**Early work (1972-1984)**

Yoshihisa Yamamoto’s early work includes the fabrication and characterization of a liquid-core single-mode fiber, a single-mode metal-clad dielectric waveguide on Si substrate and a tapered directional coupler for single-mode fiber splicing (thesis work at the University of Tokyo, Japan, from 1972 to 1978). Those experimental studies demonstrated the fundamental principles of tight optical field confinement by surface plasmons [1] and loss-less adiabatic evolution of photonic wavefunctions [2]. After joining the NTT Basic Research Laboratories (Tokyo, Japan), he has proposed for the first time the two optical fiber communication systems: coherent optical communication [3] and optical amplifier on-line repeaters [4], which constitute a contemporary optical communication platform supporting, for example, under-sea and terrestrial internet links. During this period, he has also focused his research on the quantum noise of laser oscillators [5,6] and injection-locked oscillators [7].

[1] Y. Yamamoto, T. Kamiya, and H. Yanai, "Propagation characteristics of partially metal clad optical waveguide: metal clad optical strip line," Appl. Opt. 14, 322-326 (February 1975).

[2] Y. Yamamoto, Y. Naruse, T. Kamiya, and H. Yanai, "A Large-tolerant single mode fiber coupler with a tapered structure," Proc. IEEE 64, 1013-1014 (June 1976).

[3] Y. Yamamoto and T. Kimura, "Coherent optical fiber transmission systems," IEEE J. Quantum Electron. QE-17, 919-935 (June 1981).

[4] Y. Yamamoto, "Noise and error rate performance of semiconductor laser amplifiers in PCM-IM optical transmission systems," IEEE J. Quantum Electron. QE-16, 1073-1081 (October 1980).

[5] Y. Yamamoto, "AM and FM quantum noise in semiconductor lasers - Part I: Theoretical analysis," IEEE J. Quantum Electron. QE-19, 34-46 (January 1983).

[6] Y. Yamamoto, S. Saito, and T. Mukai, "AM and FM quantum noise in semiconductor lasers - Part II: Comparison of theoretical and experimental results for AlGaAs lasers," IEEE J. Quantum Electron. QE-19, 47-58 (January 1983).

[7] H. A. Haus and Y. Yamamoto, "Quantum noise of an injection-locked laser oscillator," Phys. Rev. A 29, 1261-1274 (March 1984).

**Squeezed states and quantum non-demolition measurements (1985-1993) **

A standard method of generating a squeezed state of light is the use of phase sensitive amplification/deamplification in optical degenerate parametric amplifier or degenerate four-wave mixer. Yamamoto and his colleagues proposed for the first time an alternative scheme for squeezed state generation using a negative-feedback oscillator, in which the intensity noise is reduced to below shot noise limit [8,9]. A similar technique is also capable of reducing the laser spectral linewidth to below the Schawlow-Townes limit [10]. For the case of a semiconductor laser, in particular, quantum non-demolition (QND) measurements of photon number and negative feedback to injection current can be naturally and inherently realized in the dynamical coupling between junction voltage and junction current via a high-impedance current source. This is the principle of a high-impedance suppression of pump fluctuation and amplitude squeezing of a semiconductor laser [11,12]. This theoretical prediction was experimentally demonstrated for the first time in 1987 [13]. Even though a degree of squeezing was less than 0.5 dB in the initial experiment, much larger squeezing of more than 10 dB as well as the quantum correlation between junction voltage and intensity fluctuations were demonstrated in later experiments [14,15]. Subsequently, the squeezed states of light from diode lasers were applied to enhance the signal-to-noise ratio in FM spectroscopy of cold atoms [16-18] and optical interferometers [19].

A concept of quantum non-demolition (QND) measurement (or back action evading measurement) was coined in early 1970’s as a means of gravitational wave detection using a free mass antenna or mechanical harmonic oscillator. Yamamoto and his colleagues proposed a practical scheme for QND measurement of photon number using cross-phase modulation in a Kerr nonlinear medium [20]. Control of quantum states of light by QND measurements was subsequently studied for lasers [21] and parametric oscillators [22]. The use of soliton collision in an optical fiber was analyzed as a means of QND measurement of photon number [23] and experimentally demonstrated [24].

The research fields on squeezed states and QND measurements during this period were reviewed in the two articles [25,26].

[8] Y. Yamamoto, N. Imoto, and S. Machida, "Amplitude squeezing in a semiconductor laser using quantum nondemolition measurement and negative feedback," Phys. Rev. A 33, 3243-3261 (May 1986).

[9] H. A. Haus and Y. Yamamoto, "Theory of feedback-generated squeezed states," Phys. Rev. A 34, 270-292 (July 1986).

[10] Y. Yamamoto, O. Nilsson, and S. Saito, "Theory of a negative frequency feedback semiconductor laser," IEEE J. Quantum Electron. QE-21, 1919-1928 (December 1985).

