# Computational Complexity of Some Quantum Theories in $1+1$ Dimensions

@article{Mehraban2015ComputationalCO, title={Computational Complexity of Some Quantum Theories in \$1+1\$ Dimensions}, author={Saeed Adel Mehraban}, journal={ArXiv}, year={2015}, volume={abs/1512.09243} }

We study the computational complexity of certain integrable quantum theories in 1+1 dimensions. We formalize a model of quantum computation based on these theories. In this model, distinguishable particles start out with known momenta and initial superposition of different configurations. Then the label of these particles are measured at the end. We prove that additive approximation to single amplitudes of these models can be obtained by the one-clean-qubit model, if no initial superpositions…

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## References

SHOWING 1-10 OF 64 REFERENCES

### Quantum complexity theory

- Computer ScienceSTOC
- 1993

This paper gives the first formal evidence that quantum Turing machines violate the modern (complexity theoretic) formulation of the Church--Turing thesis, and proves that bits of precision suffice to support a step computation.

### Quantum computation

- Physics
- 1996

The theory of quantum computation can be constructed from the abstract study of anyonic systems. In mathematical terms, these are unitary topological modular functors. They underlie the Jones…

### Classical simulation of noninteracting-fermion quantum circuits

- PhysicsArXiv
- 2001

It is shown that a class of quantum computations that was recently shown to be efficiently simulatable on a classical computer by Valiant corresponds to a physical model of noninteracting fermions in one dimension.

### Power of One Bit of Quantum Information

- Physics
- 1998

In standard quantum computation, the initial state is pure and the answer is determined by making a measurement of some of the bits in the computational basis. What can be accomplished if the initial…

### On the hardness of classically simulating the one clean qubit model

- PhysicsPhysical review letters
- 2014

This Letter introduces a slightly modified version of DQC1, which it is shown that DZC1(k) cannot be classically efficiently simulated for any k≥3 unless the polynomial hierarchy collapses at the third level.

### Encoded Universality in Physical Implementations of a Quantum Computer

- Physics, Computer Science
- 2001

A general Lie-algebraic framework is outlined which can be used to find the encoding for universality in quantum computing and several examples relevant to solid-state quantum computing are given.

### The computational complexity of linear optics

- Computer ScienceSTOC '11
- 2011

A model of computation in which identical photons are generated, sent through a linear-optical network, then nonadaptively measured to count the number of photons in each mode is defined, giving new evidence that quantum computers cannot be efficiently simulated by classical computers.

### Quantum computation by measurements

- Physics
- 2003

We first consider various methods for the indirect implementation of unitary gates. We apply these methods to rederive the universality of 4-qubit measurements based on a scheme much simpler than…

### Encoding a qubit in an oscillator

- Physics
- 2001

Quantum error-correcting codes are constructed that embed a finite-dimensional code space in the infinite-dimensional Hilbert space of a system described by continuous quantum variables. These codes…

### 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.