Decoupling with random quantum circuits

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

Decoupling

S cannot see correlations between A and E

Decoupling theorem: how large can s be?

U S: s qubits

E

A: n qubits Sc: n-s qubits

Decoupling theorem

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

U S: s qubits

E

A: n qubits Sc: n-s qubits

Decoupling theorem: examples

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

In this talk

U S: s qubits

E

A: n qubits Sc: n-s qubits

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 thermalizationo 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 gateso 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

More details [Brown, Poulin,

soon]

Main resultRQC’s with O(n log2n) gates decouple

Depth: O(log3n)

U s

E

nn-s

Compare to Ω(log n)

Compare to Ω(n)

Almost tight

Proof stepsRecall:

o Pure input ρ, no E systemo Study decoupling directly

U s

E

nn-s

S

Proof setup

Total mass on strings with support on S

Fourier coefficient

Evolution of mass dist.

IX IZ YI XX XZ YY ZX ZZ

Distribution of masses

The Markov chain

Putting things together

Main technical contribution

Initial mass at level l

Conclusion• Summary

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

• Open questionso 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?