# Fermionic marginal constraints for hybrid algorithms

1 February 2018

by Nick Rubin

I’m excited to announce the theory project I have been collaborating on over the last year has finally hit the arXiv! Even more exciting is that this project articulates the root of many ideas that will help make hybrid quantum computation a reality for chemistry and material simulation.

So what’s the main idea? The variational quantum eigensolver (VQE), which is a member of a more general class of hybrid quantum algorithms, is essentially incomplete. There are components of the algorithm that need to be improved for this particular algorithmic strategy to provide the promised revolutions in quantum simulation. To see where we need to to tighten our understanding of the algorithm we can examine the major components of VQE: 1) measurement, 2) state ansatz, and 3) optimization. State ansatz is a parameterized programmatic representation of a quantum state that likely includes the ground state of the system we are simulating. The most famous example being the unitary coupled-cluster state ansatz. Measurement is the main mechanism for energy evaluation and is directly proportional to run time of the algorithm. In classical computing terms, measurement is our function call to the quantum computer. Lastly, classical optimization that tolerates statistical fluctuations and corrects for gate errors will be required for the algorithm to be a success. Each of these main components offers a challenge and are important areas of research on hybrid algorithms. We’ve focused on the measurement problem as a starting point in making hybrid algorithms a reality.

Each iteration in the VQE algorithm involves an energy evaluation given a quantum circuit parameterized by a polynomial number of variables. The energy is evaluated by many state preparations and measurements. For example, if I want to estimate the expected value (also known as the mean value) of a Hamiltonian represented as a sum of Pauli operators, I can prepare the same state over and over again measuring after each preparation. The measurements return an eigenstate of the the observable I am estimating and with enough samples I can estimate the mean value reliably. This is analogous to drawing samples from a parameterized distribution to calculate expectation values. One may ask: How many samples are required for a particular accuracy? What errors do statistical fluctuations induce? What does noise on the quantum computer do to the observables? Is there a way to correct the expected values we observe because of under sampling? In this new paper, we make significant progress toward a solution to the measurement problem of VQE by answers these questions.

Our strategy was to leverage geometric constraints on marginals of the state, known as n-representability conditions, partially developed by the quantum chemistry community. It turns out that the geometry constraints provide a lot of extra information on the structure of the state and can be used in a variety of ways to reduce the total number of measurements and to reduce errors from noisy quantum gates. What’s interesting for chemistry and materials simulation is that we only need access to the 2-body marginal of the system to evaluate the energy — i.e., we don’t need the entire state of the system, only the two body correlations. Therefore, if we can measure the 2-body marginal we can get the energy for free. Measuring the 2-body marginal is formally the same complexity as estimating the expected value of the Hamiltonian (i.e., the energy) but comes with extra benefits–those geometric constraints I mentioned before. These benefits led us to new ways to analyze the ansatz problem for VQE, reduce the total number of measurements required for expected value estimation, and propose projection techniques to remove noise.

Of course, all of the theory work in the paper was accompanied by validating numerics with the aid of OpenFermion[1, 2, 3]. We benchmarked the various strategies for reducing measurement and removing noise against static energy calculations of diatomic hydrogen and a hydrogen chain of length four. Molecular data was generated with OpenFermion and the OpenFermion-Psi4 plugins. One of the techniques for reducing noise has already made it into OpenFermion and is ready for use on a wider variety of molecules and situations. The other techniques from the geometry of 2-body marginals are planned additions to the repository.

The structure of 2-body marginals and how they can be used for the measurement problem is all well-and-good in theory but needs to be validated with more sophisticated numerics or a quantum simulation! You can be sure we’ll be validating these techniques with real quantum resources.

Stay tuned!

Originally published at rigetticomputing.github.io on February 1, 2018.

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