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 much easier ... more >>>

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 proves an ... more >>>

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 >>>

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 >>>

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 ... more >>>

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 states that can account ... more >>>

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 ... more >>>

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 ... more >>>

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 can later ... more >>>

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 ... more >>>

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 >>>

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 >>>

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 >>>

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 ... more >>>

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 >>>

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 >>>

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 >>>