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Classical simulation of circuits with realistic odd-dimensional Gottesman-Kitaev-Preskill states

Cameron Calcluth, Oliver Hahn, Juani Bermejo-Vega, Alessandro Ferraro, Giulia Ferrini

TL;DR

An algorithm to simulate circuits with encoded Gottesman-Kitaev-Preskill (GKP) states, specifically for odd-dimensional encoded qudits, which leverages the Zak-Gross Wigner function introduced by Davis, Fabre, and Chabaud, which represents infinitely squeezed encoded stabilizer states positively.

Abstract

Classically simulating circuits with bosonic codes is challenging due to the prohibitive cost of simulating quantum systems with many, possibly infinite, energy levels. We propose an algorithm to simulate circuits with encoded Gottesman-Kitaev-Preskill (GKP) states, specifically for odd-dimensional encoded qudits. Our approach is tailored to be especially effective in the most challenging but practically relevant regime, where the codeword states exhibit high (but finite) squeezing. Our algorithm leverages the Zak-Gross Wigner function introduced by J. Davis et al. [arXiv:2407.18394], which represents infinitely squeezed encoded stabilizer states positively. The runtime of the algorithm scales with the negativity of the Wigner function, allowing for efficient simulation of certain large-scale circuits - namely, input stabilizer GKP states undergoing generalized GKP-encoded Clifford operations followed by modular measurements - with a high degree of squeezing. For stabilizer GKP states exhibiting 12 dB of squeezing, our algorithm can simulate circuits with up to 1,000 modes with less than double the number of samples required for a single input mode, in stark contrast to existing simulators. Therefore, this approach holds significant potential for benchmarking early implementations of quantum computing architectures utilizing bosonic codes.

Classical simulation of circuits with realistic odd-dimensional Gottesman-Kitaev-Preskill states

TL;DR

An algorithm to simulate circuits with encoded Gottesman-Kitaev-Preskill (GKP) states, specifically for odd-dimensional encoded qudits, which leverages the Zak-Gross Wigner function introduced by Davis, Fabre, and Chabaud, which represents infinitely squeezed encoded stabilizer states positively.

Abstract

Classically simulating circuits with bosonic codes is challenging due to the prohibitive cost of simulating quantum systems with many, possibly infinite, energy levels. We propose an algorithm to simulate circuits with encoded Gottesman-Kitaev-Preskill (GKP) states, specifically for odd-dimensional encoded qudits. Our approach is tailored to be especially effective in the most challenging but practically relevant regime, where the codeword states exhibit high (but finite) squeezing. Our algorithm leverages the Zak-Gross Wigner function introduced by J. Davis et al. [arXiv:2407.18394], which represents infinitely squeezed encoded stabilizer states positively. The runtime of the algorithm scales with the negativity of the Wigner function, allowing for efficient simulation of certain large-scale circuits - namely, input stabilizer GKP states undergoing generalized GKP-encoded Clifford operations followed by modular measurements - with a high degree of squeezing. For stabilizer GKP states exhibiting 12 dB of squeezing, our algorithm can simulate circuits with up to 1,000 modes with less than double the number of samples required for a single input mode, in stark contrast to existing simulators. Therefore, this approach holds significant potential for benchmarking early implementations of quantum computing architectures utilizing bosonic codes.

Paper Structure

This paper contains 13 sections, 8 theorems, 99 equations, 2 figures.

Table of Contents

  1. Modified Stratonovich-Weyl Axioms
  2. Connection to Gross Wigner function
  3. Proof of Lemma 1
  4. Connection to stabilizer subsystem decomposition
  5. Proof of Corollary 1
  6. ZGW function evolution
  7. Decomposition of operations
  8. Operations
  9. Measurements
  10. Wigner function of the tensor product of two states
  11. ZGW function of finitely squeezed GKP states
  12. Zero-logical state
  13. Phase state

Key Result

Lemma 1

(Theorem 2 of Ref. davis2024.) The ZGW function of a single-mode CV state $\hat{\rho}$ is expressed in terms of the Gross Wigner function as where ${\boldsymbol\eta}=\ell (\mathbf u+ \mathbf t)$ , i.e. $\mathbf u=\tfrac{1}{\ell}{\boldsymbol\eta} \mod 1\in \mathbb T^{2}=[0,1)^{\times 2}$ and $\mathbf t=\tfrac{1}{\ell}{\boldsymbol\eta}-\mathbf u\in\mathbb Z_{d}^{2}$.

Figures (2)

  • Figure 1: Circuit class that we consider. It has arbitrary input GKP-encoded qudits with inverse squeezing parameter $\Delta$. The evolution is given by unitaries with corresponding integer symplectic matrix and arbitrary displacement (which include all encoded Clifford operations). The measurements are homodyne measurements modulo $d\ell$.
  • Figure 2: ZGW logarithmic negativity for the $0$-logical and $\pi$-logical GKP state at different levels of squeezing $\Delta$. The cross represents the negativity of the vacuum state, coinciding with the negativity of the zero-squeezing limit ($\Delta \rightarrow 1$) of both 0- and $\pi$-logical GKP states.

Theorems & Definitions (15)

  • Definition 1
  • Lemma 1
  • Corollary 1
  • Theorem 1
  • Theorem 2
  • Definition 2
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • ...and 5 more