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arXiv:1912.06463 · PRX Quantum

Cluster States from Gaussian States: Essential Diagnostic Tools for Continuous-Variable One-Way Quantum Computing

Carlos González-Arciniegas, P. Nussenzveig, M. Martinelli, Olivier Pfister

Published Sep 13, 2021• 11 citations
arXivHTMLPDF

Abstract

Continuous-variable (CV) cluster states are a universal quantum computing platform that has experimentally outscaled qubit platforms by orders of magnitude.Room-temperature implementation of CV cluster states has been achieved with quantum optics by using multimode squeezed Gaussian states.It has also been proven that fault tolerance thresholds for CV quantum computing can be reached at realistic squeezing levels.In this paper, we show that standard approaches to design and characterize CV cluster states can miss entanglement present in the system.Such hidden entanglement may be used to increase the power of a quantum computer but it can also, if undetected, hinder the successful implementation of a quantum algorithm.By a detailed analysis of the structure of Gaussian states, we derive an algorithm that reveals hidden entanglement in an arbitrary Gaussian state and optimizes its use for one-way quantum computing.

DOI
10.1103/prxquantum.2.030343

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Continuous-variable (CV) cluster states are a universal quantum computing platform that has experimentally outscaled qubit platforms by orders of magnitude.Room-temperature implementation of CV cluster states has been achieved with quantum optics by using multimode squeezed Gaussian states.It has also been proven that fault tolerance thresholds for CV quantum computing can be reached at realistic squeezing levels.In this paper, we show that standard approaches to design and characterize CV cluster states can miss entanglement present in the system.Such hidden entanglement may be used to increase the power of a quantum computer but it can also, if undetected, hinder the successful implementation of a quantum algorithm.By a detailed analysis of the structure of Gaussian states, we derive an algorithm that reveals hidden entanglement in an arbitrary Gaussian state and optimizes its use for one-way quantum computing.
Continuous-variable (CV) cluster states are a universal quantum computing platform that has experimentally out-scaled qubit platforms by orders of magnitude. The implementation of CV cluster states uses multimode squeezed Gaussian states and fault tolerance thresholds for CV quantum computing have been predicted for realistic squeezing. An outstanding question has been whether any Gaussian state can be expressed as a cluster state. We show here that Gaussian states can present hidden entanglement, which is not captured by the corresponding cluster state graph and therefore invalidates this graph. Hidden entanglement vanishes in the infinite squeezing limit, which should therefore be used with extreme caution. We derive a general procedure to detect hidden entanglement in an arbitrary Gaussian state and validate it as a CV cluster state. These results are relevant to all experimental implementations of CV cluster states to date.
Continuous-variable (CV) cluster states offer a promising platform for large-scale measurement-based quantum computations (MBQC). However, finite squeezing inevitably introduces Gaussian noise during MBQC. While fault-tolerant MBQC schemes exist in principle, they require the scalable incorporation of non-Gaussian resources, such as GKP states, which remain experimentally challenging. Consequently, a central question at this stage is how finite squeezing fundamentally constrains the intrinsic computational power of CV cluster states themselves. In this work, we address this question by analyzing the classical complexity of measurement-based linear optics (MBLO) implemented with such states, motivated by its near-term feasibility and recent experimental progress. We develop an explicit MBLO framework and examine how the squeezing level governs the complexity of the classical simulation of the resulting output states. Specifically, we identify squeezing-level thresholds that delineate classically tractable and intractable regimes, thereby revealing a squeezing-driven complexity phase transition. These findings advance our understanding of the squeezing resources necessary for meaningful quantum computation in current experimental regimes. Furthermore, they underscore the critical need to either scale the squeezing level or integrate error-correction schemes to achieve reliable, large-scale quantum computation with CV cluster states.
Continuous-variable cluster states (CVCSs) can be supplemented with Gottesman-Kitaev-Preskill (GKP) states to form a hybrid cluster state with the power to execute universal, fault-tolerant quantum computing in a measurement-based fashion. As the resource states that comprise a hybrid cluster state are of a very different nature, a natural question arises: Why do GKP states interface so well with CVCSs? To answer this question, we apply the recently introduced subsystem decomposition of a bosonic mode, which divides a mode into logical and gauge-mode subsystems, to three types of cluster state: CVCSs, GKP cluster states, and hybrid continuous-variable (CV)-GKP cluster states. We find that each of these contains a ``hidden'' qubit cluster state across their logical subsystems, which lies at the heart of their utility for measurement-based quantum computing. To complement the analytical approach, we introduce a simple graphical description of these CV-mode cluster states that depicts precisely how the hidden qubit cluster states are entangled with the gauge modes, and we outline how these results would extend to the case of finitely squeezed states. This work provides important insight that is both conceptually satisfying and helps to address important practical issues like when a simpler resource (such as a Gaussian state) can stand in for a more complex one (like a GKP state), leading to more efficient use of the resources available for CV quantum computing.
Gaussian continuous-variable cluster states are a unique quantum information resource as they currently scale up experimentally to much larger sizes than any qubit platform. We derive the first general criterion for a pure Gaussian state not to be expressible as a canonical cluster state, and show that such Gaussian states must then display hidden entanglement. This applies to experimentally relevant deviations from Gaussian cluster states, such as variations of the squeezing parameter from mode to mode, which cannot be captured by the infinite squeezing limit.
Continuous-variable (CV) cluster states are a universal resource for fault-tolerant quantum computation when supplemented with the Gottesman-Kitaev-Preskill (GKP) bosonic code. We generalize the recently introduced subsystem decomposition of a bosonic mode [G. Pantaleoni et al., Phys. Rev. Lett. 125, 040501 (2020)], and we use it to analyze CV cluster-state quantum computing with GKP states. Specifically, we decompose squeezed-vacuum states and approximate GKP states to reveal their encoded logical information, and we decompose several gates crucial to CV cluster-state quantum computing. Then, we use the subsystem decomposition to quantify damage to the logical information in approximate GKP states teleported through noisy CV cluster states. Each of these studies uses the subsystem decomposition to circumvent complications arising from the full CV nature of the mode in order to focus on the encoded qubit information.
Measurement-based one-way quantum computation, which uses cluster states as resources, provides an efficient model to perform computation. However, few of the continuous variable (CV) quantum algorithms and classical algorithms based on one-way quantum computation were proposed. In this work, we propose a method to implement the classical Hadamard transform algorithm utilizing the CV cluster state. Compared with classical computation, only half operations are required when it is operated in the one-way CV quantum computer. As an example, we present a concrete scheme of four-mode classical Hadamard transform algorithm with a four-partite CV cluster state. This method connects the quantum computer and the classical algorithms, which shows the feasibility of running classical algorithms in a quantum computer efficiently.