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
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!!
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 }
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.
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]
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]
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
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?
[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
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.