Strong simulation of tracking single photons with which-way-detectors in linear optics

Abstract Which-way-detectors (WWDs) are path-entangled detectors characterizing mutual exclusivity between path information and interference visibility in wave-particle duality experiments. We show surprisingly that WWDs allow to utilize single photons distinguishable in time domain to realize linear optical circuits where tracking their paths is exponentially hard for strong simulation analogous to rectangular lattice based Ising models. Distinguishable photons have scalability advantages of generation and detection compared with indistinguishable photons by promising both theoretical and experimental improvements in linear optical computing including boson sampling. We calculate strong simulation complexities by using variable elimination (VE) method for undirected graphs related to tensor network contraction for quantum circuits and recursive Feynman path-integral (RFPI) method to reduce space complexity. Two designs include either a single photon touring m times or m single photons propagating sequentially through an optical circuit composed of n beam splitters and phase shifters entangled with n WWDs. VE method for tracking results in undirected graphs matching with (2 m − 1) × ( n + 1) and m × ( n + 1) lattice Ising models with computational complexities of <?CDATA $\mathcal{O}(m\,n\,{2}^{\mathrm{min}(2\,m\,-\,1,\,n\,+\,1)})$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi mathvariant="script">O</mml:mi> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi>m</mml:mi> <mml:mspace width="0.17em" /> <mml:mi>n</mml:mi> <mml:mspace width="0.17em" /> <mml:msup> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> <mml:mrow> <mml:mi>min</mml:mi> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mn>2</mml:mn> <mml:mspace width="0.17em" /> <mml:mi>m</mml:mi> <mml:mspace width="0.17em" /> <mml:mo>−</mml:mo> <mml:mspace width="0.17em" /> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mspace width="0.17em" /> <mml:mi>n</mml:mi> <mml:mspace width="0.17em" /> <mml:mo>+</mml:mo> <mml:mspace width="0.17em" /> <mml:mn>1</mml:mn> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:mrow> </mml:msup> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> and <?CDATA $\mathcal{O}(m\,n\,{2}^{\mathrm{min}(m,\,n\,+\,1)})$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi mathvariant="script">O</mml:mi> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi>m</mml:mi> <mml:mspace width="0.17em" /> <mml:mi>n</mml:mi> <mml:mspace width="0.17em" /> <mml:msup> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> <mml:mrow> <mml:mi>min</mml:mi> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>,</mml:mo> <mml:mspace width="0.17em" /> <mml:mi>n</mml:mi> <mml:mspace width="0.17em" /> <mml:mo>+</mml:mo> <mml:mspace width="0.17em" /> <mml:mn>1</mml:mn> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:mrow> </mml:msup> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> in time and <?CDATA $\mathcal{O}({2}^{\mathrm{m}\mathrm{i}\mathrm{n}(2\;m\;-\;1,\;n\;+\;1)})$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi mathvariant="script">O</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:msup> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> <mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">m</mml:mi> <mml:mi mathvariant="normal">i</mml:mi> <mml:mi mathvariant="normal">n</mml:mi> </mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>2</mml:mn> <mml:mstyle> <mml:mspace width="thinmathspace" /> </mml:mstyle> <mml:mi>m</mml:mi> <mml:mstyle> <mml:mspace width="thinmathspace" /> </mml:mstyle> <mml:mo>−</mml:mo> <mml:mstyle> <mml:mspace width="thinmathspace" /> </mml:mstyle> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mstyle> <mml:mspace width="thinmathspace" /> </mml:mstyle> <mml:mi>n</mml:mi> <mml:mstyle> <mml:mspace width="thinmathspace" /> </mml:mstyle> <mml:mo>+</mml:mo> <mml:mstyle> <mml:mspace width="thinmathspace" /> </mml:mstyle> <mml:mn>1</mml:mn> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:math> and <?CDATA $\mathcal{O}({2}^{\mathrm{min}(m,\,n\,+\,1)})$?> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mi mathvariant="script">O</mml:mi> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:msup> <mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> <mml:mrow> <mml:mi>min</mml:mi> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mrow> <mml:mi>m</mml:mi> <mml:mo>,</mml:mo> <mml:mspace width="0.17em" /> <mml:mi>n</mml:mi> <mml:mspace width="0.17em" /> <mml:mo>+</mml:mo> <mml:mspace width="0.17em" /> <mml:mn>1</mml:mn> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:mrow> </mml:msup> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> in space for single and multi-photon based designs, respectively. We exploit RFPI method for m ≫ n to reduce space complexities to polynomial levels with respect to n and log m . Probability amplitude of specific cases of multi-photon design is represented in terms of Ising partition function with purely imaginary weights to characterize sampling complexity. Open issues about sampling complexity and experimental implementation of multi-WWD set-ups are discussed.