Product-state approximations to quantum ground states

Abstract

The local Hamiltonian problem consists of estimating the ground-state energy (given by the minimum eigenvalue) of a local quantum Hamiltonian. It can be considered as a quantum generalization of constraint satisfaction problems (CSPs) and has a central role both in quantum many-body physics and quantum complexity theory. A key feature that distinguishes quantum Hamiltonians from classical CSPs is that the solutions may involve complicated entangled states. In this paper, we demonstrate several large classes of Hamiltonians for which product (i.e. unentangled) states can approximate the ground state energy to within a small extensive error.First, we show the existence of a good product-state approximation for the ground-state energy of 2-local Hamiltonians with one or more of the following properties: (1) high degree, (2) small expansion, or (3) a ground state with sublinear entanglement with respect to some partition into small pieces. The approximation based on degree is a surprising difference between quantum Hamiltonians and classical CSPs, since in the classical setting, higher degree is usually associated with harder CSPs. The approximation based on low entanglement, in turn, was previously known only in the regime where the entanglement was close to zero.Estimating the energy is NP-hard by the PCP theorem, but whether it is QMA-hard is an important open question in quantum complexity theory. A positive solution would represent a quantum analogue of the PCP theorem. Since the existence of a low-energy product state can be checked in NP, the result implies that any Hamiltonian used for a quantum PCP theorem should have: (1) constant degree, (2) constant expansion, (3) a "volume law" for entanglement with respect to any partition into small parts. The result also gives a no-go to a quantum version of the PCP theorem in conjunction with parallel repetition for quantum CSPs.Second, we show that in several cases, good product-state approximations not only exist, but can be found in polynomial time: (1) 2-local Hamiltonians on any planar graph, solving an open problem of Bansal, Bravyi, and Terhal, (2) dense k-local Hamiltonians for any constant k, solving an open problem of Gharibian and Kempe, and (3) 2-local Hamiltonians on graphs with low threshold rank, via a quantum generalization of a recent result of Barak, Raghavendra and Steurer.Our work introduces two new tools which may be of independent interest. First, we prove a new quantum version of the de Finetti theorem which does not require the usual assumption of symmetry. Second, we describe a way to analyze the application of the Lasserre/Parrilo SDP hierarchy to local quantum Hamiltonians.

Quantum de Finetti Theorem under Fully-One-Way Adaptive Measurements

“…In a beautiful work [10], Brandão and Harrow recently proved a de Finetti theorem under locally restricted measurements, generalizing a similar result for the case k ¼ 2 [16]. Both [10] and [16] have overcome the limitation of the standard de Finetti theorem regarding the dimension dependence.…”

“…A series of works have established analogs of this theorem in the quantum domain [3][4][5][6][7][8][9][10], where a classical probability distribution is replaced by a quantum state and the situation is more complicated and interesting due to entanglement and the existence of many different ways to distinguish states of multipartite systems. These quantum de Finetti theorems are appealing not only due to their own elegance on the characterization of symmetric states, but also because of the successful applications in many-body physics [5,11,12], quantum information [9,13,14], and computational complexity theory [10,15,16].…”

“…These quantum de Finetti theorems are appealing not only due to their own elegance on the characterization of symmetric states, but also because of the successful applications in many-body physics [5,11,12], quantum information [9,13,14], and computational complexity theory [10,15,16].…”

“…Both [10] and [16] have overcome the limitation of the standard de Finetti theorem regarding the dimension dependence. The basic idea is to relax the measure of approximation by employing an operational norm associated with measurements that are implementable by a kind of one-way local operations and classical communication (LOCC).…”

Strong Monogamies of No-Signaling Violations for Bipartite Correlation Bell Inequalities

“…This approach follows the idea of symmetric extensions and shareability of correlations which can also be recast in terms of the existence of joint probability distributions [12]. See also [13] for a novel method to deriving monogamies based on the powerful machinery of de Finetti theorems for no-signaling probability distributions. In this paper, we follow a different approach and derive a strengthened version of the monogamy relation for twoparty inequalities that holds in many flagship scenarios involving correlation expressions, in particular to the wide class of binary output correlation scenarios known as XOR games and to a restricted set of the so-called unique games.…”