[11] Y. Yamamoto, S. Machida, and O. Nilsson, "Amplitude squeezing in a pump-noise-suppressed laser oscillator," Phys. Rev. A 34, 4025-4042 (November 1986).

[12] Y. Yamamoto and S. Machida, "High-impedance suppression of pump fluctuation and amplitude squeezing in semiconductor lasers," Phys. Rev. A 35, 5114-5130 (June 1987).

[13] S. Machida, Y. Yamamoto, and Y. Itaya, "Observation of amplitude squeezing in a constant-current-driven semiconductor laser," Phys. Rev. Lett. 58, 1000-1003 (March 1987).

[14] W. H. Richardson, S. Machida, and Y. Yamamoto, "Squeezed photon number noise and sub-poissonian electrical partition noise in a semiconductor laser," Phys. Rev. Lett. 66, 2867-2870 (June 1991).

[15] W. H. Richardson and Y. Yamamoto, "Quantum correlation between the junction-voltage fluctuation and the photon-number fluctuation in a semiconductor laser," Phys. Rev. Lett. 66, 1963-1966 (April 1991).

[16] S. Lathi, S. Kasapi, and Y. Yamamoto, "Phase-sensitive frequency-modulation noise spectroscopy with a diode laser", Opt. Lett. 21, 1600-1602 (October 1996).

[17] S. Kasapi, S. Lathi, and Y. Yamamoto, "Sub-shot-noise FM noise spectroscopy of trapped rubidium atoms", J. Opt. Soc. Am. B 15, 2626-2630 (October 1998).

[18] S. Kasapi, S. Lathi, and Y. Yamamoto, "Sub-shot-noise frequency-modulation spectroscopy by use of amplitude-squeezed light from semiconductor lasers," J. Opt. Soc. Am. B 17, 275-279 (February 2000).

[19] S. Inoue and Y. Yamamoto, "Gravitational wave detection using dual input Michelson interferometer", Phys. Lett. A 236, 183-187 (December 1997).

[20] N. Imoto, H. A. Haus, and Y. Yamamoto, "Quantum nondemolition measurement of the photon number via the optical Kerr effect," Phys. Rev. A 32, 2287-2292 (October 1985).

[21] M. Kitagawa, N. Imoto, and Y. Yamamoto, "Realization of number-phase minimum-uncertainty states and number states by quantum nondemolition measurement," Phys. Rev. A 35, 5270-5273 (June 1987).

[22] K. Watanabe and Y. Yamamoto, "Quantum correlation and state reduction of photon twins produced by a parametric amplifier," Phys. Rev. A 38, 3556-3565 (October 1988).

[23] H. A. Haus, K. Watanabe, and Y. Yamamoto, "Quantum nondemolition measurement of optical solitons," J. Opt. Soc. Am. B 6, 1138-1148 (June 1989).

[24] S. R. Friberg, S. Machida, and Y. Yamamoto, "Quantum-nondemolition measurement of the photon number of an optical soliton," Phys. Rev. Lett. 69, 3165-3168 (November 1992).

[25] Y. Yamamoto, S. Machida, and W. H. Richardson, "Photon number squeezed states in semiconductor lasers," Science 255, 1219-1224 (March 1992).

[26] P. D. Drummond, R. M. Shelby, S. R. Friberg, and Y. Yamamoto, "Quantum solitons in optical fibers", Nature 365, 307-313 (September 1993).

**Semiconductor cavity QED and Bose-Einstein condensation of exciton-polaritons (1988-2016) **

Spontaneous emission of an atom is not an inherent property of an atom but the consequence of atom-vacuum field interaction. Therefore, if the spectral and spatial distributions of a vacuum field at a location of the atom are modified by cavity walls, the rate and radiation pattern of a spontaneous emission can be altered. Such a technique of modulating a spontaneous emission rate and radiation pattern by use of cavity walls is called cavity quantum electrodynamics (cavity QED). The first cavity QED experiment for a semiconductor quantum well exciton, which is a bound-pair of electron and hole, with a semiconductor planar microcavity was reported in 1991[27]. This first experiment of semiconductor cavity QED was performed in a so-called weak coupling regime or low-Q regime. That is, the spontaneous emission is still an irreversible process with a modified but exponential decay due to larger cavity loss than an exciton-field interaction strength. In 1992, C. Weisbuch and his colleagues observed a partially reversible spontaneous emission or split emission spectrum of a quantum well exciton in a similar but lower-loss cavity structure. The elementary excitation (or quasi-particles) generated in this second experiment is called an exciton-polariton, which is a hybridized excitation of exciton and photon. A comprehensive review on semiconductor cavity QED experiments in both low-Q and high-Q regimes was presented in 1993 [28].

