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Decoupling with random quantum circuits

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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  1. Decoupling with random quantum circuits S Omar Fawzi (ETH Zürich) Joint work with Winton Brown (University College London)

  2. Random unitaries • Encoding for almost any quantum information transmission problem • Entanglement generation • Thermalization • Scrambling (black hole dynamics) • Uncertainty relations / information locking • Data hiding • … Decoupling

  3. 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

  4. 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

  5. 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

  6. 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?

  7. Random quantum circuits • Random gate on random pair of qubits • Complexity measures: • Number of gates • Depth

  8. 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)

  9. 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]

  10. 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)

  11. Proof steps n-s Recall: • Pure input ρ, no E system • Study decoupling directly S U n s E

  12. Proof setup Fourier coefficient Total mass on strings with support on S

  13. Evolution of mass dist.

  14. The Markov chain

  15. Putting things together Initial mass at level l Main technical contribution

  16. 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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