Quantum to Classical Transition in Noisy Classical Environment JACOPO TRAPANI.
GENERALIZED STABILIZERS Ted Yoder. Quantum/Classical Boundary How do we study the power of quantum...
-
Upload
hana-miller -
Category
Documents
-
view
226 -
download
0
Transcript of GENERALIZED STABILIZERS Ted Yoder. Quantum/Classical Boundary How do we study the power of quantum...
![Page 1: GENERALIZED STABILIZERS Ted Yoder. Quantum/Classical Boundary How do we study the power of quantum computers compared to classical ones? Compelling problems.](https://reader035.fdocuments.us/reader035/viewer/2022081502/56649c745503460f949275c3/html5/thumbnails/1.jpg)
GENERALIZED STABILIZERSTed Yoder
![Page 2: GENERALIZED STABILIZERS Ted Yoder. Quantum/Classical Boundary How do we study the power of quantum computers compared to classical ones? Compelling problems.](https://reader035.fdocuments.us/reader035/viewer/2022081502/56649c745503460f949275c3/html5/thumbnails/2.jpg)
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
![Page 3: GENERALIZED STABILIZERS Ted Yoder. Quantum/Classical Boundary How do we study the power of quantum computers compared to classical ones? Compelling problems.](https://reader035.fdocuments.us/reader035/viewer/2022081502/56649c745503460f949275c3/html5/thumbnails/3.jpg)
Schrödinger
C
~ What is the probability of measuring the first qubit to be 0?
![Page 4: GENERALIZED STABILIZERS Ted Yoder. Quantum/Classical Boundary How do we study the power of quantum computers compared to classical ones? Compelling problems.](https://reader035.fdocuments.us/reader035/viewer/2022081502/56649c745503460f949275c3/html5/thumbnails/4.jpg)
Heisenberg
C
~ What set of operators do we choose?
~ Require
![Page 5: GENERALIZED STABILIZERS Ted Yoder. Quantum/Classical Boundary How do we study the power of quantum computers compared to classical ones? Compelling problems.](https://reader035.fdocuments.us/reader035/viewer/2022081502/56649c745503460f949275c3/html5/thumbnails/5.jpg)
Examples
~ By analogy to the first, we can write any stabilizer as
~ And the state it stabilizes as
![Page 6: GENERALIZED STABILIZERS Ted Yoder. Quantum/Classical Boundary How do we study the power of quantum computers compared to classical ones? Compelling problems.](https://reader035.fdocuments.us/reader035/viewer/2022081502/56649c745503460f949275c3/html5/thumbnails/6.jpg)
Destabilizer, Tableaus, Stabilizer Bases
~ We have . What is ?
~ Collect all in a group,
~ A tableau defines a stabilizer basis,
![Page 7: GENERALIZED STABILIZERS Ted Yoder. Quantum/Classical Boundary How do we study the power of quantum computers compared to classical ones? Compelling problems.](https://reader035.fdocuments.us/reader035/viewer/2022081502/56649c745503460f949275c3/html5/thumbnails/7.jpg)
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
![Page 8: GENERALIZED STABILIZERS Ted Yoder. Quantum/Classical Boundary How do we study the power of quantum computers compared to classical ones? Compelling problems.](https://reader035.fdocuments.us/reader035/viewer/2022081502/56649c745503460f949275c3/html5/thumbnails/8.jpg)
C1 C2
![Page 9: GENERALIZED STABILIZERS Ted Yoder. Quantum/Classical Boundary How do we study the power of quantum computers compared to classical ones? Compelling problems.](https://reader035.fdocuments.us/reader035/viewer/2022081502/56649c745503460f949275c3/html5/thumbnails/9.jpg)
The Interaction Picture
![Page 10: GENERALIZED STABILIZERS Ted Yoder. Quantum/Classical Boundary How do we study the power of quantum computers compared to classical ones? Compelling problems.](https://reader035.fdocuments.us/reader035/viewer/2022081502/56649c745503460f949275c3/html5/thumbnails/10.jpg)
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 .
![Page 11: GENERALIZED STABILIZERS Ted Yoder. Quantum/Classical Boundary How do we study the power of quantum computers compared to classical ones? Compelling problems.](https://reader035.fdocuments.us/reader035/viewer/2022081502/56649c745503460f949275c3/html5/thumbnails/11.jpg)
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
![Page 12: GENERALIZED STABILIZERS Ted Yoder. Quantum/Classical Boundary How do we study the power of quantum computers compared to classical ones? Compelling problems.](https://reader035.fdocuments.us/reader035/viewer/2022081502/56649c745503460f949275c3/html5/thumbnails/12.jpg)
References
![Page 13: GENERALIZED STABILIZERS Ted Yoder. Quantum/Classical Boundary How do we study the power of quantum computers compared to classical ones? Compelling problems.](https://reader035.fdocuments.us/reader035/viewer/2022081502/56649c745503460f949275c3/html5/thumbnails/13.jpg)
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?
![Page 14: GENERALIZED STABILIZERS Ted Yoder. Quantum/Classical Boundary How do we study the power of quantum computers compared to classical ones? Compelling problems.](https://reader035.fdocuments.us/reader035/viewer/2022081502/56649c745503460f949275c3/html5/thumbnails/14.jpg)
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.
![Page 15: GENERALIZED STABILIZERS Ted Yoder. Quantum/Classical Boundary How do we study the power of quantum computers compared to classical ones? Compelling problems.](https://reader035.fdocuments.us/reader035/viewer/2022081502/56649c745503460f949275c3/html5/thumbnails/15.jpg)
Channels~ Define a Pauli channel as,
for Pauli operators
~ Define as a measure of its complexity.