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.
