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

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

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

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

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

C1C1 C2C2

The Interaction Picture

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.

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

References

Stabilizer Circuits ~ Clifford gates can be simulated in time ~ 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?

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.

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