Jean-Pierre Seifert

While quantum computers might speed up in principle

certain computations dramatically, in pratice, though

quantum computing technology is still in its infancy.

Even we cannot clearly envison at present what the

hardware of that machine will be like.

Nevertheless, we can be quite confident that it will be

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Iordanis Kerenidis, Ronald de Wolf

We prove exponential lower bounds on the length of 2-query

locally decodable codes. Goldreich et al. recently proved such bounds

for the special case of linear locally decodable codes.

Our proof shows that a 2-query locally decodable code can be decoded

with only 1 quantum query, and then ...
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Scott Aaronson

We revisit the oft-neglected 'recursive Fourier sampling' (RFS) problem, introduced by Bernstein and Vazirani to prove an oracle separation between BPP and BQP. We show that the known quantum algorithm for RFS is essentially optimal, despite its seemingly wasteful need to uncompute information. This implies that, to place BQP outside ... more >>>

Scott Aaronson

Given a Boolean function f, we study two natural generalizations of the certificate complexity C(f): the randomized certificate complexity RC(f) and the quantum certificate complexity QC(f). Using Ambainis' adversary method, we exactly characterize QC(f) as the square root of RC(f). We then use this result to prove the new relation ... more >>>

Scott Aaronson

The problem of finding a local minimum of a black-box function is central

for understanding local search as well as quantum adiabatic algorithms.

For functions on the Boolean hypercube {0,1}^n, we show a lower bound of

Omega(2^{n/4}/n) on the number of queries needed by a quantum computer to

solve this ...
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Scott Aaronson

Several researchers, including Leonid Levin, Gerard 't Hooft, and

Stephen Wolfram, have argued that quantum mechanics will break down

before the factoring of large numbers becomes possible. If this is

true, then there should be a natural "Sure/Shor separator" -- that is,

a set of quantum ...
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Hartmut Klauck, Robert Spalek, Ronald de Wolf

A strong direct product theorem says that if we want to compute

k independent instances of a function, using less than k times

the resources needed for one instance, then our overall success

probability will be exponentially small in k.

We establish such theorems for the classical as well as ...
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Scott Aaronson

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 ... more >>>

Scott Aaronson

This paper introduces a new technique for removing existential quantifiers

over quantum states. Using this technique, we show that there is no way

to pack an exponential number of bits into a polynomial-size quantum

state, in such a way that the value of any one of those bits ...
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Scott Aaronson, Greg Kuperberg

This paper studies whether quantum proofs are more powerful than

classical proofs, or in complexity terms, whether QMA=QCMA. We prove

two results about this question. First, we give a "quantum oracle

separation" between QMA and QCMA. More concretely, we show that any

quantum algorithm needs order sqrt(2^n/(m+1)) queries to find ...
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Scott Aaronson

Traditional quantum state tomography requires a number of measurements that grows exponentially with the number of qubits n. But using ideas from computational learning theory, we show that "for most practical purposes" one can learn a state using a number of measurements that grows only linearly with n. Besides possible ... more >>>

Andris Ambainis, Joseph Emerson

A t-design for quantum states is a finite set of quantum states with the property of simulating the Haar-measure on quantum states w.r.t. any test that uses at most t copies of a state. We give efficient constructions for approximate quantum t-designs for arbitrary t.

We then show that an ... more >>>

Thomas Watson, Dieter van Melkebeek

We obtain the first nontrivial time-space lower bound for quantum algorithms solving problems related to satisfiability. Our bound applies to MajSAT and MajMajSAT, which are complete problems for the first and second levels of the counting hierarchy, respectively. We prove that for every real $d$ and every positive real $\epsilon$ ... more >>>

Scott Aaronson, Salman Beigi, Andrew Drucker, Bill Fefferman, Peter Shor

The class QMA(k), introduced by Kobayashi et al., consists

of all languages that can be verified using k unentangled quantum

proofs. Many of the simplest questions about this class have remained

embarrassingly open: for example, can we give any evidence that k

quantum proofs are more powerful than one? Can ...
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Scott Aaronson

Whether the class QMA (Quantum Merlin Arthur) is equal to QMA1, or QMA with one-sided error, has been an open problem for years. This note helps to explain why the problem is difficult, by using ideas from real analysis to give a "quantum oracle" relative to which QMA and QMA1 ... more >>>

Scott Aaronson, John Watrous

While closed timelike curves (CTCs) are not known to exist, studying their consequences has led to nontrivial insights in general relativity, quantum information, and other areas. In this paper we show that if CTCs existed, then quantum computers would be no more powerful than classical computers: both would have the ... more >>>

Andrew Drucker, Ronald de Wolf

Alongside the development of quantum algorithms and quantum complexity theory in recent years, quantum techniques have also proved instrumental in obtaining results in classical (non-quantum) areas. In this paper we survey these results and the quantum toolbox they use.

more >>>Ben Reichardt

Quantum query complexity measures the number of input bits that must be read by a quantum algorithm in order to evaluate a function. Hoyer et al. (2007) have generalized the adversary semi-definite program that lower-bounds quantum query complexity. By giving a matching algorithm, we show that the general adversary lower ... more >>>

