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arXiv:2205.12325

Perturbation Theory and the Sum of Squares

Matthew B. Hastings

Published May 24, 2022• 1 citation
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Abstract

The sum-of-squares (SoS) hierarchy is a powerful technique based on semi-definite programming that can be used for both classical and quantum optimization problems. This hierarchy goes under several names; in particular, in quantum chemistry it is called the reduced density matrix (RDM) method. We consider the ability of this hierarchy to reproduce weak coupling perturbation theory for three different kinds of systems: spin (or qubit) systems, bosonic systems (the anharmonic oscillator), and fermionic systems with quartic interactions. For such fermionic systems, we show that degree-$4$ SoS (called $2$-RDM in quantum chemsitry) does not reproduce second order perturbation theory but degree-$6$ SoS ($3$-RDM) does (and we conjecture that it reproduces third order perturbation theory). Indeed, we identify a fragment of degree-$6$ SoS which can do this, which may be useful for practical quantum chemical calculations as it may be possible to implement this fragment with less cost than the full degree-$6$ SoS. Remarkably, this fragment is very similar to one studied by Hastings and O'Donnell for the Sachdev-Ye-Kitaev (SYK) model.

DOI
10.48550/arxiv.2205.12325

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The sum-of-squares (SoS) hierarchy is a powerful technique based on semi-definite programming that can be used for both classical and quantum optimization problems. This hierarchy goes under several names; in particular, in quantum chemistry it is called the reduced density matrix (RDM) method. We consider the ability of this hierarchy to reproduce weak coupling perturbation theory for three different kinds of systems: spin (or qubit) systems, bosonic systems (the anharmonic oscillator), and fermionic systems with quartic interactions. For such fermionic systems, we show that degree-444 SoS (called 222-RDM in quantum chemsitry) does not reproduce second order perturbation theory but degree-666 SoS (333-RDM) does (and we conjecture that it reproduces third order perturbation theory). Indeed, we identify a fragment of degree-666 SoS which can do this, which may be useful for practical quantum chemical calculations as it may be possible to implement this fragment with less cost than the full degree-666 SoS. Remarkably, this fragment is very similar to one studied by Hastings and O'Donnell for the Sachdev-Ye-Kitaev (SYK) model.
In Ref. 1, a method was given for self-consistently generating sum-of-squares decompositions of quartic fermionic Hamiltonians. Perturbation theory was used to generate a useful choice of cubic operators in this sum-of-squares. On a range of model problems, this method, which is only a fragment of degree-six sum-of-squares, was able to outperform the full degree-four sum-of-squares in both speed and accuracy. Unfortunately for applications, many problems in chemistry have strong density-density interaction terms, as well as moderately strong density-dependent hopping and spin-spin interaction terms, limiting the power of the perturbative choice of the cubic operators. Here we propose a method for generating these decompositions in the presence of these strong interaction terms, hopefully extending the range of applicability of this method.
This is a collection of various result and notes, addressing the sum-of-squares hierarchy for spin and fermion systems using some ideas from quantum field theory, including higher order perturbation theory, critical phenomena, nonlocal coupling in time, and auxiliary field Monte Carlo. This paper should be seen as a sequel to Refs. 1,2. Additionally in this paper, we consider the difficulty of approximating the ground state energy of the Sachdev-Ye-Kitaev (SYK) model using other methods. We provide limitations on the power of the Lanczos method, starting with a Gausian wavefunction, and on the power of a sum of Gaussian wavefunctions (in this case under an assumption).
Hamiltonian representations based on the sum-of-squares (SOS) hierarchy provide rigorous lower bounds on ground-state energies and facilitate the design of efficient classical and quantum simulation algorithms. This work presents a unified framework connecting SOS decompositions with variational two-particle reduced density matrix (v2RDM) theory. We demonstrate that the ``weighted'' SOS ansatz naturally recovers the dual of the v2RDM program, enabling the strict enforcement of symmetry constraints such as particle number and spin. We provide explicit SOS constructions for the Hubbard model and electronic structure Hamiltonians, ranging from spin-free approximations to full rank-2 expansions. We also highlight theoretical connections to block-invariant symmetry shifts. Numerical benchmarks on molecular systems and Iron-Sulfur clusters validate these near frustration-free representations, demonstrating their utility in improving spectral gap amplification and reducing block encoding costs in quantum algorithms.
