Co-Designing Spectral Transformation Oracles with Hybrid Oscillator-Qubit Quantum Processors: From Algorithms to Compilation

TL;DR

This work introduces a co-design approach that leverages continuous-variable qumodes together with discrete qubits to construct and compile eigenvalue transformation oracles as continuous linear combinations of unitaries. A Gaussian imaginary-time evolution filter is encoded via a Hubbard-Stratonovich transform, enabling a block-encoded, CV-DV spectral oracle whose action equals a Gaussian filter $\hat{P}_{\hat{H}'}(\tau)= e^{-{1}/{2}\hat{H}'^2 \tau^2}$. The authors provide an end-to-end compilation for Heisenberg/XXZ spin models in 1D and 2D, including 1D/2D SWAP networks and multi-oscillator parallelization that achieve $\mathcal{O}(\sqrt{M})$ runtime speedups while bounding approximation errors. They also analyze leading-order physical errors, discuss randomized benchmarking-inspired protocols for error characterization, and outline practical pathways toward near-term experimental realization and future CV-DV compiler automation.

Abstract

We co-design a family of quantum eigenvalue transformation oracles that can be efficiently implemented on hybrid discrete/continuous-variable (qubit/qumode) hardware. To illustrate the oracle's representation-theoretic power and near-term experimental accessibility, we encode a Gaussian imaginary time evolution spectral filter. As a result, we define a continuous linear combination of unitaries block-encoding. Due to the ancillary qumode's infinite-dimensional nature, continuous variable qumodes constitute a powerful compilation tool for encoding continuous spectral functions without discretization errors while minimizing resource requirements. We then focus on the ubiquitous task of preparing eigenstates in quantum spin models. For completeness, we provide an end-to-end compilation which expresses high-level oracles in terms of an experimentally realizable instruction set architecture in both 1D and 2D. Finally, we examine the leading-order effects of physical errors and highlight open research directions. Our algorithms scale linearly with the spatial extent of the target system and are applicable to both near-term and large-scale quantum processors.

Figures (10)

• Figure 1: Top panel: Schematic illustration of a displacement $\alpha$ controlled on a local interaction $H_{ij}$. Middle panel: The spin spectrum decays as a result of spectrum-conditioned oscillator translations and measurement which results in the bottom panel's approximate ground state.
• Figure 2: Hardware layout of 2D superconducting quantum processor (left panel), illustrating the microwave (green) and trasmon qubit (purple) sublattices. See Figure 2 of Liu2024 for additional trapped-ion and neutral-atom CV/DV architectures. The right column indicates embedding 1D geometries with open (OBC) and periodic (PBC) boundary conditions and a 2D generalization. Links between lattice sites are highlighted in green to emphasize that all spin-spin interactions are mediated via the oscillators.
• Figure 3: Compilation workflow where each object represents a relevant quantum operation. The arrows transform the operations, corresponding to the labeled equations, in the Heisenberg picture.
• Figure 4: Compiled circuits for (a) $D_{Z_1 Z_2}^{(1)}$, (b) $D_{Z_1 Z_2}^{(1,2)}$, and (c) $D_{\hat{H}_{1,2}}^{(1,2)} = D_{Z_1 Z_2}^{(1,2)} D_{Y_1 Y_2}^{(1,2)} D_{X_1 X_2}^{(1,2)}$, where $\hat{H}_{1,2}$ is defined in \ref{['eq:Heisenberg_M=1']}. Thicker wires denote two oscillators (need not be adjacent) in the respective quantum state $\ket{\Psi_{1}}$ and $\ket{\Psi_{2}}$; $\ket{\psi_{1}}$ and $\ket{\psi_{2}}$ denote the respective states of two qubits linked to their corresponding oscillator.
• Figure 5: Trotter orderings for 1D (a) and 2D (b) spin models. Each color denotes spin-spin interaction terms for which conditional oscillator displacements can be parallelized. (a) In 1D, this results in alternating sequence of displacements corresponding to red ($\hat{A}$) and blue ($\hat{B}$) links. In 2D, interaction terms are grouped by red ($\hat{A}$), blue ($\hat{B}$), green ($\hat{C}$), and purple ($\hat{D}$) links.

Theorems & Definitions (5)

• Lemma 1: CV-DV phase kick-back
• Theorem 2: CV-DV-HS Transformation Oracle
• Theorem 3: Trotter-Hubbard-Stratonovich Formula
• Lemma 4: Spacetime Parallelization
• Theorem 5: Parallelized LTHS Formula