# Quantum Computing:

## Quantum Fault Tolerance:

#### I am interested in understanding how to reliably determine error thresholds that are both necessary and sufficient to support fault tolerant quantum computing. The standard approach to quantum fault tolerance is to calculate error thresholds on basic gates in the limit of arbitrarily many concatenation levels. I want to be able to take the number of qubits and the target implementation accuracy as a constraint and then provide a framework for engineering the constrained quantum system to the required tolerance. I have developed a different approach which is based on complete analytical solutions to the dynamics of quantum error correction. I believe it will result in better integration of hardware and software since the full error manifold clarifies what is easier to do in physics and what is easier to do in error control algorithms. As an example, the image on the right shows the manifold of initial error probabilities that are consistent with a given implementation inaccuracy when the 7-qubit code is concatenated a specified number of times. My research program was funded by DARPA under the highly competitive QuEST Program.

- V. Aggarwal, A. R. Calderbank, G. Gilbert and Y. S. Weinstein, "Volume Thresholds for Quantum Fault Tolerance",
Quantum Information Processing, Volume 9 Issue 5, October 2010.
- V. Aggarwal, A. R. Calderbank, G. Gilbert and Y. S. Weinstein, "Analyses of Volume Thresholds in Quantum Fault Tolerance," in Proc. APS March Meeting 2010.
- V. Aggarwal, A. R. Calderbank, G. Gilbert and Y. S. Weinstein, "Engineering Fault Tolerance for Realistic Quantum Systems via the Full Error Dynamics of Quantum Codes," in Proc. ISIT, Seoul, Korea, Jun.-Jul. 2009.
- G. Gilbert, Y. S. Weinstein, V. Aggarwal and A. Robert Calderbank, "Operator Theoretic Quantum Fault Tolerance," in Proc. APS March Meeting 2009.
- V. Aggarwal,
A. R. Calderbank, G. Gilbert, M. Hamrick and Y.
S. Weinstein, "A Universal
Operator Theoretic Framework for Quantum Fault Tolerance," in
Proc. APS March Meeting 2008.
- G. Gilbert, M. Hamrick,
Y. S. Weinstein, V. Aggarwal and A. R. Calderbank, "A
Universal Operator Theoretic Framework for Quantum Fault Tolerance",
arXiv:0709.0128, Sept. 2007.

## Boolean functions and Projection Operators:

#### There is a fundamental correspondence between Boolean functions and projection operators in Hilbert space. I have used it to provide a common mathematical framework for the design of both additive and non-additive quantum error correcting codes and to construct new families of error correcting codes. The correspondence is widely applicable and I have also used it to construct and analyze new families of non-coherent space-time codes for applications in wireless communication.

- V. Aggarwal
and A. R. Calderbank, "Boolean
Functions, Projection Operators, and Quantum Error Correcting Codes," IEEE Trans. on
Inf. Th., vol. 54, no. 4, Apr. 2008, pp. 1700-1707.
- V. Aggarwal
and A. R. Calderbank, "Boolean Functions, Projection Operators
and Quantum Error Correcting Codes," in Proc. International Symposium on Information
Theory, pp.2091-2095, June 2007, Nice.
- V. Aggarwal, A. Ashikhmin and A. R. Calderbank, "A
Grassmannian Packing Based on the Nordstrom-Robinson Code", in Proc. Information Theory Workshop, pp. 1-5, Oct. 2006, Chengdu,
China.

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