Leibniz international proceedings in informatics

Improved Approximation Algorithms for the EPR Hamiltonian

Ju, Nathan, Nagda, Ansh

Published Jan 1, 2025• 0 citations

Abstract

The EPR Hamiltonian is a family of 2-local quantum Hamiltonians introduced by King [King, 2023]. We introduce a polynomial time (1+√5)/4≈0.809-approximation algorithm for the problem of computing the ground energy of the EPR Hamiltonian, improving upon the previous state of the art of 0.72 [Jorquera et al., 2024].

DOI: 10.4230/lipics.approx/random.2025.24

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Leibniz international proceedings in informatics

Improved Approximation Algorithms for the EPR Hamiltonian

Ju, Nathan, Nagda, Ansh

Published Jan 1, 2025• 0 citations

Publisher

Abstract

DOI: 10.4230/lipics.approx/random.2025.24

Related papers

Improved approximation algorithms for the EPR Hamiltonian

Nathan Ju, Ansh Nagda

The EPR Hamiltonian is a family of 2-local quantum Hamiltonians introduced by King (arXiv:2209.02589). We introduce a polynomial time $\frac{1+\sqrt{5}}{4}\approx 0.809$ -approximation algorithm for the problem of computing the ground energy of the EPR Hamiltonian, improving upon the previous state of the art of $0.72$ (arXiv:2410.15544). As a special case, this also implies a $\frac{1+\sqrt{5}}{4}$ -approximation for Quantum Max Cut on bipartite instances, improving upon the approximation ratio of $3/4$ that one can infer in a relatively straightforward manner from the work of Lee and Parekh (arXiv:2401.03616).The EPR Hamiltonian is a family of 2-local quantum Hamiltonians introduced by King (arXiv:2209.02589). We introduce a polynomial time $\frac{1+\sqrt{5}}{4}\approx 0.809$-approximation algorithm for the problem of computing the ground energy of the EPR Hamiltonian, improving upon the previous state of the art of $0.72$ (arXiv:2410.15544). As a special case, this also implies a $\frac{1+\sqrt{5}}{4}$-approximation for Quantum Max Cut on bipartite instances, improving upon the approximation ratio of $3/4$ that one can infer in a relatively straightforward manner from the work of Lee and Parekh (arXiv:2401.03616).

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The EPR Hamiltonian is a family of 2-local quantum Hamiltonians introduced by King (arXiv:2209.02589). We introduce a polynomial time

\frac{1+\sqrt{5}}{4}\approx 0.809

-approximation algorithm for the problem of computing the ground energy of the EPR Hamiltonian, improving upon the previous state of the art of

0.72

(arXiv:2410.15544). As a special case, this also implies a

\frac{1+\sqrt{5}}{4}

-approximation for Quantum Max Cut on bipartite instances, improving upon the approximation ratio of

3/4

that one can infer in a relatively straightforward manner from the work of Lee and Parekh (arXiv:2401.03616).

We construct classical algorithms computing an approximation of the ground state energy of an arbitrary

k

-local Hamiltonian acting on

n

qubits. We first consider the setting where a good ``guiding state'' is available, which is the main setting where quantum algorithms are expected to achieve an exponential speedup over classical methods. We show that a constant approximation (i.e., an approximation with constant relative accuracy) of the ground state energy can be computed classically in

\mathrm{poly}\left(1/χ,n\right)

time and

\mathrm{poly}(n)

space, where

χ

denotes the overlap between the guiding state and the ground state (as in prior works in dequantization, we assume sample-and-query access to the guiding state). This gives a significant improvement over the recent classical algorithm by Gharibian and Le Gall (SICOMP 2023), and matches (up a to polynomial overhead) both the time and space complexities of quantum algorithms for constant approximation of the ground state energy. We also obtain classical algorithms for higher-precision approximation. For the setting where no guided state is given (i.e., the standard version of the local Hamiltonian problem), we obtain a classical algorithm computing a constant approximation of the ground state energy in

2^{O(n)}

time and

\mathrm{poly}(n)

space. To our knowledge, before this work it was unknown how to classically achieve these bounds simultaneously, even for constant approximation. We also discuss complexity-theoretic aspects of our results.

6

-local Hamiltonians when the guiding state has fidelity (inverse-polynomially) close to

1/2

with a ground state. In this paper, we optimally improve both the locality and the fidelity parameter: we show that the BQP-completeness persists even with 2-local Hamiltonians, and even when the guiding state has fidelity (inverse-polynomially) close to 1 with a ground state. Moreover, we show that the BQP-completeness also holds for 2-local physically motivated Hamiltonians on a 2D square lattice or a 2D triangular lattice. Beyond the hardness of estimating the ground state energy, we also show BQP-hardness persists when considering estimating energies of excited states of these Hamiltonians instead. Those make further steps towards establishing practical quantum advantage in quantum chemistry.