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arXiv:2205.09781 · Physical Review A

Vacuum provides quantum advantage to otherwise simulatable architectures

Cameron Calcluth, Alessandro Ferraro, Giulia Ferrini

Published Jun 15, 2023• 13 citations
arXivHTMLPDF

Abstract

We consider a computational model composed of ideal Gottesman-Kitaev-Preskill stabilizer states, Gaussian operations including all rational symplectic operations and all real displacements, and homodyne measurement. We prove that such architecture is classically efficiently simulatable by explicitly providing an algorithm to calculate the probability density function of the measurement outcomes of the computation. We also provide a method to sample when the circuits contain conditional operations. This result is based on an extension of the celebrated Gottesman-Knill theorem, via introducing proper stabilizer operators for the code at hand. We conclude that the resource enabling quantum advantage in the universal computational model considered by Baragiola et al. [Phys. Rev. Lett. 123, 200502 (2019)], composed of a subset of the elements given above augmented with a provision of vacuum states, is indeed the vacuum state.

DOI
10.1103/physreva.107.062414

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We consider a computational model composed of ideal Gottesman-Kitaev-Preskill stabilizer states, Gaussian operations including all rational symplectic operations and all real displacements, and homodyne measurement. We prove that such architecture is classically efficiently simulatable by explicitly providing an algorithm to calculate the probability density function of the measurement outcomes of the computation. We also provide a method to sample when the circuits contain conditional operations. This result is based on an extension of the celebrated Gottesman-Knill theorem, via introducing proper stabilizer operators for the code at hand. We conclude that the resource enabling quantum advantage in the universal computational model considered by Baragiola et al. [Phys. Rev. Lett. 123, 200502 (2019)], composed of a subset of the elements given above augmented with a provision of vacuum states, is indeed the vacuum state.
Physical Gottesman-Kitaev-Preskill (GKP) states are inherently noisy as ideal ones would require infinite energy. While this is typically considered as a deficiency to be actively corrected, this work demonstrates that imperfect GKP stabilizer states can be leveraged in order to apply non-Clifford gates using only linear optical elements. In particular, Gaussian operations on normalizable GKP states, combined with homodyne measurements, permit two key primitives: clean projection onto Pauli eigenstates in the normalizable GKP codespace, thereby implementing Clifford gates with high fidelity; and probabilistic projection of unmeasured modes onto non-Pauli eigenstates. These results demonstrate that normalizable GKP stabilizer states combined with Gaussian operations provide a practical framework for computational universality within the measurement-based model of quantum computation in a realistic continuous-variable setting.
Continuous-variable bosonic systems stand as prominent candidates for implementing quantum computational tasks. While various necessary criteria have been established to assess their resourcefulness, sufficient conditions have remained elusive. We address this gap by focusing on promoting circuits that are otherwise simulatable to computational universality. The class of simulatable, albeit non-Gaussian, circuits that we consider is composed of Gottesman-Kitaev-Preskill (GKP) states, Gaussian operations, and homodyne measurements. Based on these circuits, we first introduce a general framework for mapping a continuous-variable state into a qubit state. Subsequently, we cast existing maps into this framework, including the modular and stabilizer subsystem decompositions. By combining these findings with established results for discrete-variable systems, we formulate a sufficient condition for achieving universal quantum computation. Leveraging this, we evaluate the computational resourcefulness of a variety of states, including Gaussian states, finite-squeezing GKP states, and cat states. Furthermore, our framework reveals that both the stabilizer subsystem decomposition and the modular subsystem decomposition (of position-symmetric states) can be constructed in terms of simulatable operations. This establishes a robust resource-theoretical foundation for employing these techniques to evaluate the logical content of a generic continuous-variable state, which can be of independent interest. Published by the American Physical Society 2024
We study the classical simulatability of Gottesman-Kitaev-Preskill (GKP) states in combination with arbitrary displacements, a large set of symplectic operations and homodyne measurements. For these types of circuits, neither continuous-variable theorems based on the non-negativity of quasi-probability distributions nor discrete-variable theorems such as the Gottesman-Knill theorem can be employed to assess the simulatability. We first develop a method to evaluate the probability density function corresponding to measuring a single GKP state in the position basis following arbitrary squeezing and a large set of rotations. This method involves evaluating a transformed Jacobi theta function using techniques from analytic number theory. We then use this result to identify two large classes of multimode circuits which are classically efficiently simulatable and are not contained by the GKP encoded Clifford group. Our results extend the set of circuits previously known to be classically efficiently simulatable.
Discriminating between quantum computing architectures that can provide quantum advantage from those that cannot is of crucial importance.From the fundamental point of view, establishing such a boundary is akin to pinpointing the resources for quantum advantage; from the technological point of view, it is essential for the design of nontrivial quantum computing architectures.Wigner negativity is known to be a necessary resource for computational advantage in several quantum-computing architectures, including those based on continuous variables (CVs).However, it is not a sufficient resource, and it is an open question under which conditions CV circuits displaying Wigner negativity offer the potential for quantum advantage.In this work we identify vast families of circuits that display large, possibly unbounded, Wigner negativity, and yet are classically efficiently simulatable, although they are not recognized as such by previously available theorems.These families of circuits employ bosonic codes based on either translational or rotational symmetries (e.g., Gottesman-Kitaev-Preskill or cat codes) and can include both Gaussian and non-Gaussian gates and measurements.Crucially, within these encodings, the computational basis states are described by intrinsically negative Wigner functions, even though they are stabilizer states if considered as codewords belonging to a finite-dimensional Hilbert space.We derive our results by establishing a link between the simulatability of high-dimensional discrete-variable quantum circuits and bosonic codes.
Quantum resource theories identify the features of quantum computers that provide their computational advantage over classical systems. We investigate the resources driving the complexity of classical simulation in the standard model of continuous-variable quantum computing, and their interplay enabling computational universality. Specifically, we uncover a new property in continuous-variable circuits, analogous to coherence in discrete-variable systems, termed , defined as the ability of Gaussian gates to mix position and momentum operators. Using quadrature propagation across multiple computational paths, we develop an efficient classical simulation algorithm for continuous-variable computations with symplectic coherence constant in the size of the circuit. This establishes symplectic coherence as a necessary resource for universality in continuous-variable quantum computing, alongside non-Gaussianity and entanglement. Via the Gottesman-Kitaev-Preskill encoding, we show that the interplay of these three continuous-variable quantum resources mirrors the discrete-variable relationship between coherence, magic, and entanglement.
We study classical simulation of quantum computation, taking the Gottesman-Knill theorem as a starting point. We show how each Clifford circuit can be reduced to an equivalent, manifestly simulatable circuit (normal form). This provides a simple proof of the Gottesman-Knill theorem without resorting to stabilizer techniques. The normal form highlights why Clifford circuits have such limited computational power in spite of their high entangling power. At the same time, the normal form shows how the classical simulation of Clifford circuits fits into the standard way of embedding classical computation into the quantum circuit model. This leads to simple extensions of Clifford circuits which are classically simulatable. These circuits can be efficiently simulated by classical sampling (``weak simulation'') even though the problem of exactly computing the outcomes of measurements for these circuits (``strong simulation'') is proved to be #P\#\mathbf{P}#P-complete---thus showing that there is a separation between weak and strong classical simulation of quantum computation.