Continuous-variable quantum neural networks Killoran, Nathan; Bromley, Thomas R.; Arrazola, Juan Miguel ...
Physical review research,
10/2019, Volume:
1, Issue:
3
Journal Article
Peer reviewed
Open access
We introduce a general method for building neural networks on quantum computers. The quantum neural network is a variational quantum circuit built in the continuous-variable (CV) architecture, which ...encodes quantum information in continuous degrees of freedom such as the amplitudes of the electromagnetic field. This circuit contains a layered structure of continuously parameterized gates which is universal for CV quantum computation. Affine transformations and nonlinear activation functions, two key elements in neural networks, are enacted in the quantum network using Gaussian and non-Gaussian gates, respectively. The non-Gaussian gates provide both the nonlinearity and the universality of the model. Due to the structure of the CV model, the CV quantum neural network can encode highly nonlinear transformations while remaining completely unitary. We show how a classical network can be embedded into the quantum formalism and propose quantum versions of various specialized models such as convolutional, recurrent, and residual networks. Finally, we present numerous modeling experiments built with the strawberry fields software library. These experiments, including a classifier for fraud detection, a network which generates tetris images, and a hybrid classical-quantum autoencoder, demonstrate the capability and adaptability of CV quantum neural networks.
We introduce an exact classical algorithm for simulating Gaussian boson sampling (GBS). The complexity of the algorithm is exponential in the number of photons detected, which is itself a random ...variable. For a fixed number of modes, the complexity is in fact equivalent to that of calculating output probabilities, up to constant prefactors. The simulation algorithm can be extended to other models such as GBS with threshold detectors, GBS with displacements, and sampling linear combinations of Gaussian states. In the specific case of encoding non-negative matrices into a GBS device, our method leads to an approximate sampling algorithm with polynomial runtime. We implement the algorithm, making the code publicly available as part of Xanadu's The Walrus library and benchmark its performance on GBS with random Haar interferometers and with encoded Erdős-Renyi graphs.
We gather and examine in detail gate decomposition techniques for continuous-variable quantum computers and also introduce some new techniques which expand on these methods. Both exact and ...approximate decomposition methods are studied and gate counts are compared for some common operations. While each having distinct advantages, we find that exact decompositions have lower gate counts whereas approximate techniques can cover decompositions for all continuous-variable operations but require significant circuit depth for a modest precision.
Photonics is a promising platform for demonstrating a quantum computational advantage (QCA) by outperforming the most powerful classical supercomputers on a well-defined computational task. Despite ...this promise, existing proposals and demonstrations face challenges. Experimentally, current implementations of Gaussian boson sampling (GBS) lack programmability or have prohibitive loss rates. Theoretically, there is a comparative lack of rigorous evidence for the classical hardness of GBS. In this work, we make progress in improving both the theoretical evidence and experimental prospects. We provide evidence for the hardness of GBS, comparable to the strongest theoretical proposals for QCA. We also propose a QCA architecture we call high-dimensional GBS, which is programmable and can be implemented with low loss using few optical components. We show that particular algorithms for simulating GBS are outperformed by high-dimensional GBS experiments at modest system sizes. This work thus opens the path to demonstrating QCA with programmable photonic processors.
Parametrized quantum optical circuits are a class of quantum circuits in which the carriers of quantum information are photons and the gates are optical transformations. Classically optimizing these ...circuits is challenging due to the infinite dimensionality of the photon number vector space that is associated to each optical mode. Truncating the space dimension is unavoidable, and it can lead to incorrect results if the gates populate photon number states beyond the cutoff. To tackle this issue, we present an algorithm that is orders of magnitude faster than the current state of the art, to recursively compute the exact matrix elements of Gaussian operators and their gradient with respect to a parametrization. These operators, when augmented with a non-Gaussian transformation such as the Kerr gate, achieve universal quantum computation. Our approach brings two advantages: first, by computing the matrix elements of Gaussian operators directly, we don't need to construct them by combining several other operators; second, we can use any variant of the gradient descent algorithm by plugging our gradients into an automatic differentiation framework such as TensorFlow or PyTorch. Our results will find applications in quantum optical hardware research, quantum machine learning, optical data processing, device discovery and device design.
Gaussian Boson Sampling (GBS) is a model of photonic quantum computing where single-mode squeezed states are sent through linear-optical interferometers and measured using single-photon detectors. In ...this work, we employ a recent exact sampling algorithm for GBS with threshold detectors to perform classical simulations on the Titan supercomputer. We determine the time and memory resources as well as the amount of computational nodes required to produce samples for different numbers of modes and detector clicks. It is possible to simulate a system with 800 optical modes postselected on outputs with 20 detector clicks, producing a single sample in roughly 2 h using 40% of the available nodes of Titan. Additionally, we benchmark the performance of GBS when applied to dense subgraph identification, even in the presence of photon loss. We perform sampling for several graphs containing as many as 200 vertices. Our findings indicate that large losses can be tolerated and that the use of threshold detectors is preferable over using photon-number-resolving detectors postselected on collision-free outputs.
Gaussian boson sampling is a model of photonic quantum computing that has attracted attention as a platform for building quantum devices capable of performing tasks that are out of reach for ...classical devices. There is therefore significant interest, from the perspective of computational complexity theory, in solidifying the mathematical foundation for the hardness of simulating these devices. We show that, under the standard Anti-Concentration and Permanent-of-Gaussians conjectures, there is no efficient classical algorithm to sample from ideal Gaussian boson sampling distributions (even approximately) unless the polynomial hierarchy collapses. The hardness proof holds in the regime where the number of modes scales quadratically with the number of photons, a setting in which hardness was widely believed to hold but that nevertheless had no definitive proof.
Crucial to the proof is a new method for programming a Gaussian boson sampling device so that the output probabilities are proportional to the permanents of submatrices of an arbitrary matrix. This technique is a generalization of Scattershot BosonSampling that we call BipartiteGBS. We also make progress towards the goal of proving hardness in the regime where there are fewer than quadratically more modes than photons (i.e., the high-collision regime) by showing that the ability to approximate permanents of matrices with repeated rows/columns confers the ability to approximate permanents of matrices with no repetitions. The reduction suffices to prove that GBS is hard in the constant-collision regime.
We introduce an algorithm for the classical simulation of Gaussian boson sampling that is quadratically faster than previously known methods. The complexity of the algorithm is exponential in the ...number of photon pairs detected, not the number of photons, and is directly proportional to the time required to calculate a probability amplitude for a pure Gaussian state. The main innovation is to use auxiliary conditioning variables to reduce the problem of sampling to the computation of the pure-state probability amplitudes, for which the most computationally expensive step is the calculation of a loop hafnian. We implement and benchmark an improved loop-hafnian algorithm and show that it can be used to compute pure-state probabilities, the dominant step in the sampling algorithm, of events involving up to 50 photons in a single workstation, i.e., without the need of a supercomputer.
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 behavior 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 continuous-variable 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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