# Quantum computing, postselection, and probabilistic polynomial-time

@article{Aaronson2005QuantumCP, title={Quantum computing, postselection, and probabilistic polynomial-time}, author={Scott Aaronson}, journal={Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences}, year={2005}, volume={461}, pages={3473 - 3482} }

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 solve PP-complete problems efficiently. The result also implies, as an easy corollary, a celebrated theorem of Beigel, Reingold and Spielman…

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## 276 Citations

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- Computer ScienceACM-SE 44
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This paper offers an exposition of a theorem originally due to Adleman, Demarrais and Huang that shows that the quantum complexity class BQP is contained in the classical counting class PP (Probabilistic Polynomial time).

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- Mathematics, Computer ScienceArXiv
- 2015

It is shown that the gap-definable counting classes which bound exact and zero-error quantum algorithms can be characterised in terms of "quantum-like" algorithms involving nonunitary gates, and that postselection and nonunitarity have equivalent power for exact quantum computation only if these classes collapse.

Classical simulation of commuting quantum computations implies collapse of the polynomial hierarchy

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- Physics, Computer ScienceITCS
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This paper shows complexity theoretic evidence of hardness that is on par with the strongest theoretical proposals for supremacy, and shows that RCS satisfies an average-case hardness condition - computing output probabilities of typical quantum circuits is as hard as computing them in the worst-case, and therefore #P-hard.

Anti-concentration theorems for schemes showing a quantum computational supremacy

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One of the main milestones in quantum information science is to realize quantum devices that exhibit an exponential computational advantage over classical ones without being universal quantum…

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A critical milestone on the path to useful quantum computers is the demonstration of a quantum computation that is prohibitively hard for classical computers—a task referred to as quantum supremacy.…

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- Mathematics, Computer ScienceArXiv
- 2011

We prove that endowing a real-time probabilistic or quantum computer with the ability of postselection increases its computational power. For this purpose, we provide a new model of finite automata…

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- Computer Science, PhysicsProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
- 2007

This theorem has the conceptual implication that quantum states, despite being exponentially long vectors, are nevertheless ‘reasonable’ in a learning theory sense and has two applications to quantum computing: first, a new simulation of quantum one-way communication protocols and second, the use of trusted classical advice to verify untrusted quantum advice.

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The complexity of approximate counting, the problem of multiplicatively estimating the size of a nonempty set S ⊆ [N], is resolved in two natural generalizations of quantum query complexity.

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