Decoupling with
random quantum
circuits
Omar Fawzi (ETH Zürich)
Joint work with Winton Brown (University College London)
Random unitaries
• Encoding for almost any quantum information
transmission problem
• Entanglement generation
• Thermalization
• Scrambling (black hole dynamics)
• Uncertainty relations / information locking
• Data hiding
• …
Decoupling
Sc: n-s qubits
A: n qubits
r AE
U
S: s qubits
E
»
idS
Ä rE
s
2
S cannot see correlations between A and E
Decoupling theorem: how large can s be?
Decoupling theorem
Sc: n-s qubits
A: n qubits
r AE
U
S: s qubits
E
»
idS
Ä rE
s
2
[Schumacher, Westmoreland, 2001],…, [Abeyesinghe, Devetak,
Hayden, Winter, 2009],…, [Dupuis, Berta, Wullschleger, Renner, 2012]
Decoupling theorem:
examples
Sc: n-s qubits
A: n qubits
U
r AE
In this talk
S: s qubits
E
»
idS
Ä rE
s
2
• A Pure:
• Max. entanglement:
• k EPR pairs:
Computational efficiency
• A typical unitary needs exponential time!
• Two-design is sufficient: O(n2) gates
• O(n) gates possible?
• Physics motivation:
o Time scale for thermalization
o Fast scramblers (black hole information)
• How fast can typical “local” dynamics decouple?
Random quantum circuits
• Random gate on random pair of qubits
• Complexity measures:
o Number of gates
o Depth
Random quantum circuits
• RQCs of size O(n2) are approximate two-designs
[Harrow, Low, 2009]
• Approx two-designs decouple
[Szehr, Dupuis, Tomamichel, Renner, 2013]
=> RQCs of size O(n2) decouple
Objective: Improve to O(n)
Decoupling vs. approx.
two-designs
• Approx. two design ≠ decoupling
•
[Szehr, Dupuis, Tomamichel, Renner, 2013]
•
[Dankert, Cleve, Emerson, Livine, 2006]
o Random circuit model:
e-approx two-design with O(n log(1/e)) gates
o Does NOT decouple unless Ω(n2)
Cannot use route
n-s
n
r AE
U
s
E
Main result
» p S Ä rE
Compare to Ω(n)
RQC’s with O(n log2n) gates decouple
Depth: O(log3n)
Almost tight
Compare to Ω(log n)
n-s
n
r AE
U
s
E
» p S Ä rE
Proof steps
Recall:
o Pure input ρ, no E system
o Study decoupling directly
r (0)
S
r (t) r (t +1)
Proof setup
Fourier coefficient
Total mass on strings with
support on S
Evolution of mass dist.
Distribution of masses
IX
IZ
YI
XX
XZ
YY
ZX
ZZ
The Markov chain
Putting things together
Initial mass at level l
Main technical
contribution
Conclusion
• Summary
o Random quantum circuits with O(n log2 n) gates
and depth O(log3 n) decouple
• Open questions
o Depth improved to O(log n)?
o Quantum analogue of randomness extractors
• Explicit constructions of efficient unitaries?
• Number of unitaries?
o Geometric locality, d-dimensional lattice?
o Hamiltonian evolutions?