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 result

RQC’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?