The complexity of Gottesman-Kitaev-Preskill states
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arXiv
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| Format: | Preprint |
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2024
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| author | Brenner, Lukas Caha, Libor Coiteux-Roy, Xavier Koenig, Robert |
| author_facet | Brenner, Lukas Caha, Libor Coiteux-Roy, Xavier Koenig, Robert |
| contents | We initiate the study of state complexity for continuous-variable quantum systems. Concretely, we consider a setup with bosonic modes and auxiliary qubits, where available operations include Gaussian one- and two-mode operations, single- and two-qubit operations, as well as qubit-controlled phase-space displacements. We define the (approximate) complexity of a bosonic state by the minimum size of a circuit that prepares an $L^1$-norm approximation to the state. We propose a new circuit which prepares an approximate Gottesman-Kitaev-Preskill (GKP) state $|\mathsf{GKP}_{κ,Δ}\rangle$. Here $κ^{-2}$ is the variance of the envelope and $Δ^2$ is the variance of the individual peaks. We show that the circuit accepts with constant probability and -- conditioned on acceptance -- the output state is polynomially close in $(κ,Δ)$ to the state $|\mathsf{GKP}_{κ,Δ}\rangle$. The size of our circuit is linear in $(\log 1/κ,\log 1/Δ)$. To our knowledge, this is the first protocol for GKP-state preparation with fidelity guarantees for the prepared state. We also show converse bounds, establishing that the linear circuit-size dependence of our construction is optimal. This fully characterizes the complexity of GKP states. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_19610 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The complexity of Gottesman-Kitaev-Preskill states Brenner, Lukas Caha, Libor Coiteux-Roy, Xavier Koenig, Robert Quantum Physics We initiate the study of state complexity for continuous-variable quantum systems. Concretely, we consider a setup with bosonic modes and auxiliary qubits, where available operations include Gaussian one- and two-mode operations, single- and two-qubit operations, as well as qubit-controlled phase-space displacements. We define the (approximate) complexity of a bosonic state by the minimum size of a circuit that prepares an $L^1$-norm approximation to the state. We propose a new circuit which prepares an approximate Gottesman-Kitaev-Preskill (GKP) state $|\mathsf{GKP}_{κ,Δ}\rangle$. Here $κ^{-2}$ is the variance of the envelope and $Δ^2$ is the variance of the individual peaks. We show that the circuit accepts with constant probability and -- conditioned on acceptance -- the output state is polynomially close in $(κ,Δ)$ to the state $|\mathsf{GKP}_{κ,Δ}\rangle$. The size of our circuit is linear in $(\log 1/κ,\log 1/Δ)$. To our knowledge, this is the first protocol for GKP-state preparation with fidelity guarantees for the prepared state. We also show converse bounds, establishing that the linear circuit-size dependence of our construction is optimal. This fully characterizes the complexity of GKP states. |
| title | The complexity of Gottesman-Kitaev-Preskill states |
| topic | Quantum Physics |
| url | https://arxiv.org/abs/2410.19610 |