In 1996, we theoretically predicted that the exciton-polaritons in a semiconductor microcavity can be condensed into a ground state by a similar principle of Bose-Einstein condensation [29]. Due to the open-dissipative nature of a semiconductor microcavity, the stimulated scattering of high-energy excitons into the ground state exciton-polariton must exceed the radiation decay rate of the ground state exciton-polariton in order to sustain a macroscopic papulation in the ground state. A detailed theoretical analysis of such dynamical condensation processes was presented in 1999 [30].

Observation of a macroscopic population and coherent exciton-polariton wavefunction in the ground state were successfully demonstrated via g(2) measurement [31]. Observation of the exciton-polariton condensation and the conventional photon lasing in the same microcavity structure was also reported [32]. In 2006, the degenerate Bose-Einstein distribution of exciton-polaritons was finally observed [33]. The measured chemical potential of an exciton-polariton gas was indeed detuned from the ground state energy in less than kBT. In 2007, we also observed the dynamical condensation in mutually coupled exciton-polaritons confined in a square lattice potential [34]. We observed not only s-wave superfluid state but also p-wave superfluid state. Subsequently, we obtained two more evidences of the exciton-polariton condensation effect, which are the observation of a linear (phonon-like) dispersion spectrum of excited states, a so-called Bogoliubov excitation spectrum [35] and the observation of a vortex-antivortex bound pair, i.e. the elementary excitation of a two-dimensional Berezinskii-Kosterlitz-Thouless (BKT) phase [36]. Two review articles on the exciton-polariton condensation were published in 2010 and 2016 [37,38].

More recently, an exciton-polariton condensate was produced with the current injection across a pn junction [39]. This experiment opened a door toward a practical exciton-polariton light source.

One of the promising applications for exciton-polariton condensates is the quantum simulation of many body systems with strong interactions. One example is the physics of BKT phase transition and dynamics of vortices and vortex-pairs in a two-dimensional bosonic system [40,41]. Another example is the physics of anyons and dynamics of braiding and topological qubit-qubit interaction [38]. It is not straightforward to study those many body physics in a pure light wave system nor pure matter wave system due to its very weak nonlinearity and limited spatial coherence, respectively. Exciton-polariton condensates offer relatively strong nonlinear interaction and large spatial coherence simultaneously, so that they are unique candidates of future quantum simulators for many body physics, condensed-matter physics and statistical mechanics.

[27] G. Björk, S. Machida, Y. Yamamoto, and K. Igeta, “Modification of spontaneous emission rate in planar dielectric microcavity structures,” Phys. Rev. A 44, 669-681 (July 1991).

[28] Y. Yamamoto and R. E. Slusher, “Optical processes in microcavities,” Physics Today 46, 66-73 (June 1993).

[29] A. Imamoglu, R. J. Ram, S. Pau, and Y. Yamamoto, “Nonequilibrium condensates and lasers without inversion: Exciton-polariton lasers,” Phys. Rev. A 53, 4250-4253 (June 1996).

[30] F. Tassone and Y. Y amamoto, “Exciton-exciton scattering dynamics in a semiconductor microcavity and stimulated scattering into polaritons,”Phys. Rev. B 59, 10830-10842 (April 1999).

[31] H. Deng, G. Weihs, C. Santori, J. Bloch, and Y. Yamamoto, “Condensation of semiconductor microcavity exciton polaritons,” Science 298, 199-202 (October 2002).

[32] H. Deng, G. Weihs, D. Snoke, J. Bloch, and, Y. Yamamoto, “Polariton lasing vs. photon lasing in a semiconductor microcavity,” PNAS 100, 15318-15323 (December 2003).

[33] H. Deng, D. Press, S. Goetzinger, G. S. Solomon, R. Hey, K. H. Ploog, and Y. Yamamoto, “Quantum degenerate exciton-polaritons in thermal equilibrium,” Phys. Rev. Lett. 97, 146402 (October 2006).

[34] C. W. Lai, N. Y. Kim, S. Utsunomiya, G. Roumpos, H. Deng, M. D. Fraser, T. Byrnes, P. Recher, N. Kumada, T. Fujisawa, Y. Yamamoto, “Coherent zero-state and π-state in an exciton-polariton condensate array,” Nature 450, 529-532 (November 2007).

[35] S. Utsunomiya, L. Tian, G. Roumpos, C. W. Lai, N. Kumada, T. Fujisawa, M. Kuwata-Gonokami, A. Löffler, S. Höfling, A. Forchel, and Y. Yamamoto “Observation of Bogoliubov excitations in exciton-polariton condensates,” Nature Physics 4, 700-705 (September 2008).

