Abstract
Reducing the complexity of quantum algorithms to treat quantum chemistry problems is essential to demonstrate an eventual quantum advantage of Noisy-Intermediate Scale Quantum (NISQ) devices over their classical counterpart. Significant improvements have been made recently to simulate the time-evolution operator $U(t) = e^{i\mathcal{\hat{H}}t}$ where $\mathcal{\hat{H}}$ is the electronic structure Hamiltonian, or to simulate $\mathcal{\hat{H}}$ directly (when written as a linear combination of unitaries) by using block encoding or "qubitization" techniques. A fundamental measure quantifying the practical implementation complexity of these quantum algorithms is the so-called "1-norm" of the qubit-representation of the Hamiltonian, which can be reduced by writing the Hamiltonian in factorized or tensor-hypercontracted forms for instance. In this work, we investigate the effect of classical pre-optimization of the electronic structure Hamiltonian representation, via single-particle basis transformation, on the 1-norm. Specifically, we employ several localization schemes and benchmark the 1-norm of several systems of different sizes (number of atoms and active space sizes). We also derive a new formula for the 1-norm as a function of the electronic integrals, and use this quantity as a cost function for an orbital-optimization scheme that improves over localization schemes. This paper gives more insights about the importance of the 1-norm in quantum computing for quantum chemistry, and provides simple ways of decreasing its value to reduce the complexity of quantum algorithms.
Highlights
Quantum chemistry has been identified as the killer application of quantum computers, which promises to solve problems of high industrial impact that are not tractable for their classical counterparts [1,2,3,4]
We localize only inside the active space (i.e., the localizing unitary in Eq (26) only has indices corresponding to active orbitals) such that the subspace spanned by the active space remains invariant under the unitary rotations
This means that the expectation values of observables remain the same inside the active space when an exact solver is used, one can converge to different expectation values when approximate solvers are considered, such as the truncated unitary coupled cluster ansatz
Summary
Quantum chemistry has been identified as the killer application of quantum computers, which promises to solve problems of high industrial impact that are not tractable for their classical counterparts [1,2,3,4]. [12,13,14,15] and references therein) and excited-state energies [16,17,18,19,20,21,22,23,24,25,26], as well as molecular properties [27,28,29,30] It remains unclear if these algorithms can provide a clear quantum advantage in the long run, especially due to the difficulty in circuit optimization [31] and to the large overhead in the number of measurements required to achieve sufficient accuracy [32,33], though significant progress has been made recently [33,34,35,36,37,38,39,40,41,42,43,44,45].
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