The Quantum Approximate Optimization Algorithm (QAOA) is a general-purpose algorithm for combinatorial optimization problems whose performance can only improve with the number of layers
p
. While ...QAOA holds promise as an algorithm that can be run on near-term quantum computers, its computational power has not been fully explored. In this work, we study the QAOA applied to the Sherrington-Kirkpatrick (SK) model, which can be understood as energy minimization of
n
spins with all-to-all random signed couplings. There is a recent classical algorithm by Montanari that, assuming a widely believed conjecture, can efficiently find an approximate solution for a typical instance of the SK model to within
(
1
−
ϵ
)
times the ground state energy. We hope to match its performance with the QAOA.Our main result is a novel technique that allows us to evaluate the typical-instance energy of the QAOA applied to the SK model. We produce a formula for the expected value of the energy, as a function of the
2
p
QAOA parameters, in the infinite size limit that can be evaluated on a computer with
O
(
16
p
)
complexity. We evaluate the formula up to
p
=
12
, and find that the QAOA at
p
=
11
outperforms the standard semidefinite programming algorithm. Moreover, we show concentration: With probability tending to one as
n
→
∞
, measurements of the QAOA will produce strings whose energies concentrate at our calculated value. As an algorithm running on a quantum computer, there is no need to search for optimal parameters on an instance-by-instance basis since we can determine them in advance. What we have here is a new framework for analyzing the QAOA, and our techniques can be of broad interest for evaluating its performance on more general problems where classical algorithms may fail.
A quantum system will stay near its instantaneous ground state if the Hamiltonian that governs its evolution varies slowly enough. This quantum adiabatic behavior is the basis of a new class of ...algorithms for quantum computing. We tested one such algorithm by applying it to randomly generated hard instances of an NP-complete problem. For the small examples that we could simulate, the quantum adiabatic algorithm worked well, providing evidence that quantum computers (if large ones can be built) may be able to outperform ordinary computers on hard sets of instances of NP-complete problems.
The probability that a random permutation in $S_n$ is a derangement is well known to be $\displaystyle\sum\limits_{j=0}^n (-1)^j \frac{1}{j!}$. In this paper, we consider the conditional probability ...that the $(k+1)^{st}$ point is fixed, given there are no fixed points in the first $k$ points. We prove that when $n \neq 3$ and $k \neq 1$, this probability is a decreasing function of both $k$ and $n$. Furthermore, it is proved that this conditional probability is well approximated by $\frac{1}{n} - \frac{k}{n^2(n-1)}$. Similar results are also obtained about the more general conditional probability that the $(k+1)^{st}$ point is fixed, given that there are exactly $d$ fixed points in the first $k$ points.
Quantum computation and decision trees Farhi, Edward; Gutmann, Sam
Physical review. A, Atomic, molecular, and optical physics,
08/1998, Letnik:
58, Številka:
2
Journal Article
The probability that a random permutation in \(S_n\) is a derangement is well known to be \(\displaystyle\sum\limits_{j=0}^n (-1)^j \frac{1}{j!}\). In this paper, we consider the conditional ...probability that the \((k+1)^{st}\) point is fixed, given there are no fixed points in the first \(k\) points. We prove that when \(n \neq 3\) and \(k \neq 1\), this probability is a decreasing function of both \(k\) and \(n\). Furthermore, it is proved that this conditional probability is well approximated by \(\frac{1}{n} - \frac{k}{n^2(n-1)}\). Similar results are also obtained about the more general conditional probability that the \((k+1)^{st}\) point is fixed, given that there are exactly \(d\) fixed points in the first \(k\) points.
The Quantum Approximate Optimization Algorithm (QAOA) is designed to maximize a cost function over bit strings. While the initial state is traditionally a uniform superposition over all strings, it ...is natural to try expediting the QAOA: first use a classical algorithm to produce some good string, and then run the standard QAOA starting in the computational basis state associated with that string. Here we report numerical experiments that show this method of initializing the QAOA fails dramatically, exhibiting little to no improvement of the cost function. We provide multiple analytical arguments for this lack of improvement, each of which can be made rigorous under different regimes or assumptions, including at nearly linear depths. We emphasize that our negative results only apply to our simple incarnation of the warm-start QAOA and may not apply to other approaches in the literature. We hope that our theoretical analysis will inform future algorithm design.
We construct a completely cyclic Minimalist theory of syntactic derivations. A derivation consists of a sequence of cycles. Each cycle starts with the introduction of a new head and merger of the ...head's selected arguments, followed by satisfaction (via checking) of the head's removable features. The theory includes no acyclic devices such as lexical arrays or comparison of derivations. Satisfaction of features is accompanied by full category movement whenever it is not blocked by morphology or constraints barring multiple specifiers. The Minimal Link Condition is viewed computationally and is naturally incorporated into Satisfy. Our precise notion of checking involves sets of features interacting in the same checking relation, and yields an account of successive cyclic movement, the distribution of expletives, EPP, and quirky case phenomena. The paper can be read as empirical evidence that the core syntactic algorithm is computationally efficient.
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