Distributed inner product estimation (DIPE) is a fundamental task in quantum information, aiming to estimate the inner product between two unknown quantum states prepared on distributed quantum platforms. For multimode continuous-variable (CV) systems, the infinite-dimensional Hilbert space poses an additional challenge. A CV-DIPE protocol using only local homodyne measurements at shared phase settings and classical communication has been developed.

With a known bound on the mean energy per mode, explicit finite-sample guarantees and sufficient copy counts for estimating the full inner product to prescribed accuracy and confidence are derived. The energy constraint yields an inverse-square root scaling of sample complexity with respect to desired precision.

Polylogarithmic Scaling Enables Feasible Quantum State Benchmarking via CV-DIPE

Researchers at Southeast University made key progress in continuous-variable distributed inner product estimation (CV-DIPE). The team reduced required quantum state copies for accurate estimations from O ε−(2M+2) log(2/δ) to O ε−2[log(2M/ε)]2M log(2/δ) when using Gaussian states, a shift representing an improvement by replacing exponential scaling with polylogarithmic dependence on the system’s mode number M and desired precision ε. Previously impossible without exponentially increasing resources as complexity grew, benchmarking of independent platforms preparing complex quantum states is now feasible even if those states lack easily understood classical descriptions.

This protocol relies solely on local homodyne measurements, a standard technique involving measuring amplitude and phase of light, combined with conventional communication methods between quantum systems. The advancement hinges on leveraging the specific structure of Gaussian states to reduce accuracy costs from exponential dependence on system mode number M to polylogarithmic terms.

Numerical simulations confirmed a predictable balance between truncation bias (errors introduced by approximating infinite quantities) and statistical error using both Gaussian and non-Gaussian quantum states. These tests also revealed that accurate estimations depend upon knowing an upper bound for energy per mode, while scaling in realistic scenarios where this constraint is relaxed or unknown remains an open challenge.

Benchmarking advantages presently favour continuous variable systems utilising Gaussian states

Enabling benchmarking between independent quantum platforms marks a step forward for complex systems, though significant accuracy gains appear limited to Gaussian states; the effectiveness with more exotic phenomena remains uncertain. This limitation stems from reliance on specific properties inherent in Gaussian distributions, particularly their control of statistical tails and reduction of estimation errors, suggesting alternative approaches may be needed when dealing with non-Gaussian states where these benefits diminish. Still, acknowledging that improvements are currently most pronounced for bell-curve shaped probability distributions, this work represents a vital advance in quantum networking.

The development establishes a new method for verifying quantum state preparation across separate devices using continuous variables without needing full characterisation of the states themselves. By employing only local measurements alongside standard communication techniques, researchers have created a protocol allowing independent platforms to benchmark performance against one another. Accurately describing complex quantum states classically presented an insurmountable barrier and could unlock new avenues for distributed quantum computing and networking in the coming decade, beginning a new era of interoperability.

This research developed a method for estimating the inner product between two unknown quantum states on different continuous-variable systems using local homodyne measurements and classical communication. The approach provides finite guarantees for estimation accuracy when there is a known bound on energy per mode, reducing dependence on system size. Numerical simulations with both Gaussian and non-Gaussian states demonstrated this balance between truncation bias and statistical error; however, current results suggest greater precision is achieved with Gaussian states. This allows independent platforms to benchmark their state preparations against one another without fully describing those states classically.

👉 More information
🗞 Distributed Inner-Product Estimation for Continuous-Variable Quantum States
✍️ Jun Tang, Congcong Zheng, Xutao Yu and Zaichen Zhang (Southeast University)
🧠 ArXiv: https://arxiv.org/abs/2610.40141

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