Mini Workshop on the Frontiers of Quantum Information

We will host a Mini Workshop on the Frontiers of Quantum Information at the University of Guelph, on 8th April 2014.

- David Kribs (University of Guelph)
- Bei Zeng (University of Guelph)

- Chi-Kwong Li (The College of William and Mary) abstract
- Mary Beth Ruskai (IQC, University of Waterloo) abstract Slides
- Barry Sanders (University of Calgary) abstract

- Jianxin Chen (University of Guelph & IQC, University of Waterloo) abstract
- Isaac Kim (Perimeter Institute) abstract
- Nengkun Yu (University of Guelph & IQC, University of Waterloo) abstract

Apr. 8, 2014 (Tuesday) | |
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10:00 - 11:00 | Mary Beth Ruskai.
Contraction Coefficients for Noisy Quantum Channels |

11:00 - 11:30 | Isaac Kim.
On The Informational Completeness of Local Observables |

11:30 - 13:00 | Lunch |

13:00 - 14:00 | Barry Sanders.
BosonSampling with Controllable Distinguishability of Photons |

14:00 - 14:30 | Jianxin Chen.
A Complete Characterization of Anti-degradable Qubit Channels |

14:30 - 15:00 | Break |

15:00 - 16:00 | Chi-Kwong Li.
Some Recent Results on Quantum Error Correction |

16:00 - 16:30 | Nengkun Yu.
Optimal Simulation of Toffoli gate, Deutsch Gates and Fredkin Gate. |

Room 1504, Science Complex Building, University of Guelph, 50 Stone Road East, Guelph, Ontario, N1G 2W1 Canada

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*Jianxin Chen (University of Guelph & IQC, University of Waterloo):*

**A Complete Characterization of Anti-degradable Qubit Channels**

It is well-known that degradable and anti-degradable
channels are among the few for which the quantum capacity is known
explicitly. However, determining the structure of these channels is
very challenging, even for the qubit case. Though the degradable qubit
channels has already been completely characterized, the structure of
anti-degradable qubit channels remains unclear until very recently.
Inspired by our recent work on symmetric extension of two-qubit
states, we are able to characterize the structure of anti-degradable
qubit channels.

*Isaac Kim (Perimeter Institute):*

**On The Informational Completeness of Local Observables**

For a general multipartite quantum state, we formulate a locally checkable condition, under which the expectation values of certain nonlocal observables are completely determined by the expectation values of some local observables. The condition is satisfied for ground states of gapped quantum many-body systems in one and two spatial dimensions, assuming a widely conjectured form of area law is correct. Its implications on quantum state tomography, quantum state verification, and quantum information storage capacity is discussed.

*Chi-Kwong Li (The College of William and Mary):*

**Some Recent Results on Quantum Error Correction**

We discuss some recent results on quantum error correction.
The focus will be on optimizing the implementation process.
Open problems will be mentioned.

*Mary Beth Ruskai (IQC, University of Waterloo):*

** Contraction Coefficients for Noisy Quantum Channels**

It is well known that certain operator convex functions can be used to define a large class of generalizations of relative entropy, sometimes called "quasi-entriopies" or "f-digvergences". It is also known that a closely related class of operator convex functions can be used to define a bilinear form which can be regarded as a Riemannian metric on the manifold of trace zero matrices. Each of these can also be obtained as the Hession of a generalized relative entropy, in which case it can be seen as a quantum analogue of the Fisher information, with a corresponding relationship between the operator convex functions involved. Furthermore, each Riemannian metric can be used to definite a geodesic distance between a pair of positive operators of trace one.

All of these quantities are non-increasing under the action of a quantum channel, i.e., a completely positive trace-preserving map, as is the trace distance. The ratio of any of these quantities after and before the action of the quantum channel is known as a contraction coefficient. We thus have three families of contraction coefficients associated with operator convex functions. A number of inequalities were known and several conjectures were open. Recently, Hiai showed that for a fixed function the contraction coefficients for the Riemannian metric and geodesic distance are identical. For unital qubit channels these contraction coefficients have a simple form, independent of the function. For CQ qubit channels explicit expressions can be obtained in some cases, and shown to depend on the function. These expressions can also be used to resolve some open questions.

It is also interesting that the contraction coefficient for a Riemannian metric can be expressed as an eigenvalue problem.

*Barry Sanders (University of Calgary):*

**BosonSampling with Controllable Distinguishability of Photons**

The BosonSampling Problem requires samples of output photon-coincidence probability distributions for an interferometer with either single photons or nothing injected into each input port [1]. This computational problem is classically hard to simulate as these probabilities are weighted by permanents of sub-matrices of the interferometer transition matrix yet the problem appears to be efficiently solvable just by doing the experiment. The classical hardness and quantum easiness of this problem makes it special in studies of quantum computing because a demonstrable exponential computational speedup by quantum means would be evidence that the strong Church-Turing thesis of computer science is false. Our innovation [2,3,4] introduces distinguishability between photons by controlling arrival times of otherwise identical photons in order to test the model, assess sampling errors and generalize BosonSampling beyond permanent-weighted to immanant-weighted probabilities.

[1] S. Aaronson and A. Arkhipov, in Proceedings of the 43rd Annual ACM Symposium on Theory of Computing (ACM, New York, 2011).

[2] S.-H. Tan, Y. Y. Gao, H. de Guise and B. C. Sanders, “SU(3) Quantum Interferometry with single-photon input pulses”, Physical Review Letters110 (11): 113603 (5 pp.), 12 March 2013, arXiv.org:1208.5677.

[3] H. de Guise, S.-H. Tan, I. P. Poulin and B. C. Sanders, “Immanants for three-channel linear optical networks”, arXiv.org:1402.2391.

[4] M. Tillman, S.-H. Tan, S. E. Stoeckl, B. C. Sanders, H. de Guise, R. Hellmann, S. Nolte, A. Szameit and P. Walther, “BosonSampling with controllable distinguishability”, arXiv.org:1403.3433.

*Nengkun Yu (IQC, University of Waterloo & University of Guelph):*

**Optimal Simulation of Toffoli Gate, Deutsch Gates and Fredkin Gate**

We study the optimal simulation of three-qubit unitary by using two-qubit gates. First, we completely characterize the two-qubit gate cost of simulating Toffoli gate and Deutsch gates (controlled controlled gate) by generalizing our result on the cost of Toffoli gate. The function of any Deutsch Gate is simply a three-qubit controlled unitary gate and can be intuitively explained as follows: the gate will output the states of the two control qubit directly, and apply the given one-qubit unitary $u$ on the target qubit only if both the states of the control are $\ket{1}$. Previously, it is only known that five two-qubit gates is sufficient for implementing such a gate [Sleator and Weinfurter, Phys. Rev. Lett. 74, 4087 (1995)]. We show that if the determinant of $u$ is 1, four two-qubit gates is achievable optimal. Otherwise, five is optimal. Thenm we prove that five two-qubit gates are necessary and sufficient for implementing the Fredkin gate(the controlled swap gate), which settles the open problem introduced in [Smolin and DiVincenzo, Phys. Rev. A, 53, 2855 (1996)]. Before our work, a five two-qubit gates decomposition of the Fredkin gate was already known, and only numerical evidence of showing five is optimal is found.