arXiv subjects

Bobak Kiani

Publications and source records attributed to Bobak Kiani.

7 recordsLinked to original sources

arXiv2023

Improving the speed of variational quantum algorithms for quantum error correction

We consider the problem of devising a suitable Quantum Error Correction (QEC) procedures for a generic quantum noise acting on a quantum circuit. In general, there is no analytic universal procedure to obtain the encoding and correction unitary gates, and the problem is even harder if the noise is unknown and has to be reconstructed. The existing procedures rely on Variational Quantum Algorithms (VQAs) and are very difficult to train since the size of the gradient of the cost function decays exponentially with the number of qubits. We address this problem using a cost function based on the Quantum Wasserstein distance of order 1 ($QW_1$). At variance with other quantum distances typically adopted in quantum information processing, $QW_1$ lacks the unitary invariance property which makes it a suitable tool to avoid to get trapped in local minima. Focusing on a simple noise model for which an exact QEC solution is known and can be used as a theoretical benchmark, we run a series of numerical tests that show how, guiding the VQA search through the $QW_1$, can indeed significantly increase both the probability of a successful training and the fidelity of the recovered state, with respect to the results one obtains when using conventional approaches.

quant-ph↗

arXiv2023

Neural Networks for Programming Quantum Annealers

Quantum machine learning has the potential to enable advances in artificial intelligence, such as solving problems intractable on classical computers. Some fundamental ideas behind quantum machine learning are similar to kernel methods in classical machine learning. Both process information by mapping it into high-dimensional vector spaces without explicitly calculating their numerical values. We explore a setup for performing classification on labeled classical datasets, consisting of a classical neural network connected to a quantum annealer. The neural network programs the quantum annealer's controls and thereby maps the annealer's initial states into new states in the Hilbert space. The neural network's parameters are optimized to maximize the distance of states corresponding to inputs from different classes and minimize the distance between quantum states corresponding to the same class. Recent literature showed that at least some of the "learning" is due to the quantum annealer, connecting a small linear network to a quantum annealer and using it to learn small and linearly inseparable datasets. In this study, we consider a similar but not quite the same case, where a classical fully-fledged neural network is connected with a small quantum annealer. In such a setting, the fully-fledged classical neural-network already has built-in nonlinearity and learning power, and can already handle the classification problem alone, we want to see whether an additional quantum layer could boost its performance. We simulate this system to learn several common datasets, including those for image and sound recognition. We conclude that adding a small quantum annealer does not provide a significant benefit over just using a regular (nonlinear) classical neural network.

quant-ph↗

arXiv2023

Analog quantum variational embedding classifier

Quantum machine learning has the potential to provide powerful algorithms for artificial intelligence. The pursuit of quantum advantage in quantum machine learning is an active area of research. For current noisy, intermediate-scale quantum (NISQ) computers, various quantum-classical hybrid algorithms have been proposed. One such previously proposed hybrid algorithm is a gate-based variational embedding classifier, which is composed of a classical neural network and a parameterized gate-based quantum circuit. We propose a quantum variational embedding classifier based on an analog quantum computer, where control signals vary continuously in time. In our algorithm, the classical data is transformed into the parameters of the time-varying Hamiltonian of the analog quantum computer by a linear transformation. The nonlinearity needed for a nonlinear classification problem is purely provided by the analog quantum computer through the nonlinear dependence of the final quantum state on the control parameters of the Hamiltonian. We performed numerical simulations that demonstrate the effectiveness of our algorithm for performing binary and multi-class classification on linearly inseparable datasets such as concentric circles and MNIST digits. Our classifier can reach accuracy comparable with the best classical classifiers. We find the performance of our classifier can be increased by increasing the number of qubits until the performance saturates and fluctuates. Moreover, the number of optimization parameters of our classifier scales linearly with the number of qubits. The increase of number of training parameters when the size increases is therefore not as fast as that of neural network. Our algorithm presents the possibility of using current quantum annealers for solving practical machine-learning problems, and it could also be useful to explore quantum advantage in quantum machine learning.

