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Generalized Stabilizers

Generalized Stabilizers. Ted Yoder. Quantum/Classical Boundary. How do we study the power of quantum computers compared to classical ones? Compelling problems Shor’s factoring Grover’s search Oracle separations Quantum resources Entanglement Discord Classical simulation. Schrödinger.

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Generalized Stabilizers

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  1. Generalized Stabilizers Ted Yoder

  2. Quantum/Classical Boundary • How do we study the power of quantum computers compared to classical ones? • Compelling problems • Shor’s factoring • Grover’s search • Oracle separations • Quantum resources • Entanglement • Discord • Classical simulation

  3. Schrödinger C ~ What is the probability of measuring the first qubit to be 0?

  4. Heisenberg C ~ What set of operators do we choose? ~ Require

  5. Examples ~ By analogy to the first, we can write any stabilizer as ~ And the state it stabilizes as

  6. Destabilizer, Tableaus, Stabilizer Bases ~ We have . What is ? ~ Collect all in a group, ~ A tableau defines a stabilizer basis,

  7. Generalized Stabilizer ~ Take any quantum state and write it in a stabilizer basis, ~ Then all the information about can be written as the pair ~ Any state can be represented ~ Any operation can be simulated - Unitary gates - Measurements - Channels

  8. C1 C2

  9. The Interaction Picture

  10. Update Efficiencies ~ For updates can be done with the following efficiency: ~ Gottesman-Knill 1997 On stabilizer states, we have the update efficiencies - Clifford gates: - Pauli measurements: ~ Note the correspondence when .

  11. Conclusion • New (universal) state representation • Combination of stabilizer and density matrix representation • Features dynamic basis that allows efficient simulation of Clifford gates • The interaction picture for quantum circuit simulation • Leads to a sufficient condition on states easily simulatable through any stabilizer circuit

  12. References

  13. Stabilizer Circuits ~ Recall that stabilizer circuits are those made from and a final measurement of the operator . ~ What set of states can be efficiently simulated by a classical computer through any stabilizer circuit? ~ Clifford gates can be simulated in time

  14. Measurements ~ We’ll measure the complexity of by ~ The complexity of a state can be defined as ~ Simulating measurement of takes time ~ What set of states can be efficiently simulated by a classical computer through any stabilizer circuit? is sufficient.

  15. Channels ~ Define a Pauli channel as, for Pauli operators ~ Define as a measure of its complexity.

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