Quantum error correction of qudits beyond break-even

Abstract

Hilbert space dimension is a key resource for quantum information processing^1,2. Not only is a large overall Hilbert space an essential requirement for quantum error correction, but a large local Hilbert space can also be advantageous for realizing gates and algorithms more efficiently^3-7. As a result, there has been considerable experimental effort in recent years to develop quantum computing platforms using qudits (d-dimensional quantum systems with d > 2) as the fundamental unit of quantum information^8-19. Just as with qubits, quantum error correction of these qudits will be necessary in the long run, but so far, error correction of logical qudits has not been demonstrated experimentally. Here we report the experimental realization of an error-corrected logical qutrit (d = 3) and ququart (d = 4), which was achieved with the Gottesman-Kitaev-Preskill bosonic code²⁰. Using a reinforcement learning agent^21,22, we optimized the Gottesman-Kitaev-Preskill qutrit (ququart) as a ternary (quaternary) quantum memory and achieved beyond break-even error correction with a gain of 1.82 ± 0.03 (1.87 ± 0.03). This work represents a novel way of leveraging the large Hilbert space of a harmonic oscillator to realize hardware-efficient quantum error correction.

Fig. 1. Stabilizing GKP qudits.

a, Schematic of the experimental device. b, Geometric structure of the displacement operators that define the single-mode square GKP code. c, Circuit for one round of finite-energy GKP qudit stabilization, generalizing the SBS protocol. The big ECD gate of amplitude ${\mathit{\ell }}_{d,\Delta }=\sqrt{\mathrm{\pi }d}\cosh \left({\Delta }^2\right)$ is approximately the stabilizer length. The small ECD gates of amplitude ${\epsilon }_d/2=\sqrt{\mathrm{\pi }d}\sinh \left({\Delta }^2\right)/2$ account for the envelope size Δ. At the end of SBS round j, the cavity phase is updated by ϕ_j (Methods). d, Measured real part of the characteristic function of the maximally mixed GKP qudit state for d = 1 to 4 with Δ = 0.3, prepared by performing 300 SBS rounds starting from the cavity in its vacuum state |0⟩.

Fig. 2. Realization of a logical GKP qutrit.

a, State preparation of qutrit Pauli eigenstates |P₀⟩₃ with Δ = 0.32. b, Circuit for measuring a qutrit in the basis of Pauli operator P₃ using an ancilla qubit, where ${\theta }_0=2\arctan \left(1/\sqrt{2}\right)$. The first measurement distinguishes between the state |P₀⟩₃ and the subspace {|P₁⟩₃, |P₂⟩₃}, whereas the second distinguishes between |P₁⟩₃ and {|P₀⟩₃, |P₂⟩₃}. The Bloch spheres depict the trajectories taken by the ancilla when the qutrit is in each Pauli eigenstate. c, Backaction of the qutrit Pauli measurement in the Z₃ basis, applied to the maximally mixed qutrit state. d, Decay of qutrit Pauli eigenstates |P₀⟩₃ under the optimized QEC protocol. The dashed black lines indicate a probability of 1/3. The solid grey lines are exponential fits. From left to right, we found ${\gamma }_{X_0}^{-1}=\mathrm{1,153}\pm 13\mu \mathrm{s}$, ${\gamma }_{Z_0}^{-1}=\mathrm{1,120}\pm 15\mu \mathrm{s}$, ${\gamma }_{X{Z}_0}^{-1}=743\pm 10\mu \mathrm{s}$ and ${\gamma }_{X^2{Z}_0}^{-1}=727\pm 11\mu \mathrm{s}$.

Fig. 3. Realization of a logical GKP ququart.

a, State preparation of ququart Pauli eigenstates |P₀⟩₄ and parity state |+, 0⟩₄ with Δ = 0.32. b, Circuit for measuring a ququart in the basis of Pauli operator P₄ using an ancilla qubit. The first measurement distinguishes between the even and odd states |P_even/odd⟩₄, and the second measurement distinguishes between the remaining two states. The Bloch spheres depict the trajectories taken by the ancilla when the ququart is in each Pauli eigenstate. c, Backaction of the GKP ququart Pauli measurement in the Z₄ basis applied to the maximally mixed ququart state. d, Circuit for measuring a ququart in the parity basis {|±, m⟩₄: m = 0, 1}, where $X_4^2{\mid \pm ,m⟩}_4=\pm {\mid \pm ,m⟩}_4$ and $Z_4^2{\mid \pm ,m⟩}_4={(-1)}^m{\mid \pm ,m⟩}_4$. The first measurement determines the eigenvalue of $X_4^2$, and the second determines that of $Z_4^2$. e, Backaction of the GKP ququart parity measurement applied to the maximally mixed ququart state. f, Decay of ququart Pauli eigenstates |P₀⟩₄ and parity state |+, 0⟩₄ under the optimized QEC protocol. The dashed black lines indicate a probability of 1/4. The solid grey lines are exponential fits. From left to right, we found ${\gamma }_{X_0}^{-1}=840\pm 8\mathrm{\mu }\mathrm{s}$, ${\gamma }_{Z_0}^{-1}=836\pm 9\mu \mathrm{s}$, ${\gamma }_{X{Z}_0}^{-1}=519\pm 6\mu \mathrm{s}$, ${\gamma }_{X^2{Z}_0}^{-1}=507\pm 9\mu \mathrm{s}$, ${\gamma }_{X^3{Z}_0}^{-1}=571\pm 7\mu \mathrm{s}$, ${\gamma }_{X{Z}_0^2}^{-1}=562\pm 9\mu \mathrm{s}$ and ${\gamma }_{+,0}^{-1}=607\pm 8\mu \mathrm{s}$.

Fig. 4. Comparing GKP qudits.

a, Effective lifetime of the physical cavity Fock qudit and logical GKP qudit for d ∈ {2, 3, 4}. The arrows indicate the QEC gain. b, Effective envelope size Δ_eff of the optimized GKP qudit for d ∈ {2, 3, 4}. c, Mean number of photons in the cavity for the optimized GKP qudit for d ∈ {2, 3, 4}.