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)

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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

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)


Proof steps

n-s

Recall:

  • Pure input ρ, no E system

  • Study decoupling directly

S

U

n

s

E


Proof setup

Fourier coefficient

Total mass on strings with support on S


Evolution of mass dist.


The Markov chain


Putting things together

Initial mass at level l

Main technical contribution


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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