Ken Mochizuki,Ryusuke Hamazaki
＠RIKENKen Mochizuki and R.Hamazaki, in preparation.

・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

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:
drastic change of dynamics!

PT symmetric phase

PT broken phasethresholda parameter

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

PT symmetricPT broken

classical
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).

quantum

intensity configuration of coherent light⇔probability distribution of single photons

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

photon source

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

detector

time-evolution operator

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).

photon loss effect & post selection
conventional: single photons
many photons ⇒ non-classical behavior?

single photons ⇒ many photons: boson sampling problem
injection ofdetection ofmany photons
output photons
evolution ofcreation operators：

・・・
・・・
evolution by

optical elements
quantum walks：

Probability distribution of photons can be hard to compute⇒ computational complexity/quantum supremacy

probability distribution of many photons：：# photons：# states：p-th input/output state

・・・

general form:
hard for me!

: random matrix ⇒ hard to compute
S. Aaronson, A. Arkhipov,: positive matrix ⇒ easy
STOC’11, 333.
Deshpande, et al,: local and : small ⇒ easy A.
PRL. 121, 030501 (2018).
＊hard/easy：within polynomial time ofphysical situation ⇔ hardness for computing the distribution
may be possible
characterization of dynamics/phase by computational complexity

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).

extension from single photons to many photons

generally hard, buteasy in some casesboson sampling problem

computational complexity in PT symmetric non-Hermitian systems?

・PT symmetry breaking
・photonic experiments
・boson sampling problem
・motivation
・model and PT symmetry breaking
・short-time dynamical complexity transition
・long-time dynamical complexity transition

wave plates：changes of polarizations

L. Xiao, X. Zhan, Z. H. Bian,
beam displacers：position shift of photons
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).

partially polarizing beam splitters：
polarization dependent photon losses⇒nonunitary dynamics

Gain effect of photons is also considered fortheoretical simplicity, while only loss exists.

PT symmetry：initial state-2-1

0+1+2-2-1

0+1+2-2-1

0+1+2

Ken Mochizuki, D. Kim, and H. Obuse, PRA 93, 062116 (2016). Ken Mochizuki, D. Kim, N. Kawakami, and H. Obuse, PRA 102, 062202 (2020).

weak loss

Hermitianthreshold：strong loss

PT symmetricPT broken

・PT symmetry breaking・photonic experiments
・boson sampling problem
・motivation
・model and PT symmetry breaking
・short-time dynamical complexity transition
・long-time dynamical complexity transition

hard tocomputeactual distributionof many photons

probability distribution of easy to
distinguishable particles computetransition!

transition!

computable within→computable distribution
initial state：：neglectable overlap of single-particle distributions：deviation from the computable distribution due to quantum interference

PT symmetry breaking enlarges the computable region

Ken Mochizuki, D. Kim, and H. Obuse, PRA 93, 062116 (2016).
Ken Mochizuki, D. Kim, N. Kawakami, and H. Obuse, PRA 102, 062202 (2020).

ballisticballistic

strong loss

weak loss
Hermitiandiffusive

PT symmetricPT broken

quadratic dispersion

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

Moment generating function corresponds to that of Gaussian distribution

PT-symmetric phase：ballistic dynamics

initial state：
PT symmetricPT broken

PT-broken phase：
diffusive dynamics

PT symmetry breaking leads to ballistic dynamics⇒ quadratic system-size dependence ∴prolongation of transition times

・PT symmetry breaking
・photonic experiments
・boson sampling problem
・motivation
・model and PT symmetry breaking
・short-time dynamical complexity transition
・long-time dynamical complexity transition

probability distribution ofdistinguishable particles

easy to computablecomputablewithinadditional transition
with no counterpart
in unitary dynamics!

PT symmetricPT broken

PT symmetry breaking ⇒ The distribution becomes computable again

approximation based on
the dominant eigenstate：p-th input state：q-th output statecomputable

In the PT broken phase, the true distribution approaches
a distribution computable through the dominant eigenstatecomputable

The true distribution
can be approximatedwithin the precisiontransition time:

Ken Mochizuki and R.Hamazaki, in preparation.

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

easy ⇔ classically computable