# Mathematics of Quantum Information

### Overview

Quantum information science is an interdisciplinary research endeavour that brings together computer scientists, mathematicians, physicists, chemists, and engineers to develop revolutionary information processing and communication technologies that are infeasible without exploiting the principles of quantum mechanics. The importance of quantum information was first widely recognized in 1982 when Feynman conjectured that a quantum computer would efficiently simulate quantum systems, and a universal Turing machine (“classical computer”) could not.

In the mid‐1990s, Shor showed that the quantum computer could efficiently determine the factors of large numbers whereas this problem is believed to be intractable on a classical computer. Even earlier, in 1984, Bennett and Brassard proposed an information theoretically secure key distribution technique through public channels, as opposed to standard methods that are only computationally secure. Originally proposed in 1984, quantum cryptography has since become commercial technology.

Quantum information technology is thus “disruptive” both technically and also at a fundamental level both to physics and to computer science. Quantum information leads to a violation of the strong Church‐Turing thesis and could enable information‐theoretic security over public channels. Moreover quantum computing and quantum cryptography damage and ameliorate, respectively, information security.

Theoretical research in quantum information relies on sophisticated mathematical methods, and advances rely on concomitant developments of new mathematics, hence the need for strong mathematical research in quantum information.

### Areas of interest for the PIMS CRG on MQI

This CRG is ideally suited to address and make significant progress in the areas such as the three noted below.

**Models of quantum computing:**Quantum computing is technologically challenging so different models for implementing quantum algorithms are of great interest. The first model treated unitary gates on single qubits (quantum binary digits) and on pairs of qubits; a few gates of these types can be used to construct any quantum circuit efficiently with bounded error. Subsequently remarkably different circuits were devised that serve the same end, such as adiabatic quantum computing, topological quantum computing, quantum computing with oscillators (so‐called “continuous variable quantum computing”), and one‐way quantum computing.

Proving that these models are equivalent, especially under the fault‐tolerant conditions of error correction, is generally difficult. Not only are models of quantum computing valuable in exploring different physical realizations, but also these models are conceptually quite different and have inspired new quantum algorithms by forcing new ways to think about circuits. In addition, for one‐way quantum computation, beautiful and unexpected connections with graph theory are beginning to emerge, and they require further exploration.

**Error correction:**The theory of quantum error correction showed that quantum computing errors could be efficiently corrected despite Heisenberg limits to quantum measurements. Quantum error correction was then shown to follow easily from “classical” error correction theory, which made devising error correction protocols relatively straightforward. Quantum error correction was also shown to underlie the security of quantum key distribution by proving that “classical” cryptographic privacy amplification of quantum keys protected against quantum eavesdroppers.

Although quantum error correction is accepted as strategy against errors and faults, the overhead of error correction is typically high, and new strategies are sought to reduce the resource overhead. For example quantum error correction can be assisted by providing consumable entanglement resources thereby reducing the number of extra qubits and gates. Also more sophisticated methods such as belief propagation can be used to correct errors. This CRG will develop significantly better quantum error correction protocols.

**Quantum algorithms:**Quantum computers are of revolutionary importance because they would make some computational problems efficiently solvable that are regarded as intractable on a classical computer. So far the class of problems that are converted from intractable to tractable by a quantum computer is small, and this CRG will seek to expand the class of such problems by developing new quantum algorithms. The graph isomorphism problem is of special interest to members of the CRG, and its development will link to topological models of quantum computing.

In summary members of the proposed CRG have expertise and active research in many areas of quantum information including the three important topics listed above: models of quantum computing, error correction, and algorithms. The CRG will pursue these areas by bringing together their complimentary expertise and holding workshops with the world’s leading scientists in the field.

### Participants

**CRG Leaders:**

- Barry Sanders (UCalgary)
- Robert Raussendorf (UBC)
- Petr Lisonek (SFU)
- Aram Harrow (UWashington)

**UBC:**

- Philip Stamp
- William Unruh

**U Calgary:**

- David Feder
- Gilad Gour
- Peter Høyer
- Alex Lvovsky
- Christoph Simon
- Wolfgang Tittel

**SFU:**

- Leslie Ballentine
- Paul Haljan

**Washington:**

- Boris Blinov
- Subhadeep Gupta
- Mark Oskin
- Steve Flammia

### Scientific Activities:

- PIMS Calgary CRG Launch 2010, April 6, 2010, University of Calgary
- 10th Canadian Summer School on Quantum Information, University of British Columbia, July 17-30, 2010, Organizers: Robert Raußendorf (UBC) and Petr Lisonek (SFU)
- 13th Canadian Summer School on Quantum Information, University of Calgary, June 17-21 2013, Organizer: Barry Sanders (UCalgary)
- Workshop on Quantum Information and Foundations of Quantum Mechanics, University of British Columbia, Organizers: Robert Raussendorf (UBC) and Daniel E. Browne (University College London)
- Seminar Series in the Mathematics of Quantum Information

### Visitors:

David Kribs visited University of Calgary, November 29- December 3, 2010 and gave a talk, "A family of norms with applications in entanglement theory", on December 1, 2010 at the PIMS CRG MQI Seminar.

Saleh Rahimi Keshari gave a talk at Quantum Optics V , November 15-19, 2010.

Netanel Lindner gave a talk , "Putting Floquet theory to work: Topology of time dependent Hamiltonians", on March 16, 2011 at the PIMS CRG MQI Seminar. Netanel Lindner (California Institute of Technology) visited University of British Columbia (6 – 13 March 2011) and University of Calgary (14 – 17 March 2011).

Patrick Hayden (McGill University) visited between University of British Columbia (9 – 12 March 2011) and University of Washington (13 – 20 March 2011).

Fernando G.S.L. Brandão (Universidade Federal de Minas Gerais, Belo Horizonte, Brazil) visited University of Washington (10 – 16 July 2011) and University of Calgary (17 – 23 July 2011).

Shmuel Friedland (University of Illinois at Chicago) visited University of Calgary (7 – 12 November 2011).

### Post-doctoral fellows

- Vijaykumar Singh, Supervisor: Petr Lisonek (SFU), September 2011-August 2013
- Maritza Hernandez, Supervisor: Philip Stamp (UBC), September 2011-June 2012
- Vlad Gheorghiu, Supervisor: Gilad Gour (UCalgary), September 2011-August 2013
- Collin Trail, Supervisor: Barry Sanders (UCalgary), September 2011-August 2013

### Past Graduate Students:

- Alexander Hentschel. Supervisor: Barry Sanders (UCalgary). Graduated 2011.
- Saleh Rahimi-Keshari. Supervisor: Barry Sanders (UCalgary). Graduated 2010.

### Past PDFs:

- Ben Fortescue. Supervisor: Barry Sanders (UCalgary).