84 research outputs found

### The Structure of Promises in Quantum Speedups

It has long been known that in the usual black-box model, one cannot get
super-polynomial quantum speedups without some promise on the inputs. In this
paper, we examine certain types of symmetric promises, and show that they also
cannot give rise to super-polynomial quantum speedups. We conclude that
exponential quantum speedups only occur given "structured" promises on the
input.
Specifically, we show that there is a polynomial relationship of degree $12$
between $D(f)$ and $Q(f)$ for any function $f$ defined on permutations
(elements of $\{0,1,\dots, M-1\}^n$ in which each alphabet element occurs
exactly once). We generalize this result to all functions $f$ defined on orbits
of the symmetric group action $S_n$ (which acts on an element of $\{0,1,\dots,
M-1\}^n$ by permuting its entries). We also show that when $M$ is constant, any
function $f$ defined on a "symmetric set" - one invariant under $S_n$ -
satisfies $R(f)=O(Q(f)^{12(M-1)})$.Comment: 15 page

### Quantum Tokens for Digital Signatures

The fisherman caught a quantum fish. "Fisherman, please let me go", begged
the fish, "and I will grant you three wishes". The fisherman agreed. The fish
gave the fisherman a quantum computer, three quantum signing tokens and his
classical public key. The fish explained: "to sign your three wishes, use the
tokenized signature scheme on this quantum computer, then show your valid
signature to the king, who owes me a favor".
The fisherman used one of the signing tokens to sign the document "give me a
castle!" and rushed to the palace. The king executed the classical verification
algorithm using the fish's public key, and since it was valid, the king
complied.
The fisherman's wife wanted to sign ten wishes using their two remaining
signing tokens. The fisherman did not want to cheat, and secretly sailed to
meet the fish. "Fish, my wife wants to sign ten more wishes". But the fish was
not worried: "I have learned quantum cryptography following the previous story
(The Fisherman and His Wife by the brothers Grimm). The quantum tokens are
consumed during the signing. Your polynomial wife cannot even sign four wishes
using the three signing tokens I gave you".
"How does it work?" wondered the fisherman. "Have you heard of quantum money?
These are quantum states which can be easily verified but are hard to copy.
This tokenized quantum signature scheme extends Aaronson and Christiano's
quantum money scheme, which is why the signing tokens cannot be copied".
"Does your scheme have additional fancy properties?" the fisherman asked.
"Yes, the scheme has other security guarantees: revocability, testability and
everlasting security. Furthermore, if you're at sea and your quantum phone has
only classical reception, you can use this scheme to transfer the value of the
quantum money to shore", said the fish, and swam away.Comment: Added illustration of the abstract to the ancillary file

### Sculpting Quantum Speedups

Given a problem which is intractable for both quantum and classical
algorithms, can we find a sub-problem for which quantum algorithms provide an
exponential advantage? We refer to this problem as the "sculpting problem." In
this work, we give a full characterization of sculptable functions in the query
complexity setting. We show that a total function f can be restricted to a
promise P such that Q(f|_P)=O(polylog(N)) and R(f|_P)=N^{Omega(1)}, if and only
if f has a large number of inputs with large certificate complexity. The proof
uses some interesting techniques: for one direction, we introduce new
relationships between randomized and quantum query complexity in various
settings, and for the other direction, we use a recent result from
communication complexity due to Klartag and Regev. We also characterize
sculpting for other query complexity measures, such as R(f) vs. R_0(f) and
R_0(f) vs. D(f).
Along the way, we prove some new relationships for quantum query complexity:
for example, a nearly quadratic relationship between Q(f) and D(f) whenever the
promise of f is small. This contrasts with the recent super-quadratic query
complexity separations, showing that the maximum gap between classical and
quantum query complexities is indeed quadratic in various settings - just not
for total functions!
Lastly, we investigate sculpting in the Turing machine model. We show that if
there is any BPP-bi-immune language in BQP, then every language outside BPP can
be restricted to a promise which places it in PromiseBQP but not in PromiseBPP.
Under a weaker assumption, that some problem in BQP is hard on average for
P/poly, we show that every paddable language outside BPP is sculptable in this
way.Comment: 30 page

### On Rota's Conjecture and nested separations in matroids

We prove that for each finite field $\mathbb F$ and integer $k\in \mathbb Z$
there exists $n\in \mathbb Z$ such that no excluded minor for the class of
$\mathbb F$-representable matroids has $n$ nested $k$-separations.Comment: 12 pages. This revised version includes revisions suggested by
referees. The paper will appear in JCT

### The Structure of Promises in Quantum Speedups

In 1998, Beals, Buhrman, Cleve, Mosca, and de Wolf showed that no super-polynomial quantum speedup is possible in the query complexity setting unless there is a promise on the input. We examine several types of "unstructured" promises, and show that they also are not compatible with super-polynomial quantum speedups. We conclude that such speedups are only possible when the input is known to have some structure.
Specifically, we show that there is a polynomial relationship of degree 18 between D(f) and Q(f) for any Boolean function f defined on permutations (elements of [n]^n in which each alphabet element occurs exactly once). More generally, this holds for all f defined on orbits of the symmetric group action (which acts on an element of [M]^n by permuting its entries). We also show that any Boolean function f defined on a "symmetric" subset of the Boolean hypercube has a polynomial relationship between R(f) and Q(f) - although in that setting, D(f) may be exponentially larger

### Low-Sensitivity Functions from Unambiguous Certificates

We provide new query complexity separations against sensitivity for total
Boolean functions: a power $3$ separation between deterministic (and even
randomized or quantum) query complexity and sensitivity, and a power $2.22$
separation between certificate complexity and sensitivity. We get these
separations by using a new connection between sensitivity and a seemingly
unrelated measure called one-sided unambiguous certificate complexity
($UC_{min}$). We also show that $UC_{min}$ is lower-bounded by fractional block
sensitivity, which means we cannot use these techniques to get a
super-quadratic separation between $bs(f)$ and $s(f)$. We also provide a
quadratic separation between the tree-sensitivity and decision tree complexity
of Boolean functions, disproving a conjecture of Gopalan, Servedio, Tal, and
Wigderson (CCC 2016).
Along the way, we give a power $1.22$ separation between certificate
complexity and one-sided unambiguous certificate complexity, improving the
power $1.128$ separation due to G\"o\"os (FOCS 2015). As a consequence, we
obtain an improved $\Omega(\log^{1.22} n)$ lower-bound on the
co-nondeterministic communication complexity of the Clique vs. Independent Set
problem.Comment: 25 pages. This version expands the results and adds Pooya Hatami and
Avishay Tal as author

### Randomized Query Complexity of Sabotaged and Composed Functions

We study the composition question for bounded-error randomized query complexity: Is R(f circ g) = Omega(R(f)R(g))? We show that inserting a simple function h, whose query complexity is onlyTheta(log R(g)), in between f and g allows us to prove R(f circ h circ g) = Omega(R(f)R(h)R(g)).
We prove this using a new lower bound measure for randomized query complexity we call randomized sabotage complexity, RS(f). Randomized sabotage complexity has several desirable properties, such as a perfect composition theorem, RS(f circ g) >= RS(f) RS(g), and a composition theorem with randomized query complexity, R(f circ g) = Omega(R(f) RS(g)). It is also a quadratically tight lower bound for total functions and can be quadratically superior to the partition bound, the best known general lower bound for randomized query complexity.
Using this technique we also show implications for lifting theorems in communication complexity. We show that a general lifting theorem from zero-error randomized query to communication complexity implies a similar result for bounded-error algorithms for all total functions

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