Bonneau, D.

Gentile, A. A.

McClean, J.

Morley-Short, S.

O'Brien, J. L.

Paesani, S.

Santagati, R.

Shadbolt, P. J.

Silverstone, J. W.

Tew, D. P.

Thompson, M. G.

Wang, J.

Wiebe, N.

Zhou, X.

We demonstrate a new protocol capable of finding ground and excited states of physical Hamiltonians via an eigenstate witness. The experimental test employs a silicon quantum photonic device, embedding arbitrary controlled unitary operations.</p

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                          Gentile, A. A., Santagati, R., Wang, J., Paesani, S., Wiebe, N., McClean, J.,... Thompson, M. G. (2019). Witness Assisted Eigenspectra Solver on aSilicon Quantum Photonic Simulator. In 2018 Conference on Lasers andElectro-Optics Pacific Rim, CLEO-PR 2018 [8699548] Institute of Electricaland Electronics Engineers (IEEE).https://doi.org/10.1364/CLEOPR.2018.Th1D.4Peer reviewed versionLicense (if available):OtherLink to published version (if available):10.1364/CLEOPR.2018.Th1D.4Link to publication record in Explore Bristol ResearchPDF-documentThis is the accepted author manuscript (AAM). The final published version (version of record) is available onlinevia OSA at https://doi.org/10.1364/CLEOPR.2018.Th1D.4 . Please refer to any applicable terms of use of thepublisher.University of Bristol - Explore Bristol ResearchGeneral rightsThis document is made available in accordance with publisher policies. Please cite only the publishedversion using the reference above. Full terms of use are available:http://www.bristol.ac.uk/pure/about/ebr-termsTh1D.4.pdf CLEO Pacific Rim 2018 © OSA 2018Witness assisted eigenspectra solver on a siliconquantum photonic simulatorA.A. Gentile1, R. Santagati1†, J. Wang1, S. Paesani1, N. Wiebe2, J. McClean3,D. Bonneau1, J.W. Silverstone1, S. Morley-Short1, P.J. Shadbolt1, D.P. Tew1, X. Zhou1,J. L.O’Brien1 and M. G. Thompson11 QETlabs, H. H. Wills Physics Laboratory, University of Bristol, Bristol BS8 1UB, United Kingdom.2 Quantum Architectures and Computation Group, Microsoft Research, Redmond, Washington 98052, USA.3 Google Inc., Venice, California 90291, USA.3 School of Chemistry, University of Bristol, Bristol BS8 1TS, United KingdomAbstract: We demonstrate a new protocol capable of finding ground and excited statesof physical Hamiltonians via an eigenstate witness. The experimental test employs a siliconquantum photonic device, embedding arbitrary controlled unitary operations.OCIS codes: 270.5585, 130.01301. IntroductionThe characterisation of the spectrum of a given Hamiltonian is a hard task for classical computers, yet it is requiredto study the properties of a quantum system, and unlock potential applications in a wide range of fields. Quantumsimulators offer a promising route to tackle this and other open problems in quantum chemistry. Recently, variationalquantum eigensolvers (VQE) have raised increased interest because of their intrinsic error resilience and shallow-depthcircuit requirements. These factors make them amenable to near term quantum devices, whilst retaining the peculiaradvantages of hybrid quantum-classical methods [1].Crucially, the analysis of excited states with VQE has for long remained elusive of successful experimental imple-mentations [2,3]. In this contribution we propose and demonstrate a hybrid quantum simulation protocol, representingan eigenspectra solver. Our method requires ansatz-based preparation of target states and the measurement of onlythe control qubit emerging from a controlled unitary (CU), independently from the size of the Hamiltonian. Thismeasurement provides an estimator of the proximity to an eigenstate: embedded in an objective function, it drivesthe variational search to improve the initial ansatz. The energy estimate of the approximate eigenstate prepared afterthe search is refined via Iterative Phase Estimation Algorithm (IPEA). The experimental test employs a SOI quantumphotonic chip, with integrated SFWM photon-pair sources, reconfigurable phase shifters and off-chip superconductingsingle photon detectors (Fig. 1a). Our chip is capable of implementing arbitrary CU without the need for computation-ally intensive pre-compilations, exploiting the advantages of state-of-art integrated photonic