quant-ph↗

arXiv2023

Combinatorial NLTS From the Overlap Gap Property

In an important recent development, Anshu, Breuckmann, and Nirkhe [ABN22] resolved positively the so-called No Low-Energy Trivial State (NLTS) conjecture by Freedman and Hastings. The conjecture postulated the existence of linear-size local Hamiltonians on n qubit systems for which no near-ground state can be prepared by a shallow (sublogarithmic depth) circuit. The construction in [ABN22] is based on recently developed good quantum codes. Earlier results in this direction included the constructions of the so-called Combinatorial NLTS -- a weaker version of NLTS -- where a state is defined to have low energy if it violates at most a vanishing fraction of the Hamiltonian terms [AB22]. These constructions were also based on codes. In this paper we provide a "non-code" construction of a class of Hamiltonians satisfying the Combinatorial NLTS. The construction is inspired by one in [AB22], but our proof uses the complex solution space geometry of random K-SAT instead of properties of codes. Specifically, it is known that above a certain clause-to-variables density the set of satisfying assignments of random K-SAT exhibits an overlap gap property, which implies that it can be partitioned into exponentially many clusters each constituting at most an exponentially small fraction of the total set of satisfying solutions. We establish a certain robust version of this clustering property for the space of near-satisfying assignments and show that for our constructed Hamiltonians every combinatorial near-ground state induces a near-uniform distribution supported by this set. Standard arguments then are used to show that such distributions cannot be prepared by quantum circuits with depth o(log n). Since the clustering property is exhibited by many random structures, including proper coloring and maximum cut, we anticipate that our approach is extendable to these models as well.

quant-ph↗

arXiv2022

projUNN: efficient method for training deep networks with unitary matrices

In learning with recurrent or very deep feed-forward networks, employing unitary matrices in each layer can be very effective at maintaining long-range stability. However, restricting network parameters to be unitary typically comes at the cost of expensive parameterizations or increased training runtime. We propose instead an efficient method based on rank-$k$ updates -- or their rank-$k$ approximation -- that maintains performance at a nearly optimal training runtime. We introduce two variants of this method, named Direct (projUNN-D) and Tangent (projUNN-T) projected Unitary Neural Networks, that can parameterize full $N$-dimensional unitary or orthogonal matrices with a training runtime scaling as $O(kN^2)$. Our method either projects low-rank gradients onto the closest unitary matrix (projUNN-T) or transports unitary matrices in the direction of the low-rank gradient (projUNN-D). Even in the fastest setting ($k=1$), projUNN is able to train a model's unitary parameters to reach comparable performances against baseline implementations. In recurrent neural network settings, projUNN closely matches or exceeds benchmarked results from prior unitary neural networks. Finally, we preliminarily explore projUNN in training orthogonal convolutional neural networks, which are currently unable to outperform state of the art models but can potentially enhance stability and robustness at large depth.

cs.LG↗

arXiv2020

Quantum algorithm for nonlinear differential equations

Quantum computers are known to provide an exponential advantage over classical computers for the solution of linear differential equations in high-dimensional spaces. Here, we present a quantum algorithm for the solution of nonlinear differential equations. The quantum algorithm provides an exponential advantage over classical algorithms for solving nonlinear differential equations. Potential applications include the Navier-Stokes equation, plasma hydrodynamics, epidemiology, and more.

quant-ph↗

arXiv2020

Quantum polar decomposition algorithm

The polar decomposition for a matrix $A$ is $A=UB$, where $B$ is a positive Hermitian matrix and $U$ is unitary (or, if $A$ is not square, an isometry). This paper shows that the ability to apply a Hamiltonian $\pmatrix{ 0 & A^\dagger \cr A & 0 \cr} $ translates into the ability to perform the transformations $e^{-iBt}$ and $U$ in a deterministic fashion. We show how to use the quantum polar decomposition algorithm to solve the quantum Procrustes problem, to perform pretty good measurements, to find the positive Hamiltonian closest to any Hamiltonian, and to perform a Hamiltonian version of the quantum singular value transformation.

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