Prototype implementation of qdrift algorithm for quantum chemistry simulation from https://arxiv.org/abs/1811.08017 undertaken as part of the SQUID Hackathon at ETH Zurich in September 2019.
In order to solve the important electronic structure problem, it is important to be able to calculate the energy of a given moleucle. One of the ways to do this is by preparing a state, time evolving under the Hamiltonian for a sequence of times, and measuring the phases. From this, we can infer the energy.
The time evolution is a difficult step especialy in quantum chemistry where our Hamiltonian may have O(n^4) terms.
The standard Trotter approximations U(t) ~= V(t) = \prod_j exp(-i(t)h_j H_j) for t small.
The algorithm we have implemented here approaches the problem of decomposing U(t) in a slightly different way.
We set a standard rotation angle for all the terms in the Hamiltonian, and then generate a sequence of rotations as detailed in the paper.
This method was reported to be able to reduce the gates required to simulate the time evolution by up to 1000x.
We have implemented this in Qiskit by creating a modified version of the Iterative Phase Estimation algorithm which measures the energy of the ground state with precision increasing with iterations (https://arxiv.org/pdf/quant-ph/0610214.pdf).
To get the gate counts for H2 and LiH, run the files H2_gate_counts.py and
LiH_gate_counts.py. These can be edited to find counts for other molecules.
To generate the plots of energy for H2 and compare against the reference from
the exacteigensolve method, use the run_iqpe.py script. This script accepts the
following usage options
usage: run_iqpe.py [-h] [-p PROCESSES] [-r] [-i] [-s STEPS] [-f FIRST_ATOM]
[-e ERROR] [-v]
optional arguments:
-h, --help show this help message and exit
-p PROCESSES, --processes PROCESSES
Number of processes to use (default=1)
-r, --no-ref Do not calculate reference values using exact
eigensolver.
-i, --include-standard-iqpe
Include the standard IQPE method.
-s STEPS, --steps STEPS
Number of distance steps to use between 0.5 and 1.0
(default=10).
-f FIRST_ATOM, --first_atom FIRST_ATOM
The first atom (default=H).
-e ERROR, --error ERROR
The error to use for qdrift IQPE (default=0.1).
-v, --verbose
During the project, we managed to implement the algorithm and get it producing results. We see good improvement in gate counts for larger molecules, but still require more work getting the energies right.
We also saw a massive reduction in the case of LiH but less so for H2
We have implemented qdrift in the Iterative Quantum Phase Estimation algorithm to measure the energy.
The above plot compares the energy calculated using IQPE with qdrift(orange), IQPE using trotter (red), the energy of the Hartree Fock state (blue) and the exact energy of the ground state (green).
The energies calculated with IQPE using qdrift show greater variance than those calculated using the standard trotter approach. Further work is ongoing to resolve this.
- Some more work is required to resolve the inaccuracies in the energy and variance issues
- Better control over the parameters
- Contribution of the completed algorithm to the qiskit package