Applications & constructions of random dynamics in quantum information theory YOSHIFUMI NAKATA THE UNIVERSITY OF TOKYO (UT) WITH C. HIRCHE (UAB), M. KOASHI (UT), AND A. WINTER(UAB) PRX 7, 021006 (2017) (SEE ALSO ARXIV:1609.07021) @ UTokyo-ANU Workshop on Quantum Control and Electronic Materials and Devices 

Outline 1. What is random dynamics? 2. Why should we care? 3. How to generate random dynamics by Q. circuits? 4. Conclusion & Outlook Take-home-message: 1. Random dynamics is extremely useful in QIP. 2. Main result: simple Q. circuit to generate random dynamics Experimental demonstration of many Q. protocols!! 

1. What is random dynamics? THE IDEA OF RANDOM UNITARY 

Concept of Random Unitary  Unitary time-evolution = most fundamental in Q. mechanics (e.g. −)  Random dynamics (in this talk) = random unitary dynamics.  does not mean the dynamics in open systems. {1 , 2 , 3 , … … , 6 } 1st : 3 2nd: 5 3rd: 1 4th: 6 Each with probability 1/6 E.g) Random unitary uniformly distributed over {1 , 2 , 3 , … … , 6 } 

Concept of Random Unitary 1st: 2nd: 3rd: 4th: Each with the same probability E.g) Haar random unitary uniformly distributed over the unitary group {All unitaries}  Unitary time-evolution = most fundamental in Q. mechanics (e.g. −)  Random dynamics (in this talk) = random unitary dynamics.  does not mean the dynamics in open systems. 

2. Why should we care about Haar random unitary? RANDOM UNITARY IS EXTREMELY USEFUL! 

Applications of a Haar random unitary Haar random unitary is very useful in QIP and in fundamental physics. Applications in QIP 1. Q. communication theory [Hayden et al ‘07] 2. Q. computation [Boixo et al (aka google) ‘16] 3. Randomized benchmarking [Knill et.al ‘08] 4. Quantum sensing [Oszmaniec et.al ‘16] 5. Quantum algorithm Quantum communication Quantum computation In fundamental physics 1. Disordered systems 2. Pre-thermalization [Reimann ‘16] 3. Q. black holes -scrambling- [Hayden&Preskill ‘07] 4. Q. chaos -OTOC- [Roberts&Yoshida ‘16] 5. Quantum duality? [Kitaev ‘15] 

Application 1: Haar random unitary in Q communication  Q. communication = sending “information” over quantum channels. Haar random unitary Coherent State merging (aka Fully Quantum Slepian-Wolf protocol) Mother protocol Entanglement distillation Noisy Quantum Teleportation Noisy Superdense coding Father protocol Quantum capacity Entanglement assisted classical capacity Quantum reverse Shannon theory Quantum multiple access capacities Quantum broadcast channels Distributed compression Family tree of information protocols Devetak, Harrow, &Winter ’03: Devetak & Yard ’06: Abeyesinghe, Devetak, Hayden, & Winter ’06: Dupuis, Hayden, & Li ’10: Dupuis, Berta, Wullschleger, & Renner ’14 See Hayden’s tutorial talk in QIP2011 

Application 2: Random unitary in quantum computation  Hopefully, quantum computer performs better than classical!  Shor’s algorithm, Grover’s searching algorithm, etc….  So far, only a tiny quantum computer (~20 qubits) is available (google & IBM).  without quantum error correction….. Quantum computational supremacy [Preskill ‘12]: Can a tiny quantum computer outperform the best-available supercomputer? Random unitary on ~50qubits will also do the job [Boixo et al (aka google) ’16: Man and Bremner ‘17]. IQP circuit [Bremner, Jozsa, and Shepherd ’10], Boson sampling [Aaronson and Arkhipov ’11], DQC1 [Morimae, Fujii, and Fitzsimons `13] 

Why not try to realize a Haar random unitary in experiment!! A standard tool in quantum information Haar random unitary is very useful in QIP and in fundamental physics. Applications of a Haar random unitary Applications in QIP 1. Q. communication theory [Hayden et al ‘07] 2. Q. computation [Boixo et al (aka google) ‘16] 3. Randomized benchmarking [Knill et.al ‘08] 4. Quantum sensing [Oszmaniec et.al ‘16] 5. Quantum algorithm In fundamental physics 1. Disordered systems 2. Pre-thermalization [Reimann ‘16] 3. Q. black holes -scrambling- [Hayden&Preskill ‘07] 4. Q. chaos -OTOC- [Roberts&Yoshida ‘16] 5. Quantum duality? [Kitaev ‘15] A key to understanding complex QMBSs 

