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Content archived on 2024-06-16

Applications of the Schur Transform to Quantum Information Theory


Quantum computation and quantum information offer qualitatively new capabilities; such as provably secure communication and efficient solutions to computational problems previously believed intractable. However, realizing these possibilities will require techniques for dealing with quantum noise that are efficient in their use of both computational and communication resources. Many information-theoretic solutions achieve the optimal rates of communication but are impossible to apply because they require exponentially large quantum computing resources. One promising new approach to efficiently manipulating quantum information is the Schur transform [1], an algorithm which, similarly to the quantum Fourier transform, allows a quantum computer to efficiently address non-local degrees of freedom in quantum registers.

We propose to use the Schur transform to develop practical quantum circuits for a wide range of tasks in quantum information theory, such as remote state preparation, coding for known or unknown quantum channels, mixed state data compression, optimal state estimation and hypothesis testing. In many cases, such as remote state preparation and channel coding, the information-theoretically optimal solutions are known, but we will use the Schur transform to develop novel efficient algorithms to implement them. In others, such as coding for unknown quantum channels, there are not even any computationally inefficient solutions that are currently known, and we hope to use the insights from the Schur transform to solve these problems for the first time. [1] D. Bacon, I. Chuang and A. Harrow, "Efficient Quantum Circuits for Schur and Clebsch-Gordan Transforms"; eprint quant-ph/0407082 (2004).

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