327 research outputs found
Oracles Are Subtle But Not Malicious
Theoretical computer scientists have been debating the role of oracles since
the 1970's. This paper illustrates both that oracles can give us nontrivial
insights about the barrier problems in circuit complexity, and that they need
not prevent us from trying to solve those problems.
First, we give an oracle relative to which PP has linear-sized circuits, by
proving a new lower bound for perceptrons and low- degree threshold
polynomials. This oracle settles a longstanding open question, and generalizes
earlier results due to Beigel and to Buhrman, Fortnow, and Thierauf. More
importantly, it implies the first nonrelativizing separation of "traditional"
complexity classes, as opposed to interactive proof classes such as MIP and
MA-EXP. For Vinodchandran showed, by a nonrelativizing argument, that PP does
not have circuits of size n^k for any fixed k. We present an alternative proof
of this fact, which shows that PP does not even have quantum circuits of size
n^k with quantum advice. To our knowledge, this is the first nontrivial lower
bound on quantum circuit size.
Second, we study a beautiful algorithm of Bshouty et al. for learning Boolean
circuits in ZPP^NP. We show that the NP queries in this algorithm cannot be
parallelized by any relativizing technique, by giving an oracle relative to
which ZPP^||NP and even BPP^||NP have linear-size circuits. On the other hand,
we also show that the NP queries could be parallelized if P=NP. Thus, classes
such as ZPP^||NP inhabit a "twilight zone," where we need to distinguish
between relativizing and black-box techniques. Our results on this subject have
implications for computational learning theory as well as for the circuit
minimization problem.Comment: 20 pages, 1 figur
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
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
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
The computational complexity of PEPS
We determine the computational power of preparing Projected Entangled Pair
States (PEPS), as well as the complexity of classically simulating them, and
generally the complexity of contracting tensor networks. While creating PEPS
allows to solve PP problems, the latter two tasks are both proven to be
#P-complete. We further show how PEPS can be used to approximate ground states
of gapped Hamiltonians, and that creating them is easier than creating
arbitrary PEPS. The main tool for our proofs is a duality between PEPS and
postselection which allows to use existing results from quantum compexity.Comment: 5 pages, 1 figure. Published version, plus a few extra
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