3. How to construct random unitary? HAAR IS PRICELESS….. LITERALLY. 

The price of random unitary  Haar random unitary = uniformly distributed random unitary  On qubits, it is necessary to use at least () quantum gates…very inefficient…  Better to think of approximate ones.  Unitary -design = approximation of a Haar random unitary  Unitary 1-design has the same average properties as Haar.  Unitary 2-design for variance ………  In many applications, unitary -designs for small suffice ( = 2 or 4 etc). How to generate unitary -designs on qubits by Q. circuits? 

The price of unitary -design?  When t=2, many constructions [e.g. DiVincenzo et al ‘02, Cleve et al ‘15, Nakata et al ‘17]  Only two results were known when ≥ 3. How to generate unitary -designs on qubits by Q. circuits? Harrow and Low '09 Brandao et al '13 Our result '17 Quantum circuit Q. Fourier Transformation + Toffoli-type gates Local random circuits Hadamard gates + random diagonal gates # of gates (33) [Brodsky & Hoory ’13] (102) (2) Works for ≤ (/ log ) ≤ poly() = ( ) Table: Quantum circuits to implement unitary -design on qubits 

The price of unitary -design? Harrow and Low '09 Brandao et al '13 Our result '17 Quantum circuit Q. Fourier Transformation + Toffoli-type gates Local random circuits Hadamard gates + random diagonal gates # of gates (33) [Brodsky & Hoory ’13] (102) (2) Works for ≤ (/ log ) ≤ poly() = ( ) Table: Quantum circuits to implement unitary -design on qubits How to generate unitary -designs on qubits by Q. circuits?  When t=2, many constructions [e.g. DiVincenzo et al ‘02, Cleve et al ‘15, Nakata et al ‘17]  Only two results were known when ≥ 3. 

Quantum circuits for -design  Main idea = to repeat Hadamard gates and random diagonal gates diag 1, ( ∈ {0, 2 +1 , … , 2 +1 }) diag(1, 1,1, ℓ ) (ℓ ∈ {0, 2 /2 +1 , 4 /2 +1 , … , 2 /2 +1 }). × Repeat O() times 1 1 2 3 12 13 1 23 2 1 1 Quantum circuits for unitary t-design −1, 

What is it good for?  Most efficient quantum circuits for unitary -designs ( 2 gates).  The circuit consists of mostly diagonal gates (Hadamard gates are only exception).  Using this, a number of Q. protocols may be realized!  Quantum communication, quantum computation, and fundamental physics. diag 1, ( ∈ {0, 2 +1 , … , 2 +1 }) diag(1, 1,1, ℓ ) (ℓ ∈ {0, 2 /2 +1 , 4 /2 +1 , … , 2 /2 +1 }) × Repeat O() times 1 2 3 13 1 23 2 Quantum circuits for unitary t-design 12 −1, 

4. Conclusion & Outlook WHAT’S DONE AND WHAT’S NEXT? 

Conclusion Take-home-message: 1. A random unitary is extremely useful in QIP. 1. Many Q. protocols 2. Exotic physics in QMBSs. 2. We proposed simple Q. circuits generate a random unitary. The first step towards experimental demonstration!! Mother protocol Entanglement distillation Noisy Quantum Teleportation Noisy Superdense coding Father protocol Quantum capacity Entanglement assisted classical capacity Quantum reverse Shannon theory Quantum multiple access capacities Quantum broadcast channels Coherent State merging Distributed compression 

Outlook  Experimental demonstrations of random unitary  More applications?  Symmetric random unitary [YN & Wakakuwa, in prep.]  Another implementation of designs?  Exact unitary designs [Okuda & YN, in prep.]  Random unitary in CV systems????  Not well-studied b/c they’re scary infinite dimensional systems…. THANK YOU FOR YOUR ATTENTION! PRX 7, 021006 (2017) (SEE ALSO ARXIV:1609.07021) Mother protocol Entanglement distillation Noisy Quantum Teleportation Noisy Superdense coding Father protocol Quantum capacity Entanglement assisted classical capacity Quantum reverse Shannon theory Quantum multiple access capacities Quantum broadcast channels Coherent State merging Distributed compression