# Quantum computing, postselection, and probabilistic polynomial-time

@article{Aaronson2004QuantumCP, title={Quantum computing, postselection, and probabilistic polynomial-time}, author={Scott Aaronson}, journal={Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences}, year={2004}, 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…

## 294 Citations

### Revisiting a limit on efficient quantum computation

- Computer ScienceACM-SE 44
- 2006

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).

### Computation in generalised probabilisitic theories

- MathematicsArXiv
- 2014

This work defines a circuit-based model of computation in a class of operationally-defined theories more general than quantum theory, and shows there exists a classical oracle relative to which efficient computation in any theory satisfying the causality assumption does not include .

### Rewindable Quantum Computation and Its Equivalence to Cloning and Adaptive Postselection

- Computer ScienceArXiv
- 2022

Under the strongly believed assumption that BQP (cid:43) SZK, or the shortest independent vectors problem cannot be e-ciently solved with quantum computers, it is shown that a single rewinding operator is suﬁcient to achieve tasks that are intractable for quantum computation.

### On exact counting and quasi-quantum complexity

- 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

- Computer Science, MathematicsProceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
- 2010

The class post-IQP of languages decided with bounded error by uniform families of IQP circuits with post-selection is introduced, and it is proved first that post- IQP equals the classical class PP, and that if the output distributions of uniform IQP circuit families could be classically efficiently sampled, then the infinite tower of classical complexity classes known as the polynomial hierarchy would collapse to its third level.

### Computation in generalised probabilisitic theories

- Computer Science
- 2017

This work investigates limits on computational power that are imposed by simple physical, or information theoretic, principles, and extends the computational model by allowing post-selection on measurement outcomes, and considers whether one can obtain relativized complexity results for general theories.

### Impossibility of succinct quantum proofs for collision-freeness

- MathematicsQuantum Inf. Comput.
- 2012

We show that any quantum algorithm to decide whether a function f : [n] → [n] is a permutation or far from a permutation must make Ω (n1/3/w) queries to f, even if the algorithm is given a w-qubit…

### Anti-concentration theorems for schemes showing a quantum computational supremacy

- Computer Science
- 2017

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…

### On the complexity and verification of quantum random circuit sampling

- Computer ScienceNature Physics
- 2018

Evidence is provided that quantum random circuit sampling, a near-term quantum computational task, is classically hard but verifiable, making it a leading proposal for achieving quantum supremacy.

### The learnability of quantum states

- 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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