### Generalized Stabilizers

```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
~ Collect all
. What is
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
~ Any state can be represented
~ Any operation can be simulated
- Unitary gates
- Measurements
- Channels
can be written as the pair
C1
C2
The Interaction Picture
for complex numbers φi j and Pauli operat ors B i 2 Gn . We also deﬁne ⇤(
complexity. Not e t hat any quant um operat ion can be expressed uniquely (up
a Pauli channel.
Update Efficiencies
T heor em 13. Suppose we have a generalized stabilizer on n-qubits ⌧= (χ,
~ For update efficiencies
can be done
with the following efficiency:
ministic
for quantum
operations,
• Cli ↵ ord gates: O(n) time
• Pauli measurements: O ⇤(χ) n + n 2 time
⇣
⌘
p
• Pauli channels E: O ⇤(χ) ⇤(E) n + ⇤(E)n 2 time
Note
that ⇤( ⌧) = 11997
if and only if we have a stabilizer state. I n that case, the
~ Gottesman-Knill
ment is recovered.
On stabilizer states, we have the update efficiencies
Proof. We prove each part in t urn.
- Clifford gates:
• Say
we have Cli↵ord circuit C. T hen,
- Pauli measurements:
0
X
C ⌧C = C @ χ i j
†
~ Note the correspondence when
.
X
1
i⇢
S
j
A C†
ij
†
†
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
R efer ences
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
P. W . Shor, SIA M J. Comput ., 26(5), 1484 (1994).
R. Josza and N. Linden, Proc. R. Soc. London A 459, 2011 (2003).
M . Nielson and I. Chuang, Quantum Computation and Quantum I nfor mation, Cambridge University Press, 2000.
G. Passant e, et. al., Phys. Rev. A 84, 044302 (2011).
A . M ari, J. Eisert , Phys. Rev. Let t . 109, 230503 (2012).
D. Got t esman, CalTech Ph.D. t hesis, arXiv:quant -ph/ 9705052 (1997).
D. Got t esman, Proceedings of t he X X I I Int ernat ional Colloquium on Group T heoret ical M et hods in Physics, edit ed by
S.P. Corney et al. (Int ernat ional Press, Cambridge, M A , 1999), p. 32.
[8] M . Van den Nest , Quant . Inf. Comp. 10, 0258 (2010).
[9] S. A aronson and D. Got t esman, Phys. Rev. A 70, 052328 (2004).
[10] S. A nders and H. Briegel, P, Phys. Rev. A 73, 022334 (2006).
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
~ We’ll measure the complexity of
~ The complexity of a state
~ Simulating measurement of
by
can be defined as
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.
```