All-Gaussian Universality and Fault Tolerance with the Gottesman-Kitaev-Preskill Code

The Gottesman-Kitaev-Preskill (GKP) encoding of a qubit within an oscillator is particularly appealing for fault-tolerant quantum computing with bosons because Gaussian operations on encoded Pauli eigenstates enable Clifford quantum computing with error correction. We show that applying GKP error correction to Gaussian input states, such as vacuum, produces distillable magic states, achieving universality without additional non-Gaussian elements. Fault tolerance is possible with sufficient squeezing and low enough external noise. Thus, Gaussian operations are sufficient for fault-tolerant, universal quantum computing given a supply of GKP-encoded Pauli eigenstates.

Figure 1

Wigner-function representations of the square-lattice GKP (a) Pauli eigenstates and (b) logical-Pauli operators in a single unit cell of phase space with dimensions $(2\sqrt{\pi })\times (2\sqrt{\pi })$. The states are normalized to one over one unit cell, which determines the coefficients $c$.

Figure 2

(a) GKP error correction of the vacuum: outcome-dependent fidelity $F$ with the nearest $H$-type magic state. The outcomes that do not yield a distillable magic state are marked with a white “x” (these yield GKP Pauli eigenstates). Bloch vectors for the representative outcomes (A–D) are shown on the GKP Bloch sphere in (b). (c) Probability of producing an $H$-type resource state of at least fidelity $F$ using a thermal state of mean occupation $\overline{n}$. Resource states with fidelity higher than the distillation threshold, $F>0.853$ (i.e., outside the stabilizer octahedron, or Paulihedron), can be distilled into higher-quality $|+H_L⟩$ states [43, 44]. Distillation is possible for $\overline{n}<{\overline{n}}_{\text{thresh},H}=0.366$. (d) $\widehat{U}$ maps logical square-lattice GKP states to equivalent logical states in the hexagonal-lattice GKP encoding [31]. A vacuum state on the hexagonal lattice ($1\text{−}\sigma$ error ellipse shown in blue) is mapped to a squeezed state on the square lattice under ${\widehat{U}}^{†}$. (e) Probability of producing a resource state distillable to $|+T_L^{\mathrm{hex}}⟩$ of at least fidelity $F$ by performing ${\mathrm{GKP}}^{\mathrm{hex}}$ error correction on a thermal state of mean occupation $\overline{n}$. Resource states whose Bloch vectors lie on or within the Paulihedron ($F\le 0.789$) cannot be distilled, which occurs at ${\overline{n}}_{\text{bound},T}=0.468$. For $T$-type resource states, a distillation threshold has been proven for $F>0.8273$ [45], which occurs for ${\overline{n}}_{\text{thresh},T}=0.391$.