Research ArticlePHYSICS

Witnessing eigenstates for quantum simulation of Hamiltonian spectra

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Science Advances  26 Jan 2018:
Vol. 4, no. 1, eaap9646
DOI: 10.1126/sciadv.aap9646
  • Fig. 1 The WAVES protocol.

    (A) Flowchart describing the protocol. The optimization of Embedded Image using the circuit in (B) allows one to variationally find the ground state of the Hamiltonian, preparing trial states via the ansatz Embedded Image with no perturbation Embedded Image. An initial guess for an excited state is given by a perturbation Embedded Image on the ground state and then refined using the same circuit by exploiting the eigenstate witness Embedded Image. (C) For each target eigenstate found, the eigenvalues are precisely estimated via the IPEA using the quantum logic circuit, where H is the Hadamard gate. The color coding in (B) and (C), blue for the control and red for the target, refers to the difference in wavelength between the photon in the control qubit and the one in the target register in our experimental implementation. (D) Diagram schematically representing the intuition behind the proposed approach, where initial guesses of excited states are variationally refined using the witness and IPEA returns the eigenvalues.

  • Fig. 2 Silicon quantum photonic processor.

    The quantum device enables one to produce maximally path-entangled photon states, perform arbitrary single-qubit state preparation and projective measurements, and, more importantly, perform any operation in the two-dimensional space. Photons are guided in the silicon waveguides and controlled by thermo-optical phase shifters. Photon pairs are directly generated inside the silicon spiral sources through SFWM, off-chip–filtered and postselected by AWG filters (not shown), and measured by SNSPDs. The generated signal (blue) and idler (red) photons are different in wavelength and form the control and target qubits, respectively. The quantum chip is interfaced with a classical computer. Inset: High-visibility quantum (blue) and classical (green) interference fringes obtained in the device using the photon sources part and configuring the top final interferometer. The high visibility is essential to verify the high-performance and correct characterization of the device.

  • Fig. 3 Experimental results.

    A Hamiltonian representing a single-exciton transfer between two chlorophyll units is implemented on the silicon quantum photonic device for an experimental test of the protocol. (A and B) Color-coded evolution of the particle swarm for the WAVES search of the ground state (| − 〉) and excited state (| + 〉) shown on the Bloch spheres. Different colors correspond to different steps of the search protocol. For the ground and the excited state searches, we report the evolution of Embedded Imageobj in (C) and (D) and the fidelity (F = |〈Ψ|Ψideal〉|2) versus search steps in (E) and (F), converging to a final value of 99.48 ± 0.28% and 99.95 ± 0.05%, respectively. Error bars are given by the variance of the particle distribution and photon Poissonian noise. Dashed lines are numerical simulations of the performance of the algorithm, averaged over 1000 runs, with shaded areas representing a 67.5% confidence interval. Insets: Behavior close to convergence. (G and H) Normalized photon coincidences used to calculate the 32 IPEA-estimated bits of the eigenphase for both eigenstates. The theoretical bit value is shown above each bar. Errors arising from Poissonian noise are shown as shaded areas on the bars.

  • Fig. 4 Numerical simulations for higher-dimensional Hamiltonians.

    The cases studied refer to molecular hydrogen systems Embedded Image with the full PH ansatz. (A) Variational search for the ground state of each physical system. (B) Variational search for the targeted subspace of degenerate excited states with an initial excitation perturbation Embedded Image. On the x axis, we refer to the cumulative number of trial states probed (that is, the number of particles in the swarm times the variational steps). For ease of comparison, the x-axis origin has been shifted in (A) for the various cases to have equivalent fidelity for the average initial guess. Dashed lines denote average fidelities, with the shaded areas indicating a 67.5% confidence interval. The average fidelities achieved by the particle swarm optimization for both ground and excited states are calculated for 100 independent runs of WAVES. In all simulations, a binomial noise model has been taken into account when performing projective measurements. Insets: Bar charts summarizing final fidelities obtained by each search. All the simulations converged to the same high fidelity within errors, as indicated by the dashed black line in the inset.

Supplementary Materials

  • Supplementary material for this article is available at http://advances.sciencemag.org/cgi/content/full/4/1/eaap9646/DC1

    section S1. Notes on the objective function

    section S2. Swarm optimization algorithm

    section S3. Complexity analysis of the variational protocol

    section S4. Phase estimation without quantum collapse

    section S5. Experimental details

    section S6. Robustness against experimental noise of the variational search

    section S7. Numerical simulations of hydrogen molecules

    fig. S1. Median inference error in eigenvalue inference with α = 1/2 for a distribution with two randomly chosen eigenvalues and likelihood function (37) is used.

    fig. S2. Schematic representation of the experimental setup.

    fig. S3. Numerical simulations of the variational ground state search robustness for the bosonic one-qubit Ĥ against gate infidelities using Formulaobj or Formula alone.

    fig. S4. Synopsis of numerical simulations of excited states searches for the molecules H2 and Formula.

    fig. S5. Numerical simulations of the WAVES variational search for the synthetically truncated PH ansatz are studied in the molecular hydrogen systems (H2, Formula, H3, H4).

    fig. S6. Convergence of the WAVES algorithm to a subspace of excited states for different hydrogen systems.

    fig. S7. Comparison between different ansaetze adopted in the search for excited states in the Formula system.

    fig. S8. Behavior comparison of the first part of WAVES and an equivalent implementation of the FS method when applied to the initial guess provided by the Formula excitation operator for the H2 system.

    table S1. Summary of ansätze used for simulations in the main paper and in the Supplementary Materials for the various systems investigated, along with the cardinality of their parameterization, Formula.

    table S2. Summary of possible situations occurring in numerical simulations when WAVES is performed with different ansätze and excitation operators, targeting a certain excited subspace Formula from an initial guess |Ψ0〉.

    References (4856)

  • Supplementary Materials

    This PDF file includes:

    • section S1. Notes on the objective function
    • section S2. Swarm optimization algorithm
    • section S3. Complexity analysis of the variational protocol
    • section S4. Phase estimation without quantum collapse
    • section S5. Experimental details
    • section S6. Robustness against experimental noise of the variational search
    • section S7. Numerical simulations of hydrogen molecules
    • fig. S1. Median inference error in eigenvalue inference with α = 1/2 for a distribution with two randomly chosen eigenvalues and likelihood function (37) is used.
    • fig. S2. Schematic representation of the experimental setup.
    • fig. S3. Numerical simulations of the variational ground state search robustness for the bosonic one-qubit Ĥ against gate infidelities using Fobj or E alone.
    • fig. S4. Synopsis of numerical simulations of excited states searches for the molecules H2 and H+3 .
    • fig. S5. Numerical simulations of the WAVES variational search for the synthetically truncated PH ansatz are studied in the molecular hydrogen systems ( H2, H+3, H3, H4).
    • fig. S6. Convergence of the WAVES algorithm to a subspace of excited states for different hydrogen systems.
    • fig. S7. Comparison between different ansaetze adopted in the search for excited states in the H+3 system.
    • fig. S8. Behavior comparison of the first part of WAVES and an equivalent implementation of the FS method when applied to the initial guess provided by the Êp3 excitation operator for the H2 system.
    • table S1. Summary of ansätze used for simulations in the main paper and in the Supplementary Materials for the various systems investigated, along with the cardinality of their parameterization, dim (θ) .
    • table S2. Summary of possible situations occurring in numerical simulations when WAVES is performed with different ansätze and excitation operators, targeting a certain excited subspace Eτ from an initial guess | Ψ0.
    • References (48–56)

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