GENERALIZED STABILIZERS Ted Yoder. Quantum/Classical Boundary How do we study the power of quantum computers compared to classical ones? Compelling problems.

GENERALIZED STABILIZERSTed 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

C1 C2

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 fromand 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.