Avoiding symmetry roadblocks and minimizing the measurement overhead of adaptive variational quantum eigensolvers

V. O. Shkolnikov1,2, Nicholas J. Mayhall3,2, Sophia E. Economou1,2, and Edwin Barnes1,2

1Department of Physics, Virginia Tech, Blacksburg, VA 24061, USA
2Virginia Tech Center for Quantum Information Science and Engineering, Blacksburg, VA 24061, USA
3Department of Chemistry, Virginia Tech, Blacksburg, VA 24061, USA

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Quantum simulation of strongly correlated systems is potentially the most feasible useful application of near-term quantum computers. Minimizing quantum computational resources is crucial to achieving this goal. A promising class of algorithms for this purpose consists of variational quantum eigensolvers (VQEs). Among these, problem-tailored versions such as ADAPT-VQE that build variational ansätze step by step from a predefined operator pool perform particularly well in terms of circuit depths and variational parameter counts. However, this improved performance comes at the expense of an additional measurement overhead compared to standard VQEs. Here, we show that this overhead can be reduced to an amount that grows only linearly with the number $n$ of qubits, instead of quartically as in the original ADAPT-VQE. We do this by proving that operator pools of size $2n-2$ can represent any state in Hilbert space if chosen appropriately. We prove that this is the minimal size of such "complete" pools, discuss their algebraic properties, and present necessary and sufficient conditions for their completeness that allow us to find such pools efficiently. We further show that, if the simulated problem possesses symmetries, then complete pools can fail to yield convergent results, unless the pool is chosen to obey certain symmetry rules. We demonstrate the performance of such symmetry-adapted complete pools by using them in classical simulations of ADAPT-VQE for several strongly correlated molecules. Our findings are relevant for any VQE that uses an ansatz based on Pauli strings.

Simulation of strongly correlated systems is one of the key applications envisioned for near-term quantum computers. Realizing this application requires reducing quantum and classical resources as much as possible. One of the leading approaches exploits the variational principle of quantum mechanics, but its feasibility depends crucially on finding suitable trial wavefunctions.

While adaptive algorithms that construct trial wavefunctions on the fly in a problem-tailored fashion seem particularly promising, they may come with an extra measurement cost compared to other variational algorithms. We prove that this extra cost can be reduced to being only linear in the number of qubits, and we provide explicit recipes for achieving this. We also show that it is important that these recipes must account for any symmetries in the system being simulated in order for them to work well.

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