# Computation with Unitaries and One Pure Qubit

@article{Shepherd2006ComputationWU, title={Computation with Unitaries and One Pure Qubit}, author={Dan J. Shepherd}, journal={arXiv: Quantum Physics}, year={2006} }

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 problem of estimating the trace of a large unitary matrix. We show that this problem is complete for the complexity class, and derive some further fundamental features of the class. We conclude with a discussion of some associated open conjectures and new oracle separations between classes.

## 19 Citations

Blindness and Verification of Quantum Computation with One Pure Qubit

- Computer Science, PhysicsTQC
- 2014

This paper presents the adaptation of Encoding via blindness to the one pure qubit model, and presents the first feasible scheme for the verification of delegated one purequbit model of quantum computing.

Verified Delegated Quantum Computing with One Pure Qubit

- Computer Science, Physics
- 2014

This paper presents the adaptation of this approach to the one pure qubit model, and presents the rst feasible scheme for the verication of delegated one purequbit model of quantum computing.

Normalizer Circuits and Quantum Computation

- Physics, Computer ScienceArXiv
- 2016

An efficient formalism for simulating families of quantum circuits, that are non-universal but comprise important quantum gates such as QFT or CNOT, is developed and used to design new algorithms that provide quantum speedups.

Power of Quantum Computation with Few Clean Qubits

- Mathematics, PhysicsICALP
- 2016

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.

Estimating Jones polynomials is a complete problem for one clean qubit

- MathematicsQuantum Inf. Comput.
- 2008

It is shown that evaluating a certain approximation to the Jones polynomial at a fifth root of unity for the trace closure of a braid is a complete problem for the one clean qubit complexity class.

The Quantum Complexity of Computing Schatten $p$-norms

- Computer ScienceTQC
- 2018

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.

Approximating the Turaev-Viro Invariant of Mapping Tori is Complete for One Clean Qubit

- MathematicsTQC
- 2011

It is shown that the problem of estimating the Fibonacci version of the Turaev-Viro invariant of a mapping torus is a complete problem for the one clean qubit complexity class (DQC1).

Quantum Computation Beyond the

- Physics
- 2005

The quantum circuit model is the most widely used model of quantum computation. It provides both a framework for formulating quantum algorithms and an architecture for the physical construction of…

Quantum computation beyond the circuit model

- Physics
- 2008

The quantum circuit model is the most widely used model of quantum computation. It provides both a framework for formulating quantum algorithms and an architecture for the physical construction of…

Studies on the Role of Entanglement in Mixed-state Quantum Computation

- Physics
- 2008

In this thesis, I look at the role of quantum entanglement in mixed-state quantum computation. The model we consider is the DQC1 or `power of one qubit' model. I show that there is minimal bipartite…

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