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arXiv:2004.11002 · Quantum

Fast optimization of parametrized quantum optical circuits

Filippo M. Miatto, Nicolás Quesada

Published Nov 30, 2020• 20 citations
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Abstract

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.

DOI
10.22331/q-2020-11-30-366

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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.
Linear optical quantum circuits with photon number resolving (PNR) detectors are used for both Gaussian Boson Sampling (GBS) and for the preparation of non-Gaussian states such as Gottesman-Kitaev-Preskill (GKP), cat and NOON states. They are crucial in many schemes of quantum computing and quantum metrology. Classically optimizing circuits with PNR detectors is challenging due to their exponentially large Hilbert space, and quadratically more challenging in the presence of decoherence as state vectors are replaced by density matrices. To tackle this problem, we introduce a family of algorithms that calculate detection probabilities, conditional states (as well as their gradients with respect to circuit parametrizations) with a complexity that is comparable to the noiseless case. As a consequence we can simulate and optimize circuits with twice the number of modes as we could before, using the same resources. More precisely, for an M -mode noisy circuit with detected modes D and undetected modes U , the complexity of our algorithm is O(M2∏i∈UCi2∏i∈DCi) , rather than O(M2∏i∈D∪UCi2) , where Ci is the Fock cutoff of mode i . As a particular case, our approach offers a full quadratic speedup for calculating detection probabilities, as in that case all modes are detected. Finally, these algorithms are implemented and ready to use in the open-source photonic optimization library MrMustard.
Few-mode optical quantum circuits can be simulated and optimized using gradient-descent methods, as optical gates can be parametrized by continuous parameters. However, the parameter space as seen by a nonholomorphic cost function is not Euclidean, which means that the Euclidean gradient does not generally point in the direction of steepest ascent. In order to retrieve the true steepest-ascent direction, one must take into account the local metric in what is known as the natural-gradient (NG) method. In this work we implement the NG for the optimization of optical quantum circuits. Specifically, we adapt the NG approach to a complex-valued parameter space, allowing us to apply the NG method directly within the Bargmann formalism, which we rely on for transforming between phase space and Fock space. We then compare the NG approach to vanilla gradient descent and to Adam over two standard state-preparation benchmarks: a single-photon source and a Gottesman-Kitaev-Preskill state source. We observe that the NG approach converges significantly faster (due in part to the possibility of using larger learning rates) and with a smoother decay of the cost function throughout the optimization. This result further improves our ability to design quantum circuits via classical optimization methods.
Tensor networks have been eagerly developed in recent years as a simulation method for quantum computation on classical computers. The efficiency of tensor networks for simulating quantum computation has been shown by solving a task designed for demonstrating quantum computing supremacy. However, such a task is not useful for real application. Therefore, there is the room for research of finding specific applications suitable for exploiting tensor network simulation. As such an application, we propose optimization of circuits to generate non- Gaussian states. Non-Gaussian state is one of the crucial elements for achieving universal continuous variable quantum computation. We consider this problem as a variational optimization of optical circuits to generate desired non-Gaussian states. Tensor networks efficiently simulate the optical circuits, and their compatibility with automatic differentiation provides a heuristic way to optimize the circuits. We propose and report the result of our approach that combines tensor networks and automatic differentiation to perform variational optimization of circuits such that the desired non-Gaussian state is approximately generated. The complexity of contracting tensor networks is scaled by degree of entanglement, so we consider relatively shallow optical circuit. Such a restriction is beneficial for considering devices without error correction method, which are accessible in near-term.
We propose a framework to design and optimize generic photonic quantum circuits composed of Gaussian objects (pure and mixed Gaussian states, Gaussian unitaries, Gaussian channels, Gaussian measurements) as well as non-Gaussian effects such as photon-number-resolving measurements. In this framework, we parametrize a phase space representation of Gaussian objects using elements of the symplectic group (or the unitary or orthogonal group in special cases), and then we transform it into the Fock representation using a single linear recurrence relation that computes the Fock amplitudes of any Gaussian object recursively. We also compute the gradient of the Fock amplitudes with respect to phase space parameters by differentiating through the recurrence relation. We can then use Riemannian optimization on the symplectic group to optimize M M -mode Gaussian objects, avoiding the need to commit to particular realizations in terms of fundamental gates. This allows us to “mod out” all the different gate-level implementations of the same circuit, which now can be chosen after the optimization has completed. This can be especially useful when looking to answer general questions, such as bounding the value of a property over a class of states or transformations, or when one would like to worry about hardware constraints separately from the circuit optimization step. Finally, we make our framework extendable to non-Gaussian objects that can be written as linear combinations of Gaussian ones, by explicitly computing the change in global phase when states undergo Gaussian transformations. We implemented all of these methods in the freely available open-source library MrMustard, which we use in three examples to optimize the 216-mode interferometer in Borealis, and 2- and 3-modes circuits (with Fock measurements) to produce cat states and cubic phase states.
In a recent work we presented a recursive algorithm to compute the matrix elements of a generic Gaussian transformation in the photon-number basis. Its purpose was to evolve a quantum state by building the transformation matrix and subsequently computing the matrix-vector product. Here we present a faster algorithm that computes the final state without having to generate the full transformation matrix first. With this algorithm we bring the time complexity of computing the Gaussian evolution of an NNN-dimensional MMM-mode state from O(MN2M)O(MN^{2M})O(MN2M) to O(M(N2/2)M)O(M(N^2/2)^M)O(M(N2/2)M), which is an exponential improvement in the number of modes. In the special case of high squeezing, the evolved state can be approximated with complexity O(MNM)O(MN^{M})O(MNM). Our new algorithm is differentiable, which means we can use it in conjunction with gradient-based optimizers for circuit optimization tasks. We benchmark our algorithm by optimizing circuits to produce single photons, Gottesman-Kitaev-Preskill states and NOON states, showing that it is up to one order of magnitude faster than the state of the art.
Linear optical circuits with single-photon sources offer a promising platform for quantum chemistry and machine learning. However, current applications are all based on support vector machines or gradient-free optimization methods. This paper develops classical and quantum algorithms for evaluating the analytic gradients of linear optical circuits with respect to their phase parameters. First, we set up a general framework by characterising the class of observables whose expectation values can be estimated efficiently by sampling from a passive linear optical circuit with finitely many photons. We then show how to compute the gradients of the expectation values of a special class of ``non-interacting'' observables arising in full-counting-statistics. Our differentiation algorithm uses the Halmos dilation and requires evaluating two circuits of twice the size, using one additional photon. Building on the methods of full-counting-statistics, we show how to recover the gradients of arbitrary observables from the gradient of a non-interacting characteristic function. Throughout the paper, we compare the performance of classical and quantum algorithms on the same estimation problems, analysing the sampling complexity of the algorithms and suggesting different cases for which quantum speed-ups could be obtained.