Quantum Observable Partitioning for Constrained VQE Expectation Values
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Solution Overview
Problem
Existing variational quantum eigensolvers (VQEs) face challenges in enforcing constraints such as particle number, spin multiplicity, and spatial symmetries, leading to incorrect results and inefficient use of computational resources due to errors in state preparation and readout, and methods like penalty terms exacerbate these issues.
Innovation Solution
A method involving partitioning the representation of a quantum mechanical observable into disjoint subsets and using equivalent quantum circuits to calculate the expectation value, combined with a bijective mapping to an unconstrained Hilbert space, ensures that only valid states are considered, reducing errors and resource usage.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Reliability
If penalty terms are used to enforce constraints in VQE, then constraint satisfaction is improved, but computational resources and time are worsened due to additional measurements and slower convergence
Solution Approach 1:
The Hamiltonian is segmented into constraint terms and problem terms. By separately identifying and handling constraint terms, the algorithm can enforce constraints without requiring penalty terms that slow down convergence. The constraint terms are evaluated and used to project the wavefunction onto the valid subspace, ensuring constraint satisfaction while maintaining efficient convergence.
Solution Approach 2:
The algorithm performs preliminary action by projecting the initial wavefunction and subsequent wavefunctions onto the valid subspace defined by the constraints before energy minimization begins. This preliminary projection ensures that all subsequent iterations operate within the constraint-satisfied subspace, eliminating the need for penalty terms and reducing convergence time.
2Measurement precision
If the search space includes all Fock space states, then the lowest energy state might be found, but computational resources are wasted on invalid states that violate constraints
Solution Approach 1:
The algorithm extracts and removes invalid states from the search space by projecting the wavefunction onto the valid subspace. This extraction eliminates states that violate constraints (such as incorrect particle number, spin multiplicity, or spatial symmetry) from consideration, ensuring that computational resources are focused only on physically meaningful states while maintaining accuracy in finding the lowest energy state.
3Productivity
If VQE searches without explicit constraint enforcement, then computational resources are used efficiently, but the algorithm may converge to incorrect results violating physical constraints
Solution Approach 1:
The algorithm implements self-service by automatically projecting wavefunctions onto the valid subspace during the VQE optimization process. This self-enforcing mechanism ensures that the algorithm maintains constraint satisfaction throughout the optimization without requiring external intervention or penalty terms, thereby preserving both computational efficiency and result correctness.
Data Source
AI summary
A computer-implemented method for calculating the expectation value of a Hermitian quantum mechanical observable in a quantum state prepared on a quantum computer is disclosed in which the method comprises generating a representation of the quantum mechanical observable as a sum of outer products between two computational basis states of a quantum computer, partitioning the representation into disjoint subsets of terms, generating one quantum circuit, or any equivalent circuit that performs the same transformation, for each subset, determined by the terms within each particular subset, executing the quantum circuits on the quantum computer for a plurality of repetitions to obtain a plurality of measurement results and determining the expectation value of the observable in the quantum state using the plurality of measurement results.