Estimating Jones polynomials is a complete problem for one clean qubit

@article{Shor2008EstimatingJP,
  title={Estimating Jones polynomials is a complete problem for one clean qubit},
  author={Peter W. Shor and Stephen P. Jordan},
  journal={Quantum Inf. Comput.},
  year={2008},
  volume={8},
  pages={681-714}
}
It is known that evaluating a certain approximation to the Jones polynomial for the plat closure of a braid is a BQP-complete problem. That is, this problem exactly captures the power of the quantum circuit model[13, 3, 1]. The one clean qubit model is a model of quantum computation in which all but one qubit starts in the maximally mixed state. One clean qubit computers are believed to be strictly weaker than standard quantum computers, but still capable of solving some classically intractable… 

Estimating Jones and Homfly polynomials with one clean qubit

TLDR
This work shows that one clean qubit computers can efficiently approximate the Jones and single-variable HOMFLY polynomials of the trace closure of a braid at any root of unity.

Power of Quantum Computation with Few Clean Qubits

TLDR
It is proved that the TRACE ESTIMATION problem defined with fixed constant threshold parameters is complete for the classes of problems solvable by polynomial-time quantum computations with completeness 2/3 and soundness 1/3 using logarithmically many clean qubits and just one clean qubit.

Hardness of classically sampling one clean qubit model with constant total variation distance error

TLDR
It is shown that it is indeed possible to improve the multiplicative error hardness result to a constant total variation distance error one like other sub-universal quantum computing models such as the IQP model, the Boson Sampled model, and the Fourier Sampling model if the authors accept a modified version of the average case hardness conjecture.

Experimental approximation of the Jones polynomial with one quantum bit.

TLDR
Experimental results show that for the restricted case of knots whose braid representations have four strands and exactly three crossings, identifying distinct knots is possible 91% of the time.

Impossibility of Classically Simulating One-Clean-Qubit Computation

TLDR
It is shown that the third-level collapse of the polynomial-time hierarchy can be strengthened to the second-level one and the classical simulatability of the one-clean-qubit model with further restrictions on the circuit depth or the gate types is studied.

The Quantum Complexity of Computing Schatten $p$-norms

TLDR
It is shown that the problem of approximating $\text{Tr}\, (|A|^p)$ for a log-local $n$-qubit Hamiltonian $A$ and $p=\text{poly}(n)$, up to a suitable level of accuracy, is contained in DQC1; and that approximating this quantityup to a somewhat higherlevel of accuracy is D QC1-hard.

The Power of One Clean Qubit in Communication Complexity

TLDR
There is a quantum protocol using one clean qubit only and using $O(\log n)$ qubits of communication, such that any classical protocol simulating the acceptance behaviour of the quantum protocol within additive error needs communication $\Omega(n)$.

Computing partition functions in the one-clean-qubit model

TLDR
It is proved that a version of the partition function estimation problem within additive error is complete for the so-called DQC1 complexity class, suggesting that the method provides a super-polynomial speedup for certain parameter values.

Eliminating intermediate measurements in space-bounded Quantum computation

TLDR
This work exhibits a procedure to eliminate all intermediate measurements that is simultaneously space efficient and time efficient, and shows that the definition of a space-bounded quantum complexity class is robust to allowing or forbidding intermediate measurements.

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

TLDR
It is proved that additive approximation to single amplitudes of these models can be obtained by the one-clean-qubit model, if no initial superpositions are allowed, and it is shown that conditioned on infinite polynomial hierarchy assumption it is hard to sample from the output distribution ofThese models on a classical randomized computer.
...

References

SHOWING 1-10 OF 38 REFERENCES

Entanglement and the power of one qubit

The ``power of one qubit'' refers to a computational model that has access to only one pure bit of quantum information, along with $n$ qubits in the totally mixed state. This model, though not as

The Jones polynomial: quantum algorithms and applications in quantum complexity theory

TLDR
It is concluded with direct proofs that evaluating the Jones polynomial of the plat closure at most primitive roots of unity is a #P-hard problem, while learning its most significant bit is PP-hard, circumventing the usual route through the TuttePolynomial and graph coloring.

A Polynomial Quantum Algorithm for Approximating the Jones Polynomial

TLDR
An explicit and simplePolynomial quantum algorithm to approximate the Jones polynomial of an n strands braid with m crossings at any primitive root of unity e2πi/k, where the running time of the algorithm is polynometric in m, n and k.

Computing with highly mixed states

TLDR
It is shown that unless m∈O(k + log n), oblivious (gate-by-gate) simulation of an ideal m-qubit quantum circuit by an n-qu bit circuit with k clean qubits is impossible, this indicates that there is no avoiding physical initialization of a quantity of qubits proportional to that required by the best ideal quantum circuit.

Power of One Bit of Quantum Information

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

Quantum computing and quadratically signed weight enumerators

Quantum Computation and Quadratically Signed Weight Enumerators

TLDR
It is proved that quantum computation is polynomially equivalent to classical probabilistic computation with an oracle for estimating the value of simple sums, quadratically signed weight enumerators, which can be used to define and study complexity classes and their relationships to quantum computation.

Computation with Unitaries and One Pure Qubit

We define a semantic complexity class based on the model of quantum computing with just one pure qubit (as introduced by Knill and Laflamme) and discuss its computational power in terms of the

The BQP-hardness of approximating the Jones polynomial

TLDR
The universality proof of Freedman et al (2002) is extended to ks that grow polynomially with the number of strands and crossings in the link, thus extending the BQP-hardness of Jones polynomial approximations to all values to which the AJL algorithm applies, proving that for all those values, the problems are B QP-complete.

Simulation of Topological Field Theories¶by Quantum Computers

Abstract: Quantum computers will work by evolving a high tensor power of a small (e.g. two) dimensional Hilbert space by local gates, which can be implemented by applying a local Hamiltonian H for a