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Quantum simulation of Hamiltonian spectra on a silicon chip

Raffaele Santagati, Jia-Chao Wang, Antonio A. Gentile, Stefano Paesani, Nathan Wiebe, Jarrod R. McClean, Sam Morley-Short, Peter Shadbolt, Damien Bonneau, Joshua W. Silverstone, David P. Tew, Xiaoqi Zhou, Jeremy L. O’Brien, Mark G. Thompson

Published Nov 10, 2016• 6 citations
OpenAlex

Abstract

The efficient calculation of Hamiltonian spectra, a problem often intractable on classical machines, can find application in many fields, from physics to chemistry. Here, we introduce the concept of an eigenstate witness and through it provide a new quantum approach which combines variational methods and phase estimation to approximate eigenvalues for both ground and excited states. This protocol is experimentally verified on a programmable silicon quantum photonic chip, a mass-manufacturable platform, which embeds entangled state generation, arbitrary controlled-unitary operations, and projective measurements. Both ground and excited states are experimentally found with fidelities >99%, and their eigenvalues are estimated with 32-bits of precision. We also investigate and discuss the scalability of the approach and study its performance through numerical simulations of more complex Hamiltonians. This result shows promising progress towards quantum chemistry on quantum computers.

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The efficient calculation of Hamiltonian spectra, a problem often intractable on classical machines, can find application in many fields, from physics to chemistry. Here, we introduce the concept of an eigenstate witness and through it provide a new quantum approach which combines variational methods and phase estimation to approximate eigenvalues for both ground and excited states. This protocol is experimentally verified on a programmable silicon quantum photonic chip, a mass-manufacturable platform, which embeds entangled state generation, arbitrary controlled-unitary operations, and projective measurements. Both ground and excited states are experimentally found with fidelities >99%, and their eigenvalues are estimated with 32-bits of precision. We also investigate and discuss the scalability of the approach and study its performance through numerical simulations of more complex Hamiltonians. This result shows promising progress towards quantum chemistry on quantum computers.
The efficient calculation of Hamiltonian spectra, a problem often intractable on classical machines, can find application in many fields, from physics to chemistry. We introduce the concept of an "eigenstate witness" and, through it, provide a new quantum approach that combines variational methods and phase estimation to approximate eigenvalues for both ground and excited states. This protocol is experimentally verified on a programmable silicon quantum photonic chip, a mass-manufacturable platform, which embeds entangled state generation, arbitrary controlled unitary operations, and projective measurements. Both ground and excited states are experimentally found with fidelities >99%, and their eigenvalues are estimated with 32 bits of precision. We also investigate and discuss the scalability of the approach and study its performance through numerical simulations of more complex Hamiltonians. This result shows promising progress toward quantum chemistry on quantum computers.
An integrated silicon quantum photonic device, capable of implementing non-compiled controlled unitary operations (CUs), is performing a new variational protocol for searching the ground state of an exciton-transfer Hamiltonian in chlorophyll. This protocol only requires single-qubit measurements ―amenable for scalable quantum photonic approaches ― independently from the dimension of the Hamiltonian system. A maximally path-entangled state, |0>C|0>T +|1>C|0>T, is created on chip through the SFWM effect in silicon waveguides. Then, the target qubit state (with subscript “T”) is expanded into a four-dimensional space and evolved according to two separate unitary operations, i.e. I (identity) and an arbitrary U∈SU(2), ended with a path-information erasure process. This yields a non-compiled “CU” operation, controlled by the control qubit (with subscript “C”) as |0>C ⊗I|φ>T +|1>C⊗U|φ>T. In this experiment, an exciton-transfer Hamiltonian in chlorophyll is properly mapped as U onto our circuit’s parameters. Both state preparation and CU operations are implemented using thermo-optic phase shifting heaters. Our new protocol draws on the eigenstate-generation features of the Iterative Phase Estimation Algorithm (IPEA), and relies on 1-qubit state-tomography performed on the C-qubit alone, to infer properties of the Hamiltonian. The protocol proposes a variational search of the ground state in the Hilbert space spanned by U’s eigenstates, combining quantum measurements and a classical update scheme for the trial T-states that are prepared and evolved in the quantum circuit at each step of the algorithm. The search is guided by effectively minimizing an objective function Fobj=-αP+βE. Only when the input T-qubit is an eigenstate of U, the purity (P) of the output C-qubit |0> + eiφ|1>is 1, and in this case, eigenphase φprovides an estimator for |φ>T's eigenvalue (E). The on-chip implementation granted a robust, reconfigurable setup for our variational search algorithm: within less than 10 steps, we were able to achieve 99.8% fidelity of the final T-state with the ground-state of the implemented Hamiltonian. Under reasonable assumptions, these preliminary results may pave the way towards a scalable quantum-simulation approach for the ground-state problem.
Quantum computers promise to efficiently solve important problems that are intractable on a conventional computer. For quantum systems, where the physical dimension grows exponentially, finding the eigenvalues of certain operators is one such intractable problem and remains a fundamental challenge. The quantum phase estimation algorithm efficiently finds the eigenvalue of a given eigenvector but requires fully coherent evolution. Here we present an alternative approach that greatly reduces the requirements for coherent evolution and combine this method with a new approach to state preparation based on ansätze and classical optimization. We implement the algorithm by combining a highly reconfigurable photonic quantum processor with a conventional computer. We experimentally demonstrate the feasibility of this approach with an example from quantum chemistry—calculating the ground-state molecular energy for He–H+. The proposed approach drastically reduces the coherence time requirements, enhancing the potential of quantum resources available today and in the near future. Quantum computers promise to efficiently solve problems that would be practically impossible with a normal computer. Peruzzo et al. develop a variational computation approach that uses any available quantum resources and, with a photonic quantum processing unit, find the ground-state molecular energy of He–H+.
Quantum computers promise to provide groundbreaking speed in solving complex problems. However, in the present-day noisy intermediate-scale quantum era, algorithms that require fewer resources are highly desired. In this work, we develop a new method called the variational rodeo eigensolver (VRE) for efficiently searching eigenstates and estimating eigenvalues with shallow circuits. We experimentally demonstrate this method on a programmable photonic chip with a single-qubit exciton transfer Hamiltonian whose eigenstates are searched with fidelities of more than 99% and their eigenvalues are estimated, reaching chemical accuracy. Furthermore, we experimentally estimate the ground energies of the simplified Hamiltonian of the hydrogen molecule with different atomic separations. To verify the scalability of VRE, we numerically search eigenstates of a tapered two-qubit Hamiltonian of hydrogen-helium ion. Our work provides a systematic and promising approach for the efficient estimation of Hamiltonian spectra.