Dieter van Melkebeek, Thomas Watson

We give two time- and space-efficient simulations of quantum computations with

intermediate measurements, one by classical randomized computations with

unbounded error and the other by quantum computations that use an arbitrary

fixed universal set of gates. Specifically, our simulations show that every

language solvable by a bounded-error quantum algorithm running ...
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Dmitry Gavinsky, Tsuyoshi Ito

We introduce a new type of cryptographic primitive that we call hiding fingerprinting. No classical fingerprinting scheme is hiding. We construct quantum hiding fingerprinting schemes and argue their optimality.

more >>>Andris Ambainis, Loïck Magnin, Martin Roetteler, Jérémie Roland

We introduce a new quantum adversary method to prove lower bounds on the query complexity of the quantum state generation problem. This problem encompasses both, the computation of partial or total functions and the preparation of target quantum states. There has been hope for quite some time that quantum ... more >>>

Scott Aaronson

One of the crown jewels of complexity theory is Valiant's 1979 theorem that computing the permanent of an n*n matrix is #P-hard. Here we show that, by using the model of linear-optical quantum computing---and in particular, a universality theorem due to Knill, Laflamme, and Milburn---one can give a different and ... more >>>

Scott Aaronson

One might think that, once we know something is computable, how efficiently it can be computed is a practical question with little further philosophical importance. In this essay, I offer a detailed case that one would be wrong. In particular, I argue that computational complexity theory---the field that studies the ... more >>>

Dan Boneh, Mark Zhandry

We construct the first Message Authentication Codes (MACs) that are existentially unforgeable against a quantum chosen message attack. These chosen message attacks model a quantum adversary’s ability to obtain the MAC on a superposition of messages of its choice. We begin by showing that a quantum secure PRF is sufficient ... more >>>

Scott Aaronson, Travis Hance

Around 2002, Leonid Gurvits gave a striking randomized algorithm to approximate the permanent of an n×n matrix A. The algorithm runs in O(n^2/?^2) time, and approximates Per(A) to within ±?||A||^n additive error. A major advantage of Gurvits's algorithm is that it works for arbitrary matrices, not just for nonnegative matrices. ... more >>>

Scott Aaronson, Andris Ambainis, Kaspars Balodis, Mohammad Bavarian

We study the query complexity of Weak Parity: the problem of computing the parity of an n-bit input string, where one only has to succeed on a 1/2+eps fraction of input strings, but must do so with high probability on those inputs where one does succeed. It is well-known that ... more >>>

Scott Aaronson, Adam Bouland, Joseph Fitzsimons, Mitchell Lee

We explore the space "just above" BQP by defining a complexity class PDQP (Product Dynamical Quantum Polynomial time) which is larger than BQP but does not contain NP relative to an oracle. The class is defined by imagining that quantum computers can perform measurements that do not collapse the ... more >>>

Scott Aaronson, Daniel Grier, Luke Schaeffer

We present a complete classification of all possible sets of classical reversible gates acting on bits, in terms of which reversible transformations they generate, assuming swaps and ancilla bits are available for free. Our classification can be seen as the reversible-computing analogue of Post's lattice, a central result in mathematical ... more >>>

Scott Aaronson, Shalev Ben-David

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. ... more >>>

Adam Bouland, Laura Mancinska, Xue Zhang

We classify two-qubit commuting Hamiltonians in terms of their computational complexity. Suppose one has a two-qubit commuting Hamiltonian $H$ which one can apply to any pair of qubits, starting in a computational basis state. We prove a dichotomy theorem: either this model is efficiently classically simulable or it allows one ... more >>>

Scott Aaronson, Mohammad Bavarian, Giulio Gueltrini

We ask, and answer, the question of what's computable by Turing machines equipped with time travel into the past: that is, closed timelike curves or CTCs (with no bound on their size). We focus on a model for CTCs due to Deutsch, which imposes a probabilistic consistency condition to avoid ... more >>>

Kamil Khadiev, Aliya Khadiev, Alexander Knop

In this paper, we study quantum OBDD model, it is a restricted version of read-once quantum branching programs, with respect to "width" complexity. It is known that the maximal gap between deterministic and quantum complexities is exponential. But there are few examples of functions with such a gap. We present ... more >>>

Scott Aaronson

We show that combining two different hypothetical enhancements to quantum computation---namely, quantum advice and non-collapsing measurements---would let a quantum computer solve any decision problem whatsoever in polynomial time, even though neither enhancement yields extravagant power by itself. This complements a related result due to Raz. The proof uses locally decodable ... more >>>

Shir Peleg, Amir Shpilka, Ben Lee Volk

The stabilizer rank of a quantum state $\psi$ is the minimal $r$ such that $\left| \psi \right \rangle = \sum_{j=1}^r c_j \left|\varphi_j \right\rangle$ for $c_j \in \mathbb{C}$ and stabilizer states $\varphi_j$. The running time of several classical simulation methods for quantum circuits is determined by the stabilizer rank of the ... more >>>