1,402 research outputs found
Quantum Lower Bounds for Approximate Counting via Laurent Polynomials
We study quantum algorithms that are given access to trusted and untrusted quantum witnesses. We establish strong limitations of such algorithms, via new techniques based on Laurent polynomials (i.e., polynomials with positive and negative integer exponents). Specifically, we resolve the complexity of approximate counting, the problem of multiplicatively estimating the size of a nonempty set S ? [N], in two natural generalizations of quantum query complexity.
Our first result holds in the standard Quantum Merlin - Arthur (QMA) setting, in which a quantum algorithm receives an untrusted quantum witness. We show that, if the algorithm makes T quantum queries to S, and also receives an (untrusted) m-qubit quantum witness, then either m = ?(|S|) or T = ?(?{N/|S|}). This is optimal, matching the straightforward protocols where the witness is either empty, or specifies all the elements of S. As a corollary, this resolves the open problem of giving an oracle separation between SBP, the complexity class that captures approximate counting, and QMA.
In our second result, we ask what if, in addition to a membership oracle for S, a quantum algorithm is also given "QSamples" - i.e., copies of the state |S? = 1/?|S| ?_{i ? S} |i? - or even access to a unitary transformation that enables QSampling? We show that, even then, the algorithm needs either ?(?{N/|S|}) queries or else ?(min{|S|^{1/3},?{N/|S|}}) QSamples or accesses to the unitary.
Our lower bounds in both settings make essential use of Laurent polynomials, but in different ways
Unitary property testing lower bounds by polynomials
We study unitary property testing, where a quantum algorithm is given query
access to a black-box unitary and has to decide whether it satisfies some
property. In addition to containing the standard quantum query complexity model
(where the unitary encodes a binary string) as a special case, this model
contains "inherently quantum" problems that have no classical analogue.
Characterizing the query complexity of these problems requires new algorithmic
techniques and lower bound methods.
Our main contribution is a generalized polynomial method for unitary property
testing problems. By leveraging connections with invariant theory, we apply
this method to obtain lower bounds on problems such as determining recurrence
times of unitaries, approximating the dimension of a marked subspace, and
approximating the entanglement entropy of a marked state. We also present a
unitary property testing-based approach towards an oracle separation between
and , a long standing question in quantum
complexity theory.Comment: 58 pages, v2: typos corrected, Section 6.1-6.3 revised, added some
new result
P versus NP and geometry
I describe three geometric approaches to resolving variants of P v. NP,
present several results that illustrate the role of group actions in complexity
theory, and make a first step towards completely geometric definitions of
complexity classes.Comment: 20 pages, to appear in special issue of J. Symbolic. Comp. dedicated
to MEGA 200
Unitary Property Testing Lower Bounds by Polynomials
We study unitary property testing, where a quantum algorithm is given query access to a black-box unitary and has to decide whether it satisfies some property. In addition to containing the standard quantum query complexity model (where the unitary encodes a binary string) as a special case, this model contains "inherently quantum" problems that have no classical analogue. Characterizing the query complexity of these problems requires new algorithmic techniques and lower bound methods.
Our main contribution is a generalized polynomial method for unitary property testing problems. By leveraging connections with invariant theory, we apply this method to obtain lower bounds on problems such as determining recurrence times of unitaries, approximating the dimension of a marked subspace, and approximating the entanglement entropy of a marked state. We also present a unitary property testing-based approach towards an oracle separation between QMA and QMA(2), a long standing question in quantum complexity theory
Rounding Sum-of-Squares Relaxations
We present a general approach to rounding semidefinite programming
relaxations obtained by the Sum-of-Squares method (Lasserre hierarchy). Our
approach is based on using the connection between these relaxations and the
Sum-of-Squares proof system to transform a *combining algorithm* -- an
algorithm that maps a distribution over solutions into a (possibly weaker)
solution -- into a *rounding algorithm* that maps a solution of the relaxation
to a solution of the original problem.
Using this approach, we obtain algorithms that yield improved results for
natural variants of three well-known problems:
1) We give a quasipolynomial-time algorithm that approximates the maximum of
a low degree multivariate polynomial with non-negative coefficients over the
Euclidean unit sphere. Beyond being of interest in its own right, this is
related to an open question in quantum information theory, and our techniques
have already led to improved results in this area (Brand\~{a}o and Harrow, STOC
'13).
2) We give a polynomial-time algorithm that, given a d dimensional subspace
of R^n that (almost) contains the characteristic function of a set of size n/k,
finds a vector in the subspace satisfying ,
where . Aside from being a natural relaxation, this
is also motivated by a connection to the Small Set Expansion problem shown by
Barak et al. (STOC 2012) and our results yield a certain improvement for that
problem.
3) We use this notion of L_4 vs. L_2 sparsity to obtain a polynomial-time
algorithm with substantially improved guarantees for recovering a planted
-sparse vector v in a random d-dimensional subspace of R^n. If v has mu n
nonzero coordinates, we can recover it with high probability whenever , improving for prior methods which
intrinsically required
On the Exact Evaluation of Certain Instances of the Potts Partition Function by Quantum Computers
We present an efficient quantum algorithm for the exact evaluation of either
the fully ferromagnetic or anti-ferromagnetic q-state Potts partition function
Z for a family of graphs related to irreducible cyclic codes. This problem is
related to the evaluation of the Jones and Tutte polynomials. We consider the
connection between the weight enumerator polynomial from coding theory and Z
and exploit the fact that there exists a quantum algorithm for efficiently
estimating Gauss sums in order to obtain the weight enumerator for a certain
class of linear codes. In this way we demonstrate that for a certain class of
sparse graphs, which we call Irreducible Cyclic Cocycle Code (ICCC_\epsilon)
graphs, quantum computers provide a polynomial speed up in the difference
between the number of edges and vertices of the graph, and an exponential speed
up in q, over the best classical algorithms known to date
Sum-of-Squares Certificates for Maxima of Random Tensors on the Sphere
For an -variate order- tensor , define to be the maximum value taken by the
tensor on the unit sphere. It is known that for a random tensor with i.i.d entries, w.h.p. We study the
problem of efficiently certifying upper bounds on via the natural
relaxation from the Sum of Squares (SoS) hierarchy. Our results include:
- When is a random order- tensor, we prove that levels of SoS
certifies an upper bound on that satisfies Our upper bound improves a result of Montanari and Richard
(NIPS 2014) when is large.
- We show the above bound is the best possible up to lower order terms,
namely the optimum of the level- SoS relaxation is at least
- When is a random order- tensor, we prove that levels of SoS
certifies an upper bound on that satisfies For growing , this improves upon the bound
certified by constant levels of SoS. This answers in part, a question posed by
Hopkins, Shi, and Steurer (COLT 2015), who established the tight
characterization for constant levels of SoS
- …