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arXiv:2208.11724 · Physical Review Letters

Quantum Volume for Photonic Quantum Processors

Yuxuan Zhang, Daoheng Niu, Alireza Shabani, Hassan Shapourian

Published Mar 15, 2023• 9 citations
arXivHTMLPDF

Abstract

Defining metrics for near-term quantum computing processors has been an integral part of the quantum hardware research and development efforts. Such quantitative characteristics are not only useful for reporting the progress and comparing different quantum platforms but also essential for identifying the bottlenecks and designing a technology road map. Most metrics such as randomized benchmarking and quantum volume were originally introduced for circuit-based quantum computers and were not immediately applicable to measurement-based quantum computing (MBQC) processors such as in photonic devices. In this Letter, we close this long-standing gap by presenting a framework to map physical noises and imperfections in MBQC processes to logical errors in equivalent quantum circuits, whereby enabling the well-known metrics to characterize MBQC. To showcase our framework, we study a continuous-variable cluster state based on the Gottesman-Kitaev-Preskill (GKP) encoding as a near-term candidate for photonic quantum computing, and derive the effective logical gate error channels and calculate the quantum volume in terms of the GKP squeezing and photon transmission rate.

DOI
10.1103/physrevlett.130.110602

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Defining metrics for near-term quantum computing processors has been an integral part of the quantum hardware research and development efforts. Such quantitative characteristics are not only useful for reporting the progress and comparing different quantum platforms but also essential for identifying the bottlenecks and designing a technology road map. Most metrics such as randomized benchmarking and quantum volume were originally introduced for circuit-based quantum computers and were not immediately applicable to measurement-based quantum computing (MBQC) processors such as in photonic devices. In this Letter, we close this long-standing gap by presenting a framework to map physical noises and imperfections in MBQC processes to logical errors in equivalent quantum circuits, whereby enabling the well-known metrics to characterize MBQC. To showcase our framework, we study a continuous-variable cluster state based on the Gottesman-Kitaev-Preskill (GKP) encoding as a near-term candidate for photonic quantum computing, and derive the effective logical gate error channels and calculate the quantum volume in terms of the GKP squeezing and photon transmission rate.
Encoding a qubit in the continuous degrees of freedom of an oscillator is a promising path to error-corrected quantum computation. One advantageous way to achieve this is through Gottesman-Kitaev-Preskill (GKP) grid states, whose symmetries allow for the correction of any small continuous error on the oscillator. Unfortunately, ideal grid states have infinite energy, so it is important to find finite-energy approximations that are realistic, practical, and useful for applications. In the first half of this work we investigate the impact of imperfect GKP states on computational circuits independently of the physical architecture. To this end, we analyze the behavior of the physical and logical content of normalizable GKP states through several figures of merit, employing a recently developed modular subsystem decomposition. By tracking the errors that enter into the computational circuit due to imperfections in the GKP states, we are able to gauge the utility of these states for noisy intermediate-scale quantum devices. In the second half, we focus on a state preparation approach in the photonic domain wherein photon-number-resolving measurements on some modes of Gaussian states produce non-Gaussian states in others. We produce detailed numerical results for the preparation of GKP states alongside estimating the resource requirements in practical settings and probing the quality of the resulting states with the tools we develop. Our numerical experiments indicate that we can generate any state in the GKP Bloch sphere with nearly equal resources, which has implications for magic state preparation overheads.
We present the first detailed simulation of a measurement based quantum computation based on Gottesman-Kitaev-Preskill (GKP) qubits within a quad-rail lattice (QRL) cluster state involving over 100 GKP modes. This was enabled by the recently developed functional matrix product states (FMPS) framework, with which we simulate continuous-variable (CV) quantum circuits while explicitly modelling intrinsic coherent error sources due to finite squeezing. We perform simulated randomised benchmarking across squeezing levels between 5 and 15 dB and find strong agreement with analytical estimates for high quality GKP qubits. As a demonstration of practical computation, we simulate a three-qubit Grover's algorithm within the QRL and identify a fundamental squeezing threshold -- approximately 10 dB -- beyond which the algorithm outperforms classical probability bounds.
Reliable quantum memory is essential for scalable quantum networks and fault-tolerant photonic quantum computing. We present a quantitative analysis of an all-optical quantum memory architecture in which a Gottesman-Kitaev-Preskill (GKP) encoded qubit is stored in a fibre loop and periodically stabilized using teleportation-based error correction. By modelling fibre propagation as a pure-loss channel and representing each correction round as an effective logical map acting on the Bloch vector, we obtain a compact description of the full multi-round memory channel. We show that syndrome decoder optimization plays a crucial role in the experimentally relevant finite-squeezing regime. The optimal decoder deviates from standard square-grid GKP decoder in both tile-size and tile-shape, leading to significant improved logical performance. Using this optimized decoding strategy, we identify a squeezing-dependent optimal spacing between correction nodes that maximizes the memory lifetime. Remarkably, this optimal segment length is largely independent of the desired storage time, providing a simple and practical design rule for fibre-loop quantum memory. We further find a squeezing threshold of approximately 6.7 dB below which intermediate error correction becomes counterproductive, while above threshold the achievable storage time increases approximately exponentially with squeezing. For example, at 17 dB squeezing, storage times exceeding 400 ms can be achieved with logical infidelity below 1%. These results establish clear performance benchmarks and reveal the fundamental trade-off between photon loss, squeezing, and correction frequency in continuous-variable architectures. Our findings provide actionable design principles for near-term photonic quantum memory and clarify the path toward scalable all-optical fault-tolerant quantum storage.
Encoding a qubit in the continuous degrees of freedom of an oscillator is a significant pursuit of quantum computation. One advantageous way to achieve this is through the Gottesman-Kitaev-Preskill (GKP) grid states, whose symmetries allow for the correction of any small continuous error on the oscillator. Unfortunately, ideal grid states have infinite energy, so it is important to find finite-energy approximations that are realistic, practical, and useful for applications. In the first half of the paper we investigate the impact of imperfect GKP states on computational circuits independently of the physical architecture. To this end we analyze the behaviour of the physical and logical content of normalizable GKP states through several figures of merit, employing a recently-developed modular subsystem decomposition. By tracking the errors that enter into the computational circuit due to the imperfections in the GKP states, we are able to gauge the utility of these states for NISQ (Noisy Intermediate-Scale Quantum) devices. In the second half, we focus on a state preparation approach in the photonic domain wherein photon-number-resolving measurements on some modes of Gaussian states produce non-Gaussian states in the others. We produce detailed numerical results for the preparation of GKP states along with estimating the resource requirements in practical settings and probing the quality of the resulting states with the tools we develop.
In the race to demonstrate a scalable and fault tolerant quantum computing (QC) platform, quantum photonics stands poised to have a major impact. In particular, by encoding quantum information over fields rather than photons (continuous-variable—CV, rather than discrete-variable—DV, quantum information), record-size quantum processors were demonstrated in spectral and in temporal quantum field combs. These demonstrations used the measurement-based QC paradigm, which is based on cluster entangled states. In this talk, I will outline two of our recent results: proposals for generating hypercubic cluster states by phase modulation of the quantum optical frequency comb and for generating fault-tolerant hybrid CV-DV encodings called Gottesman-Kitaev-Preskill states. The latter leverages CV cluster states along with our experimental ability to perform photon-number-resolving measurements up to 100 photons.