Quantum computing

• Peruzzo2014 : A variational eigenvalue solver on a photonic quantum processor.Nature Communications, 5(May).

Variational Quantum Eigenvalue solver

Algorithm for finding the eigenvalue of an operator using a variational approach.

Method and Algorithm

• Express the Hamiltonian of a system using creation/annihilation operators :
  • $\bb H = \sum_{pq} h_{pq}\bb a_p^{\dagger}\bb a_q + \sum_{pqrs} h_{pqrs}\bb a_p^{\dagger}\bb a_q^{\dagger}\bb a_r\bb a_s +...$.
• Turn it into a spin-chain Hamiltonian (Pauli operators) using Jordan-Wigner transformation :
  • $\bb H = \sum_{i,\alpha} h^{\alpha}_{i}(R)\sigma^{i}_{\alpha} + \sum_{ij} h_{ij\alpha\beta}\sigma_{\alpha}^{i}\sigma_{\beta}^{j} + ... = \sum_i \bb H_i$
• compute the expectation value $<\Psi|H|\Psi>/<\Psi|\Psi>$ :
  • where the Hamiltonian has been expressed as a sum of polynomial number of tensor products of Pauli operators $\bb H_i$
  • each module computes the expectation value of the tensor product of Pauli operators $<\Psi|\sigma_i\otimes\sigma_j|\Psi>$
• Use a classical minimizer to determine new parameters for $\Psi$

features

• short coherent evolution compared to QA and standard QPE
• Needs polynomial number of iterations that QPE
• Has computational and storage exponential advantage over classical algorithm

Prototype Demonstration Implementation

• Reconfigurable photonic quantum processor to evaluate a Pauli matrix tensor product
• Local measurement of each qbit is used to get the expectation value of Pauli matrix tensor products
• The wave vector is parametrized with $\phi$ achieved by phase shifters. A mapping must be defined to get $\phi$ from the optimizer parameters $\theta$.
• In the case of a 2x2 generic pauli tensor product

Application to He-H+ molecule (2 electrons => H is 2x2) the electrons may be entangled.

• PSI3 software using the STO-3G basis to obtain coefficients $h_{\alpha}^{i}, h_{ij}^{\alpha\beta}$.
• Nelder-Mead minimizer?
• Bond dissociation curve for HeH+ obtained by successive application of the VQE.

Questions