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Data re-uploading for a universal quantum classifier




Adrián Pérez-Salinas1,2, Alba Cervera-Lierta1,2, Elies Gil-Fuster3, and José I. Latorre1,2,4,5

1Barcelona Supercomputing Center
2Institut de Ciències del Cosmos, Universitat de Barcelona, Barcelona, Spain
3Dept. Física Quàntica i Astrofísica, Universitat de Barcelona, Barcelona, Spain.
4Nikhef Theory Group, Science Park 105, 1098 XG Amsterdam, The Netherlands.
5Center for Quantum Technologies, National University of Singapore, Singapore.

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A single qubit provides sufficient computational capabilities to construct a universal quantum classifier when assisted with a classical subroutine. This fact may be surprising since a single qubit only offers a simple superposition of two states and single-qubit gates only make a rotation in the Bloch sphere. The key ingredient to circumvent these limitations is to allow for multiple $textit{data re-uploading}$. A quantum circuit can then be organized as a series of data re-uploading and single-qubit processing units. Furthermore, both data re-uploading and measurements can accommodate multiple dimensions in the input and several categories in the output, to conform to a universal quantum classifier. The extension of this idea to several qubits enhances the efficiency of the strategy as entanglement expands the superpositions carried along with the classification. Extensive benchmarking on different examples of the single- and multi-qubit quantum classifier validates its ability to describe and classify complex data.

In this paper, we show how to use the computational power of a single qubit to solve non-trivial classification problems. We propose a hybrid classical-quantum algorithm based on re-uploading classical data into the angles of the single-qubit unitary gates multiple times along the circuit. Together with the data points, other parameters are introduced into the circuit and adjusted by classically minimizing a cost function. To construct this cost function, we train the circuit to distribute the data points into different regions of the Bloch sphere, one for each class. A particular division of the Bloch sphere accompanies this strategy for maximizing distinguishability between classes.
This procedure cannot provide any quantum advantage as a single qubit can be simulated classically. However, the capability of handling one qubit might be useful as a small piece of larger circuits. Besides, an extension of the algorithm for more qubits and entanglement is also presented in this work. The multi-qubit role remains unexplored and might be a candidate for quantum advantage. A first step analyzed, there exists a trade-off between the number of qubits needed and the times of data re-uploading for classifying, namely layers.
This algorithm is to be compared with a neural network with one hidden layer. Neural Networks re-upload classical data several times, once per hidden neuron, achieving the same kind of processing as in our quantum classifier. Success rates are also comparable for both models.

► BibTeX data

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Cited by

[1] Seth Lloyd, Maria Schuld, Aroosa Ijaz, Josh Izaac, and Nathan Killoran, “Quantum embeddings for machine learning”, arXiv:2001.03622.

[2] Sergi Ramos-Calderer, Adrián Pérez-Salinas, Diego García-Martín, Carlos Bravo-Prieto, Jorge Cortada, Jordi Planagumà, and José I. Latorre, “Quantum unary approach to option pricing”, arXiv:1912.01618.

The above citations are from SAO/NASA ADS (last updated successfully 2020-02-06 14:31:00). The list may be incomplete as not all publishers provide suitable and complete citation data.

Could not fetch Crossref cited-by data during last attempt 2020-02-06 14:30:58: Could not fetch cited-by data for 10.22331/q-2020-02-06-226 from Crossref. This is normal if the DOI was registered recently.



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Quantum Algorithms for Simulating the Lattice Schwinger Model




Alexander F. Shaw1,5, Pavel Lougovski1, Jesse R. Stryker2, and Nathan Wiebe3,4

1Quantum Information Science Group, Computational Sciences and Engineering Division, Oak Ridge National Laboratory, Oak Ridge, TN 37831, U.S.A.
2Institute for Nuclear Theory, University of Washington, Seattle, WA 98195-1550, U.S.A.
3Department of Physics, University of Washington, Seattle, WA 98195, U.S.A.
4Pacific Northwest National Laboratory, Richland, WA 99354, U.S.A.
5Department of Physics, University of Maryland, College Park, Maryland 20742, U.S.A.

