Quantum computers undeniably hold significant potential for modelling inherently quantum mechanical systems such as molecules and materials. However, it is proving difficult to deliver this potential for several reasons.
– A blog post by Martin Rahm, Associate Professor, Theoretical Chemistry, Chalmers University of Technology
One very important issue is that quantum computers are very sensitive to ‘noise’. It is currently practically impossible to physically eliminate the effect of noise on state-of-the-art quantum processors. Researchers have therefore focused on developing strategies to mitigate the errors, with any number having been tried. Some of these show considerable promise, especially when two or more mitigation strategies are combined. However, there is still plenty of room for improvement.
A new method for error mitigation
In a recent paper (see below), we report on a new method for mitigating errors in quantum computing of chemistry called reference-state error mitigation (REM). We think this approach has particular promise because it relies on a postprocessing step that can be done on a ‘conventional’ computer. Our method is also usable across different levels of noise and is relatively low-cost in computing terms. It can even be combined with other error mitigation strategies to allow more kinds of errors to be addressed.
REM consists of four steps. In the first step, you decide on and evaluate a reference problem that is a proxy for the real molecular system that you are trying to investigate. This reference state is chosen such that it can be quickly and exactly calculated using a ‘conventional’ computer. In the second step, you encode and perform a measurement of that same reference state on a quantum processor. The third step of the method involves comparing the energy calculated for the reference solution on the two forms of hardware – conventional hardware without error and quantum hardware subject to error due to noise. The difference in the answers gives you a value for the error.
Finally, this error term is assumed to be systematic and valid also for more complex but similar problems. You can then use this difference to correct the solutions of your original harder molecular problem that is evaluated separately with a suitable algorithm such as the variational quantum eigensolver (VQE). The VQE is a hybrid quantum algorithm that combines conventional and quantum computing resources to find the ground state of a physical system.
This sounds complicated, but the process itself is actually relatively straightforward, and it has a low cost in computing terms. The most important aspect is picking the right wave function to use as a reference. We argue that a particular kind of chemically motivated reference wave function (the Hartree-Fock state) should be sufficiently similar to the exact solution to be useful in most cases, while also being relatively easy to compute on a conventional computer. Coincidentally, this choice for reference can also be chosen as the initial point for the VQE optimisation, further reducing overhead in terms of measurement costs.
That’s the theory—but what was the result in practice? When we applied REM to a selection of test cases, small molecules such as hydrogen and lithium hydride, we managed to reduce the error and improve the computational accuracy by two orders of magnitude. That outcome was observed when we combined REM with another mitigation method, readout mitigation. Readout mitigation involves an initial calibration of the quantum circuit that prepares and measures quantum states that are unrelated to chemistry. In the absence of readout mitigation, the relative improvement offered by using REM is substantially larger. In other words, REM captures a clear majority of the error due to noise, but not all. In principle, other mitigation methods could also be combined with REM without substantially increasing the computational cost.
Even as we believe that reference-state error mitigation is a valuable addition to the quantum computing toolbox it is not a magic bullet. REM has inherent assumptions that do not always hold as well. For example, REM combined with a Hartree-Fock reference wavefunction is less effective (yet still helpful) when considering strongly correlated electronic states. The harder challenge of handling noise on the hardware level – error correction – also remains, as do many other open problems on the algorithm side. For example, and importantly, no one has yet demonstrated actual “quantum advantage” in the calculation of chemistry compared to conventional computers, or for any other practical problem.
The bottom line: setting the reference state is crucial
Anyone considering using this method should be aware that whereas REM offers a substantial improvement in accuracy compared to normal uncorrected VQE, it is less effective when applied to calculations of strongly correlated molecules. We will return in future work to the development of more robust alternative (multi-reference) reference states and see if we can improve the generality of REM further. In the meantime, REM is a strong step towards being able to make meaningful comparisons of molecular energies and achieve better accuracy in quantum computation of chemistry.