Equivalence of approximate Gottesman-Kitaev-Preskill codes

Matsuura, Takaya; Yamasaki, Hayata; Koashi, Masato

doi:10.1103/physreva.102.032408

Cited by 43 publications

(36 citation statements)

References 52 publications

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“…Many related methods can be used for defining finite-energy versions of these states [30,66]. A common approximation is to replace each delta function in the GKP wave-function with a Gaussian of width ∆, in addition to an overall Gaussian envelope of width 1/∆ that damps the peak weighting further from the origin [20]:…”

Section: Qubits Encoded Into Bosonic Modesmentioning

confidence: 99%

Blueprint for a Scalable Photonic Fault-Tolerant Quantum Computer

Bourassa

Alexander

Vasmer

et al. 2021

Quantum

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234

202

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Photonics is the platform of choice to build a modular, easy-to-network quantum computer operating at room temperature. However, no concrete architecture has been presented so far that exploits both the advantages of qubits encoded into states of light and the modern tools for their generation. Here we propose such a design for a scalable fault-tolerant photonic quantum computer informed by the latest developments in theory and technology. Central to our architecture is the generation and manipulation of three-dimensional resource states comprising both bosonic qubits and squeezed vacuum states. The proposal exploits state-of-the-art procedures for the non-deterministic generation of bosonic qubits combined with the strengths of continuous-variable quantum computation, namely the implementation of Clifford gates using easy-to-generate squeezed states. Moreover, the architecture is based on two-dimensional integrated photonic chips used to produce a qubit cluster state in one temporal and two spatial dimensions. By reducing the experimental challenges as compared to existing architectures and by enabling room-temperature quantum computation, our design opens the door to scalable fabrication and operation, which may allow photonics to leap-frog other platforms on the path to a quantum computer with millions of qubits.

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Section: Qubits Encoded Into Bosonic Modesmentioning

confidence: 99%

Blueprint for a Scalable Photonic Fault-Tolerant Quantum Computer

Bourassa

Alexander

Vasmer

et al. 2021

Quantum

Self Cite

234

202

View full text Add to dashboard Cite

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“…This normalization process is not symmetric in phase space because the peaks are constrained to remain centred at the initial positions of the ideal state [41]. A related normalization process, which has the Fock damping operator E( ) = e − n applied to the ideal state, is symmetric since the number operator n acts symmetrically in phase space; we denote such states as |k gkp .…”

Section: Gkp Statesmentioning

confidence: 99%

“…In [41] the authors provide a thorough review of the connections and mappings between these finite energy forms of GKP states, noting that they can be related to each other by a simple squeezing operation; in [42] the authors explore alternative normalization envelopes to Gaussians, demonstrating sufficient conditions for the normalization process to yield physical states. Yet another option for finite energy GKP states are comb states [43]; these correspond to taking only a finite superposition of evenlyweighted q-squeezed states centred at the location of the peaks in the ideal state.…”

Section: Gkp Statesmentioning

confidence: 99%

“…Finally, there are several avenues for future research. First, it is worth investigating the connection between other analytical representations of CV states (such as Riemann-Theta functions for GKP qubits [41,70], or the shifted Fock basis representation of cat states [2]) and the ones we have presented here to identify any additional opportunities for faster or more accurate simulation. Second, for the class of states and maps that we have considered, it would be of interest to determine the exact mathematical conditions for physicality in terms of the parameters of the states (weights, vectors of means and covariances matrices).…”

Section: Summary and Open Problemsmentioning

confidence: 99%

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Fast Simulation of Bosonic Qubits via Gaussian Functions in Phase Space

et al. 2021

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Bosonic qubits are a promising route to building fault-tolerant quantum computers on a variety of physical platforms. Studying the performance of bosonic qubits under realistic gates and measurements is challenging with existing analytical and numerical tools. We present a novel formalism for simulating classes of states that can be represented as linear combinations of Gaussian functions in phase space. This formalism allows us to analyze and simulate a wide class of non-Gaussian states, transformations and measurements. We demonstrate how useful classes of bosonic qubits-Gottesman-Kitaev-Preskill (GKP), cat, and Fock states-can be simulated using this formalism, opening the door to investigating the behaviour of bosonic qubits under Gaussian channels and measurements, non-Gaussian transformations such as those achieved via gate teleportation, and important non-Gaussian measurements such as threshold and photon-number detection. Our formalism enables simulating these situations with levels of accuracy that are not feasible with existing methods. Finally, we use a method informed by our formalism to simulate circuits critical to the study of fault-tolerant quantum computing with bosonic qubits but beyond the reach of existing techniques. Specifically, we examine how finite-energy GKP states transform under realistic qubit phase gates; interface with a CV cluster state; and transform under non-Clifford T gate teleportation using magic states. We implement our simulation method as a part of the open-source Strawberry Fields Python library.

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“…Note that in the above descriptions of finite-energy GKP qubit states, it is implicit that the conjugate quadrature is correspondingly anti-squeezed with a variance equalling that of the Gaussian outer envelope. More generally, the outer Gaussian envelope of a finite-energy GKP qubit state can have a variance ≤ 1/(2σ 2 ), so that the squeezing-anti-squeezing product is ≤ 1/4 [46].…”

Section: Gottesman-kitaev-preskill (Gkp) Qubitsmentioning

confidence: 99%

Coherent manipulation of graph states composed of finite-energy Gottesman-Kitaev-Preskill-encoded qubits

Seshadreesan,

Dhara,

Patil

et al. 2021

Preprint

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Graph states are a central resource in measurement-based quantum information processing. In the photonic qubit architecture based on Gottesman-Kitaev-Preskill (GKP) encoding, the generation of high-fidelity graph states composed of realistic, finite-energy approximate GKP-encoded qubits thus constitutes a key task. We consider the finite-energy approximation of GKP qubit states given by a coherent superposition of shifted finite-squeezed vacuum states, where the displacements are Gaussian distributed. We present an exact description of graph states composed of such approximate GKP qubits as a coherent superposition of a Gaussian ensemble of randomly displaced ideal GKPqubit graph states. We determine the transformation rules for the covariance matrix and the mean displacement vector of the Gaussian distribution of the ensemble under tools such as GKP-Steane error correction and fusion operations that can be used to grow large, high-fidelity GKP-qubit graph states. The former captures the noise in the graph state due to the finite-energy approximation of GKP qubits, while the latter relates to the possible absolute displacement shift errors on the individual qubits due to the homodyne measurements that are a part of these tools. The rules thus help in pinning down an exact coherent error model for graph states generated from truly finite-energy GKP qubits, which can shed light on their error correction properties.

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Equivalence of approximate Gottesman-Kitaev-Preskill codes

Cited by 43 publications

References 52 publications

Blueprint for a Scalable Photonic Fault-Tolerant Quantum Computer

Blueprint for a Scalable Photonic Fault-Tolerant Quantum Computer

Fast Simulation of Bosonic Qubits via Gaussian Functions in Phase Space

Coherent manipulation of graph states composed of finite-energy Gottesman-Kitaev-Preskill-encoded qubits

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