18 research outputs found
A useful inequality of inverse hyperbolic tangent
We prove an inequality related to arctanh, resolving a conjecture of Gu and
Polyanskiy [arXiv:2303.14689].Comment: 3 page
Classical algorithms and quantum limitations for maximum cut on high-girth graphs
We study the performance of local quantum algorithms such as the Quantum
Approximate Optimization Algorithm (QAOA) for the maximum cut problem, and
their relationship to that of classical algorithms.
(1) We prove that every (quantum or classical) one-local algorithm achieves
on -regular graphs of girth a maximum cut of at most for . This is the first such result
showing that one-local algorithms achieve a value bounded away from the true
optimum for random graphs, which is for
. (2) We show that there is a classical -local algorithm
that achieves a value of for -regular
graphs of girth , where . This is an
algorithmic version of the existential bound of Lyons and is related to the
algorithm of Aizenman, Lebowitz, and Ruelle (ALR) for the
Sherrington-Kirkpatrick model. This bound is better than that achieved by the
one-local and two-local versions of QAOA on high-girth graphs. (3) Through
computational experiments, we give evidence that the ALR algorithm achieves
better performance than constant-locality QAOA for random -regular graphs,
as well as other natural instances, including graphs that do have short cycles.
Our experimental work suggests that it could be possible to extend beyond our
theoretical constraints. This points at the tantalizing possibility that
-local quantum maximum-cut algorithms might be *pointwise dominated* by
polynomial-time classical algorithms, in the sense that there is a classical
algorithm outputting cuts of equal or better quality *on every possible
instance*. This is in contrast to the evidence that polynomial-time algorithms
cannot simulate the probability distributions induced by local quantum
algorithms.Comment: 1+20 pages, 2 figures, code online at https://tiny.cc/QAOAvsAL
On the Power of Nonstandard Quantum Oracles
We study how the choices made when designing an oracle affect the complexity of quantum property testing problems defined relative to this oracle. We encode a regular graph of even degree as an invertible function f, and present f in different oracle models. We first give a one-query QMA protocol to test if a graph encoded in f has a small disconnected subset. We then use representation theory to show that no classical witness can help a quantum verifier efficiently decide this problem relative to an in-place oracle. Perhaps surprisingly, a simple modification to the standard oracle prevents a quantum verifier from efficiently deciding this problem, even with access to an unbounded witness
Quantum Merlin-Arthur and proofs without relative phase
We study a variant of QMA where quantum proofs have no relative phase (i.e.
non-negative amplitudes, up to a global phase). If only completeness is
modified, this class is equal to QMA [arXiv:1410.2882]; but if both
completeness and soundness are modified, the class (named QMA+ by Jeronimo and
Wu) can be much more powerful. We show that QMA+ with some constant gap is
equal to NEXP, yet QMA+ with some *other* constant gap is equal to QMA. One
interpretation is that Merlin's ability to "deceive" originates from relative
phase at least as much as from entanglement, since QMA(2) NEXP.Comment: 18 pages, 2 figure
Fair allocation of a multiset of indivisible items
We study the problem of fairly allocating a multiset of indivisible
items among agents with additive valuations. Specifically, we introduce a
parameter for the number of distinct types of items and study fair
allocations of multisets that contain only items of these types, under two
standard notions of fairness:
1. Envy-freeness (EF): For arbitrary , , we show that a complete EF
allocation exists when at least one agent has a unique valuation and the number
of items of each type exceeds a particular finite threshold. We give explicit
upper and lower bounds on this threshold in some special cases.
2. Envy-freeness up to any good (EFX): For arbitrary , , and for , we show that a complete EFX allocation always exists. We give two different
proofs of this result. One proof is constructive and runs in polynomial time;
the other is geometrically inspired.Comment: 34 pages, 6 figures, 1 table, 1 algorith
Random Max-CSPs Inherit Algorithmic Hardness from Spin Glasses
We study random constraint satisfaction problems (CSPs) in the unsatisfiable
regime. We relate the structure of near-optimal solutions for any Max-CSP to
that for an associated spin glass on the hypercube, using the Guerra-Toninelli
interpolation from statistical physics. The noise stability polynomial of the
CSP's predicate is, up to a constant, the mixture polynomial of the associated
spin glass. We prove two main consequences:
1) We relate the maximum fraction of constraints that can be satisfied in a
random Max-CSP to the ground state energy density of the corresponding spin
glass. Since the latter value can be computed with the Parisi formula, we
provide numerical values for some popular CSPs.
2) We prove that a Max-CSP possesses generalized versions of the overlap gap
property if and only if the same holds for the corresponding spin glass. We
transfer results from Huang et al. [arXiv:2110.07847, 2021] to obstruct
algorithms with overlap concentration on a large class of Max-CSPs. This
immediately includes local classical and local quantum algorithms.Comment: 41 pages, 1 tabl
Random Max-CSPs Inherit Algorithmic Hardness from Spin Glasses
We study random constraint satisfaction problems (CSPs) in the unsatisfiable
regime. We relate the structure of near-optimal solutions for any Max-CSP to
that for an associated spin glass on the hypercube, using the Guerra-Toninelli
interpolation from statistical physics. The noise stability polynomial of the
CSP's predicate is, up to a constant, the mixture polynomial of the associated
spin glass. We prove two main consequences:
1) We relate the maximum fraction of constraints that can be satisfied in a
random Max-CSP to the ground state energy density of the corresponding spin
glass. Since the latter value can be computed with the Parisi formula, we
provide numerical values for some popular CSPs.
2) We prove that a Max-CSP possesses generalized versions of the overlap gap
property if and only if the same holds for the corresponding spin glass. We
transfer results from Huang et al. [arXiv:2110.07847, 2021] to obstruct
algorithms with overlap concentration on a large class of Max-CSPs. This
immediately includes local classical and local quantum algorithms.Comment: 41 pages, 1 tabl