A Multigraph Approach for Performing the Quantum Schur Transform

Abstract

We take inspiration from the Okounkov-Vershik approach to the representation theory of the symmetric groups to develop a new way of understanding how the Schur-Weyl duality can be used to perform the Quantum Schur Transform. The Quantum Schur Transform is a unitary change of basis transformation between the computational basis of (C d ) ⊗n and the Schur-Weyl basis of (C d ) ⊗n . We describe a new multigraph, which we call the Schur-Weyl-Young graph, that represents both standard Weyl tableaux and standard Young tableaux in the same diagram. We suggest a major improvement on Louck's formula for calculating the transition amplitudes between two standard Weyl tableaux appearing in adjacent levels of the Schur-Weyl-Young graph for the case d = 2, merely by looking at the entries in the two tableaux. The key theoretical component that underpins our results is the discovery of a branching rule for the Schur-Weyl states, which we call the Schur-Weyl branching rule. This branching rule allows us to perform the change of basis transformation described above in a straightforward manner for any n and d.

A theory of quantum error correction for permutation-invariant codes

“…There are two approaches for this. The first approach uses the inverse of a quantum Schur transform [38,39,[41][42][43][44] to bring the state back into the computational basis, after which geometric phase gates [32] bring the state back into the original PI codespace.…”

“…, which is in a T-code. The final step is to map the T-code back to the original PI code, using ideas from the quantum Schur transform [38,39,[41][42][43][44] which is a unitary transformation which maps computational basis states to states of a superposition of states |m T ⟩ over magnetic number m and SYT An error occurs on a seven-qubit PI code [12,27], and a SYT syndrome measurement obtains the SYT depicted on the upper left. This algorithm proceeds using this SYT syndrome information.…”

A theory of quantum error correction for permutation-invariant codes

“…There are two approaches for this. The first approach uses the inverse of a quantum Schur transform [38,39,[41][42][43][44] to bring the state back into the computational basis, after which geometric phase gates [32] bring the state back into the original PI codespace.…”

“…, which is in a T-code. The final step is to map the T-code back to the original PI code, using ideas from the quantum Schur transform [38,39,[41][42][43][44] which is a unitary transformation which maps computational basis states to states of a superposition of states |m T ⟩ over magnetic number m and SYT An error occurs on a seven-qubit PI code [12,27], and a SYT syndrome measurement obtains the SYT depicted on the upper left. This algorithm proceeds using this SYT syndrome information.…”

A theory of quantum error correction for permutation-invariant codes

“…There are two approaches for this. The first approach uses the inverse of a quantum Schur transform [38,39,[41][42][43][44] to bring the state back into the computational basis, after which geometric phase gates [32] bring the state back into the original PI codespace.…”

“…, which is in a T-code. The final step is to map the T-code back to the original PI code, using ideas from the quantum Schur transform [38,39,[41][42][43][44] which is a unitary transformation which maps computational basis states to states of a superposition of states |m T ⟩ over magnetic number m and SYT An error occurs on a seven-qubit PI code [12,27], and a SYT syndrome measurement obtains the SYT depicted on the upper left. This algorithm proceeds using this SYT syndrome information.…”

A theory of quantum error correction for permutation-invariant codes

“…There are two approaches for this. The first approach uses the inverse of a quantum Schur transform [38,39,[41][42][43][44] to bring the state back into the computational basis, after which geometric phase gates [32] bring the state back into the original PI codespace.…”

“…, which is in a T-code. The final step is to map the T-code back to the original PI code, using ideas from the quantum Schur transform [38,39,[41][42][43][44] which is a unitary transformation which maps computational basis states to states of a superposition of states |m T ⟩ over magnetic number m and SYT An error occurs on a seven-qubit PI code [12,27], and a SYT syndrome measurement obtains the SYT depicted on the upper left. This algorithm proceeds using this SYT syndrome information.…”