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Decoupling with random quantum circuits

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### Decoupling with random quantum circuits

S

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

Decoupling

Sc: n-s qubits

S cannot see correlations between A and E

Decoupling theorem: how large can s be?

U

A: n qubits

S: s qubits

E

Decoupling theorem

Sc: n-s qubits

[Schumacher, Westmoreland, 2001],…, [Abeyesinghe, Devetak, Hayden, Winter, 2009],…, [Dupuis, Berta, Wullschleger, Renner, 2012]

U

A: n qubits

S: s qubits

E

Decoupling theorem: examples

Sc: n-s qubits

- A Pure:
- Max. entanglement:
- k EPR pairs:

In this talk

U

A: n qubits

S: s qubits

E

Computational efficiency

- A typical unitary needs exponential time!
- Two-design is sufficient: O(n2) gates
- O(n) gates possible?
- Physics motivation:
- Time scale for thermalization
- Fast scramblers (black hole information)

- How fast can typical “local” dynamics decouple?

Random quantum circuits

- Random gate on random pair of qubits
- Complexity measures:
- Number of gates
- 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]
- Random circuit model:
e-approx two-design with O(n log(1/e)) gates

- Does NOT decouple unless Ω(n2)
Cannot use route

- Random circuit model:

More details [Brown, Poulin, soon]

Main result

n-s

Compare to Ω(n)

RQC’s with O(n log2n) gates decouple

Depth: O(log3n)

U

n

s

E

Almost tight

Compare to Ω(log n)

Conclusion

- Summary
- Random quantum circuits with O(n log2 n) gates and depth O(log3 n) decouple

- Open questions
- Depth improved to O(log n)?
- Quantum analogue of randomness extractors
- Explicit constructions of efficient unitaries?
- Number of unitaries?

- Geometric locality, d-dimensional lattice?
- Hamiltonian evolutions?

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