Ken Mochizuki (RIKEN)
Ken Mochizuki and Ryusuke Hamazaki, arXiv: 2207.12624 .

・post-selected open quantum systems
・PT symmetry breaking
・photonic experiments
introduction
・boson sampling problem
・motivation
・model and PT symmetry breaking
・short-time dynamical complexity transition results
・long-time dynamical complexity transition

unitary evolution of the total system
non-unitary evolution of the partial systemsystemenvironmentt : totals : systeme : environment

post selection:measurement on the environment⇒ choosing a specific outcome

various intriguing phenomenaabsent in isolated systems!

Parity-Time symmetry (PT symmetry)
PT symmetry of eigenstatespreserved (oscillation)：

PT symmetry of a Hamiltonianbroken (attenuation/divergence)：
C. M. Bender and S. Boettcher,
Phys. Rev. Lett, 80, 5243 (1998).

typical behavior in PT symmetric systems: unique transition in open systems!

drastic change of dynamics!

PT symmetric phase

PT broken phasethresholda parameterphoton source

direction of photon propagation ⇒ effective time
separation of paths ⇒ hoppings between lattices
detector

Xiao, Zhan, Bian, Wang, Zhang, Wang, Li,Mochizuki, Kim, Kawakami, Obuse, Sanders,Xue, Nature Physics 13, 1117 (2017).

PT symmetrictime-evolution operator

photon loss effect & post selection

conventional: single photon dynamics
many photons ⇒ boson sampling problem

PT brokeninjection ofmany photons

detection ofoutput photons

・・・・・・ byevolution
optical elements

Photon probabilities can be hard to compute by classical computers⇒ computational complexity / quantum supremacy

probability distribution of many photons：

can be hard to compute：the number of photons

properties of states,xxxx
hardness/easiness forcomputingsampling

examples：: random matrix ⇒ hard
: sparse/positive ⇒ easy

input : coherent/thermal state ⇒ easy
easy ⇔ classicalS. Aaronson, A. Arkhipov, STOC’11, 333.
S. R. Keshari, et al, PRL. 114, 060501 (2015).
W. Roga, M. Takeoka, Sci. Rep. 10, 14739 (2020).

＊hard/easy ⇔ exponential/polynomial computational time ofcharacterization of dynamics/phase by computational complexity

versatile platform ofboson sampling problem

relation between them? unique phenomenon inopen quantum systems

Open photonic systems can exhibit PT symmetry breaking transition
L. Xiao, X. Zhan, Z. H. Bian, K. K. Wang, X. Zhang, X. P. Wang, J. Li, Ken Mochizuki, D. Kim,N. Kawakami, W. Yi, H. Obuse, B. C. Sanders, and P. Xue, Nature Physics 13, 1117 (2017).

ishard/easy to computedepending uponphysical situations

computational complexity in PT symmetric open systems?

・post-selected quantum systems
・PT symmetry breaking
・photonic experiments
・boson sampling problem
・motivation(0) model and PT symmetry breaking(1) short-time dynamical complexity transition(2) long-time dynamical complexity transition

wave plates：changes of polarizations

beam displacers：position shift of photons
L. Xiao, X. Zhan, Z. H. Bian,
K. K. Wang, X. Zhang, X. P. Wang,
J. Li, Ken Mochizuki, D. Kim,
N. Kawakami, W. Yi, H. Obuse,
B. C. Sanders, and P. Xue,
Nature Physics 13, 1117 (2017).

Parity-Time (PT) symmetry：partially polarizing beam splitters：
polarization dependent photon losses⇒nonunitary dynamics

initial state-2-1

0+1+2-2-10+1+2-2-10+1+2Ken Mochizuki, D. Kim, and H. Obuse, Physical Review A 93, 062116 (2016).

PT symmetricPT broken

PT symmetry breaking
crucial effects upon
computational complexity!
threshold：photon loss effect & postselection⇒ nonunitary dynamics

・・・

・・・

1
easy!

2
easy!
PT broken
PT symmetric

Ken Mochizuki and Ryusuke Hamazaki, arXiv: 2207.12624.

・post-selected quantum systems
・PT symmetry breaking
・photonic experiments
・boson sampling problem
・motivation(0) model and PT symmetry breaking(1) short-time dynamical complexity transition(2) long-time dynamical complexity transition

Ken Mochizuki and Ryusuke Hamazaki, arXiv: 2207.12624.

probability distribution of easy todistinguishable particles computeactual distributionof many photons

threshold!

threshold!
computable within

initial state：
easy
easy
：neglectable overlap of single-particle distributions ⇒ distinguishable：deviation from the computable distribution due to quantum interference

PT symmetry breaking enlarges the computable region

Ken Mochizuki and Ryusuke Hamazaki, arXiv: 2207.12624.

single-particle dynamics with
PT symmetricballisticPT broken

diffusive

PT symmetry breaking leads to diffusive dynamics⇒ prolongation of transition times

Diffusive dynamics canbe derived analytically.

・post-selected quantum systems
・PT symmetry breaking
・photonic experiments
・boson sampling problem
・motivation(0) model and PT symmetry breaking(1) short-time dynamical complexity transition(2) long-time dynamical complexity transition

photon loss effect & postselection⇒ nonunitary dynamics

・・・

・・・

1
easy!

2
easy!
PT broken
PT symmetric

Ken Mochizuki and Ryusuke Hamazaki, arXiv: 2207.12624.

Ken Mochizuki and Ryusuke Hamazaki, arXiv: 2207.12624.

probability distribution ofdistinguishable particles

easy to compute
PT broken

computable within

PT symmetricPT broken
computable!

algorithm based on
the dominant eigenstate
PT symmetry breaking ⇒ The distribution becomes computable again

Ken Mochizuki and Ryusuke Hamazaki, arXiv: 2207.12624.

approximation based on
the dominant eigenstateubiquitous in nonunitary dynamics!

：p-th input state：q-th output statecomputable

In the PT broken phase, the true distribution approaches
a distribution computable through the dominant eigenstateKen Mochizuki and Ryusuke Hamazaki, arXiv: 2207.12624.

extending single-photonic non-Hermitian system into many-photon systemcomputational complexity of boson sampling problem
enhancement of classical nature
PT symmetry breaking
・prolongation of the threshold for the short-time complexity transition
・long-time complexity transition unique to open systems

easy!

easy!
PT broken
PT symmetriceasy ⇔ classically computablequantum system:
single photons

PT symmetricPT broken

L. Xiao, X. Zhan, Z. H. Bian, K. K. Wang, X. Zhang, X. P. Wang, J. Li, Ken Mochizuki, D. Kim,N. Kawakami, W. Yi, H. Obuse, B. C. Sanders, and P. Xue, Nature Physics 13, 1117 (2017).

probability distribution of single photons⇔intensity configuration of coherent light*Formulations
are different

classical system:
coherent light
A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov,
D. N. Christodoulides, and U. Peschel, Nature 488, 167 (2012)

Results are similar in both systems⇒ Is there a distinct feature?

Ken Mochizuki and Ryusuke Hamazaki, arXiv: 2207.12624.

quadratic dispersion

extraction of leading terms
extension of theintegral range：Hermite Polynomial：generating function

Moment generating function corresponds to that of Gaussian distribution

computableKen Mochizuki and Ryusuke Hamazaki, arXiv: 2207.12624.

・・・・・・
Relation between the computational complexity
and the entanglement entropy??
Behavior in more general non-unitary dynamics??
(time-dependent, without postselection, ・・・)