Adiabatic Quantum Computing

TL;DR

This comprehensive review surveys Adiabatic Quantum Computing (AQC) in the closed-system setting, detailing the spectrum-gap–driven run-time bounds of diverse adiabatic theorems and presenting explicit algorithms that achieve quantum speedups or universality relative to the circuit model. It covers foundational topics from Grover- and Deutsch–Jozsa–type speedups to more elaborate constructions like history-state and space-time circuit-to-Hamiltonian universality, including detailed treatment of stoquastic AQC (StoqAQC) and its complexity-theoretic limits. The article further examines how Hamiltonian quantum complexity theory, QMA-completeness, and perturbative gadgets underpin universal AQC, while critically evaluating when stoquastic Hamiltonians may or may not yield quantum advantage, including slowdown mechanisms and potential remedies such as gap amplification and catalyst terms. The discussion culminates in an outlook highlighting open problems, the relationship between entanglement and speedups, and the broad research avenues required to realize robust, scalable quantum computation via adiabatic approaches.

Abstract

Adiabatic quantum computing (AQC) started as an approach to solving optimization problems, and has evolved into an important universal alternative to the standard circuit model of quantum computing, with deep connections to both classical and quantum complexity theory and condensed matter physics. In this review we give an account of most of the major theoretical developments in the field, while focusing on the closed-system setting. The review is organized around a series of topics that are essential to an understanding of the underlying principles of AQC, its algorithmic accomplishments and limitations, and its scope in the more general setting of computational complexity theory. We present several variants of the adiabatic theorem, the cornerstone of AQC, and we give examples of explicit AQC algorithms that exhibit a quantum speedup. We give an overview of several proofs of the universality of AQC and related Hamiltonian quantum complexity theory. We finally devote considerable space to Stoquastic AQC, the setting of most AQC work to date, where we discuss obstructions to success and their possible resolutions.

Figures (11)

• Figure 1: A glued tree with $n=4$. The labeling $j$ from Eq. \ref{['eqt:col']} is depicted on top of the tree.
• Figure 2: The ground state ($\lambda_0(s)$, blue solid curve), first excited state ($\lambda_1(s)$, red dashed curve) and second excited state ($\lambda_2(s)$ yellow dot-dashed curve) of the glued-trees Hamiltonian \ref{['eq:H-glued']} for $\alpha = 1/\sqrt{8}$ and $n = 6$. Inside the region $[s_1,s_2]$ and $[s_3,s_4]$, the gap between the ground state and first excited state $\Delta_{10}$ closes exponentially with $n$. In the region $[s_2,s_3]$, the gap between the ground state and first excited state $\Delta_{10}$ and the gap between the first excited state and second excited state $\Delta_{21}$ are bounded by $n^{-3}$. Similarly, in the region $[s_4,1]$, the gap between the ground state and first excited state $\Delta_{10}$ is bounded by $n^{-3}$.
• Figure 3: (a) A $2n = 8$ qubit quantum circuit, where each grey square ($n^2 = 16$ in total) corresponds to a 2-qubit gate. (b) An equivalent representation of the quantum circuit in (a) in terms of a rotated grid. The red dashed line corresponds to an allowed string configuration for the particles. (c) The circuit is constrained such that the majority of the gates are identity except in a $k \times k$ subgrid (shown in black), located such that its left vertex is at the center of the rotated grid. A successful computation requires the $t$ position of the $2k$ particles with $w$ positions that cross the interaction region, to lie to the right of the interaction region. See also Fig. 1 in Gosset:2014rp.
• Figure 4: Known relations between complexity classes relevant for AQC. The BStoqP class defined here (Def. \ref{['def:BStoqP']}) lies in the intersection StoqMA and BQP, and includes BPP.
• Figure 5: Energy spectrum of the dimer model on an even length periodic ladder, with the dimer configurations illustrated. The $w = \pm 1$ states are at energy $E = 0$, while the $w = 0$ sector splits into a band for $\Gamma>0$. For sufficiently large $\Gamma$, the $w = 0$ sector contains the ground state of the system. An unavoided level crossing (first order quantum phase transition) occurs at $\Gamma=\Gamma_c$, which is responsible for the quantum slowdown. From Laumann:2012hs.

Theorems & Definitions (18)

• Definition 1: Adiabatic Quantum Computation
• Theorem 1
• Definition 2: Gevrey class
• Theorem 2
• Theorem 3
• Lemma 1
• proof
• Definition 3: Universal Adiabatic Quantum Computation
• Lemma 2: Nullspace Projection Lemma Childs:2013ef
• Definition 4