quantum technologies [4].2. ResultsThe device schematics is reported in Fig. 1a. The target qubit is composed of two different path-encoded componentsthat go through different unitaries, i.e. Iˆ and Uˆ , yielding a superposition of circuits as |0〉C⊗ Iˆ|φ〉T + |1〉C⊗ Uˆ |φ〉T .The final part of the target registry provides path information erasing, so that the overall circuit corresponds to a non-compiled CU [5]. The proposed variational search algorithm encodes the dynamics generated by the Hamiltonian inUˆ = e−iHˆt , thus evolving trial states prepared in the target registry as |φ〉T . Purity of the output control qubit state (P)and an estimator of the energy (E) can be both obtained via state tomography on the control qubit |φ〉C at each step.In particular, P is provably a sensible measure of the support of |φ〉T in the eigenbasis of Hˆ, behaving as a eigenstatewitness: separable two-qubit states emerging from the CU correspond to an eigenstate of Hˆ being injected as |φ〉T . Thephase acquired by |φ〉C additionally gives information on the corresponding eigenvalue [5]. This allows an objectivefunction Fob j =−αP+βE to solve for the ground state via a simultaneous energy and purity optimization (not shownhere for brevity). Ansa¨tze from classical methods can then provide trial guess states with sufficient overlap with atargeted excited state. Our protocol then maximizes P, updating the choice(s) of trial |φ〉T at every algorithm step,CLEO Pacific Rim 2018 © OSA 2018Th1D.4.pdf CLEO Pacific Rim 2018 © OSA 2018Fig. 1: a - Schematic of the 2 qubits device used to implement CU operations, along with basic off-chip components.b - Bloch sphere showing the evolution of trial input states |φT 〉, in different colors as per algorithm step, convergingto the excited state |+〉 for the experimentally investigated Hˆ. c - Test IPEA run up to 32-bits, for the excited statefound. The histogram bars provide experimental normalized coincidence counts, obtained from the projectors of |φC〉on the computational basis. d - Simulated behaviour of the variational protocol in estimating excited states for testHamiltonians of different size. Final fidelities achieved are emphasized in the inset with the same color-coding. Shadedareas indicate a 67.5% confidence interval.variationally refining the initial classical estimate. Fig. 1b shows the variational search for the excited state of a testHamiltonian, implemented in our photonic device. The swarm of trial states represented on the Bloch sphere is clearlyconverging to the correct eigenstate |+〉, up to an average fidelity of 99.8%. In Fig. 1c we demonstrate on our chiphow an additional IPEA step is able to provide 32-bits of accuracy in the energy estimate for the found eigenstate. Thepotential of our method for Hamiltonians beyond the capabilities of the photonic chip has been explored in numericalsimulations for Hydrogen molecules up to H8 (Fig 1d), yielding average fidelities higher than 99%.In conclusion, our method combines the advantages of variational protocols and quantum phase estimation, intro-ducing a concept with no classical equivalent, the eigenstate witness, that allows to extend the capabilities of variationaleigensolvers to target excited states experimentally for the first time.References1. J. R. McClean, et. al. “The theory of variational hybrid quantum-classical algorithms”. New J. Phys. 18.2, 2016.2. A. Peruzzo, et. al. A variational eigenvalue solver on a photonic quantum processor. Nat. Commun. 5, 2014.3. J. I. Colless, et. al. “Computation of Molecular Spectra on a Quantum Processor with an Error-Resilient Algo-rithm”. Phys. Rev. X 8, 2018.4. R. Santagati, et al. “Silicon photonic processor of two-qubit entangling quantum logic”. J. Opt. 19.11, 2017.5. R. Santagati, et al. “Witnessing eigenstates for quantum simulation of Hamiltonian spectra”. Sci. Adv. 4.1, 2018.

Witness Assisted Eigenspectra Solver on a Silicon Quantum Photonic Simulator

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Witness assisted eigenspectra solver on a silicon quantum photonic simulator

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Witness Assisted Eigenspectra Solver on a Silicon Quantum Photonic Simulator

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