Encoding logical quantum information in harmonic oscillator modes is a promising and hardwareefficient approach to the realization of a quantum computer.In this work, we propose to encode logical qubits in grid states of an ensemble of harmonic oscillator modes.We first discuss general results about these multimode bosonic codes; how to design them, how to practically implement them in different experimental platforms, and how lattice symmetries can be leveraged to perform logical non-Clifford operations.We then introduce in detail two two-mode grid codes based on the hypercubic and D 4 lattices, respectively, showing how to perform a universal set of logical operations.We demonstrate numerically that multimode grid codes have, compared to their single-mode counterpart, increased robustness against propagation of errors from ancillas used for error correction.Finally, we highlight some interesting links between multidimensional lattices and single-mode grid codes concatenated with qubit codes.Encoding logical quantum information in harmonic oscillator modes is a promising and hardwareefficient approach to the realization of a quantum computer.In this work, we propose to encode logical qubits in grid states of an ensemble of harmonic oscillator modes.We first discuss general results about these multimode bosonic codes; how to design them, how to practically implement them in different experimental platforms, and how lattice symmetries can be leveraged to perform logical non-Clifford operations.We then introduce in detail two two-mode grid codes based on the hypercubic and D 4 lattices, respectively, showing how to perform a universal set of logical operations.We demonstrate numerically that multimode grid codes have, compared to their single-mode counterpart, increased robustness against propagation of errors from ancillas used for error correction.Finally, we highlight some interesting links between multidimensional lattices and single-mode grid codes concatenated with qubit codes.

arXiv:2201.12337PRX QuantumMar 7, 202241 citations

Autoparametric Resonance Extending the Bit-Flip Time of a Cat Qubit up to 0.3 s

Antoine Marquet, Antoine Essig, Joachim Cohen +9 more

Cat qubits, for which logical |0⟩ and |1⟩ are coherent states |±α⟩ of a harmonic mode, offer a promising route towards quantum error correction. Using dissipation to our advantage so that photon pairs of the harmonic mode are exchanged with single photons of its environment, it is possible to stabilize the logical states and exponentially increase the bit-flip time of the cat qubit with the photon number |α|2 . A large two-photon dissipation rate κ2 ensures fast qubit manipulation and short error-correction cycles, which are instrumental to correct the remaining phase-flip errors in a repetition code of cat qubits. Here, we introduce and operate an autoparametric superconducting circuit that couples a mode containing the cat qubit to a lossy mode whose frequency is set at twice that of the cat mode. This passive coupling does not require a parametric pump, and it reaches a rate κ2/2π≈2 MHz . With such a strong two-photon dissipation, bit-flip errors of the autoparametric cat qubit are prevented for a characteristic time up to 0.3 s with only a mild impact on phase-flip errors. In addition, we illustrate how the phase of a quantum superposition between |α⟩ and |−α⟩ can be arbitrarily changed by driving the harmonic mode while keeping the engineered dissipation active. Published by the American Physical Society 2024Cat qubits, for which logical |0⟩ and |1⟩ are coherent states |±α⟩ of a harmonic mode, offer a promising route towards quantum error correction. Using dissipation to our advantage so that photon pairs of the harmonic mode are exchanged with single photons of its environment, it is possible to stabilize the logical states and exponentially increase the bit-flip time of the cat qubit with the photon number |α|2 . A large two-photon dissipation rate κ2 ensures fast qubit manipulation and short error-correction cycles, which are instrumental to correct the remaining phase-flip errors in a repetition code of cat qubits. Here, we introduce and operate an autoparametric superconducting circuit that couples a mode containing the cat qubit to a lossy mode whose frequency is set at twice that of the cat mode. This passive coupling does not require a parametric pump, and it reaches a rate κ2/2π≈2 MHz . With such a strong two-photon dissipation, bit-flip errors of the autoparametric cat qubit are prevented for a characteristic time up to 0.3 s with only a mild impact on phase-flip errors. In addition, we illustrate how the phase of a quantum superposition between |α⟩ and |−α⟩ can be arbitrarily changed by driving the harmonic mode while keeping the engineered dissipation active. Published by the American Physical Society 2024

arXiv:2307.06761Physical Review XApr 26, 202426 citations

Passive Error Correction and Gates for a Very Small Logical Qubit

Eliot Kapit

Bulletin of the American Physical SocietyMar 16, 20170 citations

Recursive Magic State Distillation on the Surface Code

Jonathan E. Moussa

I reduce the cost to prepare magic states with lattice surgery operations on the surface code by using a recursive implementation of 15-to-1 magic state distillation. On a rotated surface code with distance $d$ , $|T\rangle$ preparation requires a $d$ -by- $3 d$ grid of data qubits for up to $15 d$ error correction cycles, and $|CCZ\rangle$ preparation requires a $3 d$ -by- $2 d$ grid for up to $10.5 d$ cycles. However, a significantly lower physical error threshold than that of the underlying surface code is required to match the error probability of the output magic state with the logical error rate of the output surface code at large code distances.I reduce the cost to prepare magic states with lattice surgery operations on the surface code by using a recursive implementation of 15-to-1 magic state distillation. On a rotated surface code with distance $d$, $|T\rangle$ preparation requires a $d$-by-$3 d$ grid of data qubits for up to $15 d$ error correction cycles, and $|CCZ\rangle$ preparation requires a $3 d$-by-$2 d$ grid for up to $10.5 d$ cycles. However, a significantly lower physical error threshold than that of the underlying surface code is required to match the error probability of the output magic state with the logical error rate of the output surface code at large code distances.

quant-pharXiv:2603.05409Mar 6, 20260 citations

d

|T\rangle

preparation requires a

d

-by-

3 d

grid of data qubits for up to

15 d

error correction cycles, and

|CCZ\rangle

preparation requires a

3 d

-by-

2 d

grid for up to

10.5 d

cycles. However, a significantly lower physical error threshold than that of the underlying surface code is required to match the error probability of the output magic state with the logical error rate of the output surface code at large code distances.