}⟩ = |1_α⟩ |0_{iα}⟩, - |d_{0,1}⟩ = |1_{iα}⟩ |0_α⟩, - |d_{1,0}⟩ = |0_{iα}⟩ |1_α⟩, - |d_{1,1}⟩ = |0_α⟩ |1_{iα}⟩, where |0_α⟩ ∝ |α⟩+|−α⟩ and |1_α⟩ ∝ |α⟩−|−α⟩ are even/odd cat states. The first index ℓ labels the logical qubit, the second m labels a gauge qubit that will be used to facilitate certain logical operations.

Error‑correction properties follow from the stabilizers and Lindblad operators that naturally arise from the construction. The code space lies in the kernel of \hat a_1^4−α^4, \hat a_2^4−α^4, and \hat a_1^2\hat a_2^2+α^4, which are analogous to those of the four‑leg cat and pair‑cat codes. Importantly, the total photon‑number parity (−1)^{\hat n_1+ \hat n_2+1}=−1 is enforced, guaranteeing that a single photon loss flips the parity and can be detected and corrected. Thus the Fourier cat code can correct one photon loss exactly, just like the four‑leg cat, but without the strong bias (X‑error suppression) typical of standard cat qubits.

The universal gate set is built from the natural physical actions of the group and from additional Gaussian or weakly nonlinear operations:

• Logical X and Z are directly realized by π(X)=SWAP and π(Z)=(-1)^{\hat n_2}, respectively.
• Clifford gates S (phase) and CZ are implemented via self‑Kerr (∝ \hat n^2) and cross‑Kerr (∝ \hat n_1\hat n_2) interactions, exactly as in rotation‑symmetric cat codes.
• The Hadamard gate is more subtle. Applying π(H) acts as H_L H_M on both qubits but simultaneously deforms the code to an equivalent Fourier cat code with a different seed state |e^{iπ/4}α⟩|e^{-iπ/4}α⟩. By alternating this deformation with a phase gate S_L (H_L H_M) S_L, the net effect is a logical Hadamard H_L on the logical qubit while returning the gauge qubit to its original state.
• A non‑Clifford rotation e^{iθZ_L} is obtained via the quantum Zeno effect: a strong two‑photon drive \hat a_1^2+ \hat a_1^{†2} constrained to the code manifold yields an effective Hamiltonian θ Z_L when the gauge qubit is fixed in |0⟩ or |1⟩.

Table 2 in the paper summarizes the mapping between logical operations and feasible physical implementations (photon‑parity measurements, SNAP gates, balanced beam‑splitters, etc.). State preparation is straightforward: the four logical states are simply tensor products of two single‑mode cat states, which have already been demonstrated experimentally.

The authors compare the Fourier cat code with several existing bosonic encodings: single‑mode cat, GKP, dual‑rail, pair‑cat, and binomial codes. Compared to the standard cat, the Fourier cat has comparable loss‑tolerance but no bias, making it more suitable for universal computation. Compared to GKP, it is far easier to prepare and stabilize, though it offers less optimal protection against loss. The presence of the gauge qubit distinguishes it from the covariant code of Ref.