Source-linked AI summary
A variational eigenvalue solver on a quantum processor
Alberto Peruzzo, Jarrod McClean, Peter Shadbolt, Man-Hong Yung, Xiao-Qi Zhou, Peter J. Love, Alán Aspuru-Guzik, Jeremy L. O'Brien
TL;DR
Ground-state estimation remains QMA-complete, while available quantum resources are limited. The paper combines quantum state preparation and Hamiltonian measurement with classical optimization, concluding that one iteration can have an exponential cost advantage under stated assumptions.
Problem
Ground-state convergence remains QMA-complete, motivating methods that reduce the quantum resources required per iteration.
Method
The approach prepares quantum states with an ansatz, maps operators to Pauli Hamiltonians, measures expectation values, and optimizes parameters classically.
Results
Under stated Hamiltonian and state assumptions, the algorithm achieves an exponential cost advantage per iteration over the best known classical algorithms.
Takeaways & Limitations
The analysis supports the value of the algorithm for quantum computation with limited available quantum resources.
Takeaways & Limitations
Photon distinguishability and higher-order photon-number terms degrade entangling-gate performance and shift the measured energy.
Abstract
from arXiv · showhide
Quantum computers promise to efficiently solve important problems that are intractable on a conventional computer. For quantum systems, where the dimension of the problem space grows exponentially, finding the eigenvalues of certain operators is one such intractable problem and remains a fundamental challenge. The quantum phase estimation algorithm can efficiently find the eigenvalue of a given eigenvector but requires fully coherent evolution. We present an alternative approach that greatly reduces the requirements for coherent evolution and we combine this method with a new approach to state preparation based on ansätze and classical optimization. We have implemented the algorithm by combining a small-scale 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+, to within chemical accuracy. The proposed approach, by drastically reducing the coherence time requirements, enhances the potential of the quantum resources available today and in the near future.
APPENDIX · SUPPLEMETARY THEORY Quantum eigenvector preparation algorithm
The appendix details Algorithm 2 for quantum eigenvector preparation. The method uses a parameter-controlled circuit to prepare states and an objective function mapping experimental parameters to Hamiltonian expectation values.
- APPENDIX: The appendix introduces the implementation steps for Algorithm 2.
- SUPPLEMETARY THEORY Quantum eigenvector preparation algorithm: The supplementary theory describes a quantum circuit controlled by experimental parameters {θ_i}.
- SUPPLEMETARY THEORY Quantum eigenvector preparation algorithm: The circuit is designed to prepare a class of quantum states.
- SUPPLEMETARY THEORY Quantum eigenvector preparation algorithm: The procedure prepares an initial state |ψ_0⟩ and defines an objective function f({θ_n}).
- SUPPLEMETARY THEORY Quantum eigenvector preparation algorithm: The objective function efficiently maps experimental parameters to the Hamiltonian’s expectation value.
- SUPPLEMETARY THEORY Quantum eigenvector preparation algorithm: Algorithm 1 computes the objective function, with n denoting the current algorithm iteration.
3. Repeat until optimization is completed
Each optimization iteration evaluates the Hamiltonian expectation for the current parameters, sums the measured term values classically, and uses a classical minimizer to update the parameters. Under stated assumptions, the resulting iteration has an exponential cost advantage over the best-known classical algorithms, although overall ground-state convergence retains QMA-complete complexity.
- Optimization iteration: The procedure calls Algorithm 1 with the current parameter set {θ_i} as input.
- Optimization iteration: Measured Hamiltonian-term values are classically summed on the CPU to obtain the expectation value.
- Optimization iteration: A classical minimization algorithm, such as gradient descent or Nelder–Mead Simplex, determines the next parameters {θ_n+1}.
- Computational cost: The algorithm attains an exponential advantage in the cost of a single iteration under assumptions on the Hamiltonian and quantum state.Convergence to the final ground state still respects the QMA-Complete complexity of the task.