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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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Presentation Transcript
quantum classical boundary
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
schr dinger
Schrödinger

C

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

heisenberg
Heisenberg

C

~ What set of operators do we choose?

~ Require

examples
Examples

~ By analogy to the first, we can write any stabilizer as

~ And the state it stabilizes as

destabilizer tableaus stabilizer bases
Destabilizer, Tableaus, Stabilizer Bases

~ We have . What is ?

~ Collect all in a group,

~ A tableau defines a stabilizer basis,

generalized stabilizer
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

slide8

C1

C2

update efficiencies
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
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
stabilizer circuits
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

measurements
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
Channels

~ Define a Pauli channel as,

for Pauli operators

~ Define as a measure of its complexity.