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The Schwinger model (quantum electrodynamics in 1+1 dimensions) is a testbed for the study of quantum gauge field theories. We give scalable, explicit digital quantum algorithms to simulate the lattice Schwinger model in both NISQ and fault-tolerant settings. In particular, we perform a tight analysis of low-order Trotter formula simulations of the Schwinger model, using recently derived commutator bounds, and give upper bounds on the resources needed for simulations in both scenarios. In lattice units, we find a Schwinger model on $N/2$ physical sites with coupling constant $x^{-1/2}$ and electric field cutoff $x^{-1/2}Lambda$ can be simulated on a quantum computer for time $2xT$ using a number of $T$-gates or CNOTs in $widetilde{O}( N^{3/2} T^{3/2} sqrt{x} Lambda )$ for fixed operator error. This scaling with the truncation $Lambda$ is better than that expected from algorithms such as qubitization or QDRIFT. Furthermore, we give scalable measurement schemes and algorithms to estimate observables which we cost in both the NISQ and fault-tolerant settings by assuming a simple target observable–the mean pair density. Finally, we bound the root-mean-square error in estimating this observable via simulation as a function of the diamond distance between the ideal and actual CNOT channels. This work provides a rigorous analysis of simulating the Schwinger model, while also providing benchmarks against which subsequent simulation algorithms can be tested.

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Cited by

[1] Indrakshi Raychowdhury and Jesse R. Stryker, “Solving Gauss’s Law on Digital Quantum Computers with Loop-String-Hadron Digitization”, arXiv:1812.07554.

[2] Christopher David White, ChunJun Cao, and Brian Swingle, “Conformal field theories are magical”, arXiv:2007.01303.

[3] Anthony Ciavarella, “An Algorithm for Quantum Computation of Particle Decays”, arXiv:2007.04447.

[4] Minh C. Tran, Yuan Su, Daniel Carney, and Jacob M. Taylor, “Faster Digital Quantum Simulation by Symmetry Protection”, arXiv:2006.16248.

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On Crossref’s cited-by service no data on citing works was found (last attempt 2020-08-12 00:44:14).


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Mapping graph state orbits under local complementation




Jeremy C. Adcock1, Sam Morley-Short1, Axel Dahlberg2, and Joshua W. Silverstone1

1Quantum Engineering Technology (QET) Labs, H. H. Wills Physics Laboratory & Department of Electrical & Electronic Engineering, University of Bristol, Merchant Venturers Building, Woodland Road, Bristol BS8 1UB, UK
2QuTech – TU Delft, Lorentzweg 1, 2628CJ Delft, The Netherlands

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Graph states, and the entanglement they posses, are central to modern quantum computing and communications architectures. Local complementation – the graph operation that links all local-Clifford equivalent graph states – allows us to classify all stabiliser states by their entanglement. Here, we study the structure of the orbits generated by local complementation, mapping them up to 9 qubits and revealing a rich hidden structure. We provide programs to compute these orbits, along with our data for each of the $587$ orbits up to $9$ qubits and a means to visualise them. We find direct links between the connectivity of certain orbits with the entanglement properties of their component graph states. Furthermore, we observe the correlations between graph-theoretical orbit properties, such as diameter and colourability, with Schmidt measure and preparation complexity and suggest potential applications. It is well known that graph theory and quantum entanglement have strong interplay – our exploration deepens this relationship, providing new tools with which to probe the nature of entanglement.

Graph states are ubiquitous representations of entanglement in quantum information science, and classify the most studied set of quantum states—clifford states—by the entanglement they possess.

However, many graph states are locally equivalent to one another, that is, they possess the same type of entanglement. Graph states which are locally equivalent can be transformed into one another by successive applications of the graph operation local complementation (example shown above). Using this operation, we can analyse only graph structure of the state, which is much simpler than analysing the exponentially large quantum state vector. This equivalence of graph states has been studied previously, with all graph states up to 12 qubits classified.

However, local complementation gives us more than sets of locally equivalent graphs: it also gives us an orbit (example shown above) which tells us how different graphs are related via local complementation. In this work we study these orbits, and relate their properties to properties of the entangled quantum states they contain. We find that orbit properties, such as colourability, correlate with entanglement properties, such as schmidt measure, and discuss applications of local complementation in quantum technology.

► BibTeX data

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