The complexity of Gottesman-Kitaev-Preskill states

TL;DR

This work establishes a quantitative framework for the state complexity of continuous-variable quantum states, focusing on Gottesman-Kitaev-Preskill (GKP) states. It introduces a two-stage preparation protocol—comb-state generation followed by envelope Gaussification—whose heralded complexity scales as $O(\\log 1/\\kappa + \\log 1/\\Delta)$ and achieves fidelity guarantees. Complementary lower bounds show that this scaling is optimal for both unitary and heralded preparations, thereby fully characterizing the complexity of approximate GKP states. The results connect complexity to energy growth, showing that efficient preparation is tightly linked to controlled energy management, and they open avenues for fault-tolerant CV quantum computation using GKP resources.

Abstract

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.

Figures (12)

• Figure 1: The approximate GKP state $\ket{\mathsf{GKP}_{\kappa, \Delta}}$ in position space. The red line represents the envelope $\eta_\kappa(x)\propto e^{-\kappa^2x^2/2}$ of the state, a Gaussian with variance $\kappa^{-2}$. The GKP wavefunction is illustrated in blue (the shading is for visual emphasis only). According to our convention, this function has Gaussian peaks of variance $\Delta^2$ at all integers.
• Figure 2: An illustration of the comb state $\ket{\Sha_{L,\Delta}}$ with $L=8$ (i.e., with $8$ local maxima). For any large even integer $L\in2\mathbb{N}$, the state $\ket{\Sha_{L,\Delta}}$ is "almost centered": its peaks lie at positions $\mathcal{L}_{L}:=\{-L/2,\ldots,-1,0,\ldots,L/2-1\}$.
• Figure 3: Circuit diagram of Protocol \ref{['prot: comb state prep']}. It uses the $V$ gate described in Fig. \ref{['fig: circuit V']}. The exponent of $V$ indicates the number of applications. The squeezing unitary is realized by composing single-mode squeezing operations from the set $\mathcal{G}$ (see Section \ref{['sec: unitary state complexity']}), see the factorization given in Eq. \ref{['eq:squeezingdecomposition']}.
• Figure 4: Circuit implementing the unitary $V$ used in the comb-state-preparation protocol. It uses two qubit-controlled displacements: by $1$ in the $Q$-direction, by $\pi$-in the $P$-direction.
• Figure 5: Key to the protocol construction is that the unitary $V$ essentially doubles the number of peaks when the qubit is in the state $\ket{+}$. The output represented on the right-hand side is approximate (see Lemma \ref{['lem: doubling sha state']} for details). Note that the qubit approximately acts as a catalyst.

Theorems & Definitions (109)

• Corollary 1.1
• Lemma 2.1
• Lemma 2.2
• Corollary 2.3: Zero-error complexity of coherent states
• Theorem 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• proof
• Lemma 3.4