The sum-of-squares method can give rigorous lower bounds on the energy of quantum Hamiltonians. Unfortunately, typically using this method requires solving a semidefinite program, which can be computationally expensive. Further, the typically used degree-444 sum-of-squares (also known as the 2RDM method) does not correctly reproduce second order perturbation theory. Here, we give a general method, an analogue of Wigner's 2n+12n+12n+1 rule for perturbation theory, to compute the order of the error in a given sum-of-squares ansatz. We also give a method for finding solutions of the dual semidefinite program, based on a perturbative ansatz combined with a self-consistent method. As an illustration, we show that for a class of model Hamiltonians (with a gap in the quadratic term and quartic terms chosen as i.i.d. Gaussians), this self-consistent sum-of-squares method significantly improves over the 2RDM method in both speed and accuracy, and also improves over low order perturbation theory. We then explain why the particular ansatz we implement is not suitable for use for quantum chemistry Hamiltonians (due to presence of certain large diagonal terms), but we suggest a modified ansatz that may be suitable, which will be the subject of future work.
The noncommutative sum-of-squares (ncSoS) hierarchy was introduced by Navascués-Pironio-Acín as a sequence of semidefinite programming relaxations for approximating values of noncommutative polynomial optimization problems, which were originally intended to generalize quantum values of nonlocal games. Recent work has started to analyze the hierarchy for approximating ground energies of local Hamiltonians, initially through rounding algorithms which output product states for degree-2 ncSoS applied to Quantum Max-Cut. Some rounding methods are known which output entangled states, but they use degree-4 ncSoS. Based on this, Hwang-Neeman-Parekh-Thompson-Wright conjectured that degree-2 ncSoS cannot beat product state approximations for Quantum Max-Cut and gave a partial proof relying on a conjectural generalization of Borrell's inequality. In this work we consider a family of Hamiltonians (called the quantum rotor model in condensed matter literature or lattice O(k)O(k)O(k) vector model in quantum field theory) with infinite-dimensional local Hilbert space L2(Sk−1)L^{2}(S^{k - 1})L2(Sk−1), and show that a degree-2 ncSoS relaxation approximates the ground state energy better than any product state.
From drug design to material science, computational chemistry has become a valuable tool for many fields of applications in the past decades. Quantum chemistry in particular, can provide accurate descriptions of molecules and solids by relying on principles of quantum mechanics with the Schrödinger equation at its core. Unfortunately this equation becomes prohibitively expensive to solve for systems made of more than a few electrons and one has to rely on other approaches. Among the most popular is Density Functional Theory, relatively inexpensive and exact in principle. One has, however, to introduce approximations in practice, deteriorating its accuracy for challenging, so-called strongly correlated, systems. On the contrary, other methods like CASSCF remain reliable for these systems, but although cheaper than the solution of the Schrödinger equation, remain too expensive for extended systems. There is therefore room for methods in between, that could handle large correlated systems with reasonable accuracy. One such method is the One Body Reduced Density Matrix (1-RDM) Functional Theory (RDMFT), based on the fact that an electronic system can, in principle exactly, be described by its 1-RDM only. Although its theory dates back to the 1970s, computational hindrances have prevented this method from shining so far. Indeed, computations in RDMFT rely on a constrained optimization of the eigenvalues and eigenvectors of the 1-RDM, that can be reformulated as an unconstrained optimization by parameterizing the latter but in both cases suffers from convergence problems. In this thesis, the role of this parametrization is analyzed and motivates a now second order optimization algorithm, which improves the convergence speed by two orders of magnitude w.r.t. the state of the art algorithm. This understanding of the problem also inspires an alternative approach with the 1-RDM as the optimization variable. This however requires the development of new approximations and a first investigation in that direction, based on Machine Learning is performed. Lastly, a generalization of the method to excited states is attempted, highlighting some key computational challenges.