Quantum Hamiltonian Complexity

Abstract

Constraint satisfaction problems are a central pillar of modern computational complexity theory. This survey provides an introduction to the rapidly growing field of Quantum Hamiltonian Complexity, which includes the study of quantum constraint satisfaction problems. Over the past decade and a half, this field has witnessed fundamental breakthroughs, ranging from the establishment of a "Quantum Cook-Levin Theorem" to deep insights into the structure of 1D low-temperature quantum systems via so-called area laws. Our aim here is to provide a computer science-oriented introduction to the subject in order to help bridge the language barrier between computer scientists and physicists in the field. As such, we include the following in this survey: (1) The motivations and history of the field, (2) a glossary of condensed matter physics terms explained in computer-science friendly language, (3) overviews of central ideas from condensed matter physics, such as indistinguishable particles, mean field theory, tensor networks, and area laws, and (4) brief expositions of selected computer science-based results in the area. For example, as part of the latter, we provide a novel information theoretic presentation of Bravyi's polynomial time algorithm for Quantum 2-SAT.Comment: v4: published version, 127 pages, introduction expanded to include brief introduction to quantum information, brief list of some recent developments added, minor changes throughou

Liouville action as path-integral complexity: from continuous tensor networks to AdS/CFT

“…Let us briefly review the relevant facts about the computational complexity of a quantum state (for example, see [47][48][49][50]). In quantum information theory, a quantum state made of qubits can be constructed by a sequence of simple unitary operations acting on a simple reference state.…”

“…In that case, once we pick two given times at the two boundary CFTs, say t L and t R respectively, the WDW patch is the bulk space-time region bounded by the null surfaces and such that it is union of all possible space-like surfaces anchored at the times t L and t R in the boundary. Schematically, I WDW can be written as 49) where I bdy WDW contains the important boundary contributions coming from the null boundaries of the WDW patch M WDW , also including the joint contributions coming from the intersections of the null boundaries [35].…”

Action growth for AdS black holes

“…2 We use the conventions and results in [29] for the thermodynamics of Kerr-AdS black holes. Here we only focus on the case in four dimensions for simplicity and clarity and the results can be easily generalized to the higher dimensional case.…”

“…As a branch of theoretical computer science and mathematics, computational complexity theory [1] motivates a lot of studies in field theory [2] and gravitational physics [3][4][5][6][7]. Especially, Susskind and his collaborators' works [3][4][5][6] shed some lights on the connection between quantum computational complexity and black hole physics.…”

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“…From the formal perspective, tools from theoretical computer science have deepened our understanding of what makes a quantum system hard to simulate on a classical computer (Gharibian et al, 2015). These tools have also been used on quantum chemistry problems such as electronic structure calculation (Whitfield, Love, and Aspuru-Guzik, 2013) and density functional theory (Schuch and Verstraete, 2009) to provide a new perspective on the extent to which quantum computers may help current quantum chemical calculations.…”