Scott Aaronson

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Can Grover’s algorithm speed up search of a physical region—for example a 2-D grid of size √ n× √ n? The problem is that √ n time seems to be needed for each query, just to move amplitude across the grid. Here we show that this problem can be surmounted, refuting a claim to the contrary by Benioff. In particular, we show how to search a d-dimensional(More)
I study the class of problems efficiently solvable by a quantum computer, given the ability to ‘postselect’ on the outcomes of measurements. I prove that this class coincides with a classical complexity class called PP, or probabilistic polynomial-time. Using this result, I show that several simple changes to the axioms of quantum mechanics would let us(More)
Any proof of P!=NP will have to overcome two barriers: relativization and natural proofs. Yet over the last decade, we have seen circuit lower bounds (for example, that PP does not have linear-size circuits) that overcome both barriers simultaneously. So the question arises of whether there is a third barrier to progress on the central questions in(More)
(MATH) The collision problem is to decide whether a function <i>X</i>: { 1,&ldots;,<i>n</i>} &#8594; { 1, &ldots;,<i>n</i>} is one-to-one or two-to-one, given that one of these is the case. We show a lower bound of &#937;(<i>n</i><sup>1/5</sup>) on the number of queries needed by a quantum computer to solve this problem with bounded error probability. The(More)
We give new evidence that quantum computers -- moreover, rudimentary quantum computers built entirely out of linear-optical elements -- cannot be efficiently simulated by classical computers. In particular, we define a model of computation in which identical photons are generated, sent through a linear-optical network, then nonadaptively measured to count(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<sup><i>n</i></sup>, we show a lower bound of &#937;(2<sup><i>n</i>/4</sup>/<i>n</i>) on the number of queries needed by a quantum computer to solve this problem. More(More)
Although a quantum state requires exponentially many classical bits to describe, the laws of quantum mechanics impose severe restrictions on how that state can be accessed. This paper shows in three settings that quantum messages have only limited advantages over classical ones. First, we show that BQP/qpoly /spl sube/ PP/poly, where BQP/qpoly is the class(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 implications for experimental(More)
Can NP-complete problems be solved efficiently in the physical universe? I survey proposals including soap bubbles, protein folding, quantum computing, quantum advice, quantum adiabatic algorithms, quantum-mechanical nonlinearities, hidden variables, relativistic time dilation, analog computing, Malament-Hogarth spacetimes, quantum gravity, closed timelike(More)