[36] G. Roumpos, M. D. Fraser, A. Löffler, S. Höfling, A. Forchel, and Y. Yamamoto, “Single vortex–antivortex pair in an exciton-polariton condensate,” Nature Phys. 7, 129–133, (February 2011)

[37] H. Deng, H. Haug, and Y. Yamamoto, “Exciton-polariton Bose-Einstein condensation,” Rev. Mod. Phys. 82, 1489-1537 (May 2010).

[38] M. D. Fraser, S. Höfling, and Y. Yamamoto, “Physics and applications of exciton–polariton lasers,” Nature Materials/Commentary,” Nature Materials 15, 1049-1052 (October 2016).

[39] D. Schneider, A. Rahimi-Iman, N. Y. Kim, J. Fischer, I. G. Savenko, M. Amthor, M. Lermer, A. Wolf, L. Worschech, V. D. Kulakovskii, I. A. Shelykh, M. Kamp, S. Reitzenstein, A. Forchel, Y. Yamamoto, and S. Höfling, “An electrically pumped polariton laser,” Nature 497, 348-352 (May 2013).

[40] G. Roumpos, M. Lohse, W. H. Nitsche, J. Keeling, M. H. Szymańska , P. B. Littlewood, A. Löffler, S. Höfling, L. Worschech, A. Forchel, and Y. Yamamoto, “Power-law decay of the spatial correlation function in exciton-polariton condensates,” PNAS 109, 6467-6472 (April 2012).

[41] W. H. Nitsche, N. Y. Kim, G. Roumpos, C. Schneider, M. Kamp, S. Höfling, A. Forchel, and Y. Yamamoto, “Algebraic order and the Berezinskii-Kosterlitz-Thouless transition in an exciton-polariton gas,” Phys. Rev. B 90, 205430 (November 2014).

**Single photon and entangled photon-pair sources for quantum communication and information processing (1993-2006) **

A standard laser produces a coherent state of light or statistical mixture of coherent states, in which the number of photons fluctuates from pulse to pulse according to Poisson statistics or super-Poisson statistics. For many applications of lasers, such Poisson or super-Poisson photon number fluctuation does not impose serious drawbacks on system performance. However, for other and specifically quantum information applications such as quantum key distribution and photonic qubit based quantum information processing, it is preferred that a photon source produces one and only one photon per pulse, which is called a deterministic or heralded single photon source. We have proposed a concept of such deterministic or heralded single photon source for the first time in 1994 [42]. In this proposal, a single electron and a single hole are injected alternately into a central quantum well via the Coulomb blockade effect in a p-i-n tunnel junction. We named such a device as a single photon turnstile device, which was experimentally realized in 1999 [43]. Although this device generates a deterministic single photon, the device operates successfully only at extremely low temperatures, typically 10mK. A quantum efficiency of extracting single photons from this device is also very low. In order to increase an operational temperature, we switched a physical system from quantum wells to quantum dots [44,45]. We also incorporated a monolithic microcavity structure to enhance a quantum efficiency of extracting single photons [46].

Single photon wavepackets produced by the spontaneous emission in quantum dots have normally inhomogeneous amplitude and/or phase profiles, so that they are not identical with each other. Those single photons are called “distinguishable single photons” and do not feature a so-called two photon interference effect, which is a crucial requirement for many quantum information applications. If a quantum dot is resonantly excited by a pump laser pulse, a turn-on delay time of single photon wavepackets is considerably shorter than the dephasing time of an electron-hole pair inside a quantum dot. If a quantum dot is embedded inside a monolithic microcavity and spontaneous emission lifetime is reduced by the cavity confinement effect, a turn-off decay time of single photon wavepackets is also considerably shorter than the dephasing time of an electron-hole pair. When these two conditions are fulfilled, generated single photon pulses are Fourier-transform-limited and feature a two-photon interference effect. Such indistinguishable single photon generation from a quantum dot in monolithic microcavity was first realized in 2002 [47]. Another useful photon state for quantum information applications is a deterministic entangled photon-pair, so called an EPR-Bell pair. We have proposed such a deterministic EPR-Bell pair source by utilizing the polarization (spin) selection rule of bi-exciton to exciton cascade mission process in a quantum dot [48].

Finally, we studied various quantum information systems with those single photon sources. A gate-model quantum computer architecture with deterministic and indistinguishable single photons were theoretically studied [49]. A quantum key distribution based on BB84 protocol with deterministic single photons were experimentally demonstrated [50]. Two semiconductor quantum dot excited by resonant light pulses, and their two generated single photon pulses have very short duration of 10 - 100 ps and mutually interfere with each other [51].

[42] A. Imamoglu and Y. Yamamoto, “Turnstile device for heralded single photons: Coulomb blockade of electron and hole tunneling in quantum confined p-i-n heterojunctions,” Phys. Rev. Lett. 72, 210-213 (January 1994).

[43] J. Kim, O. Benson, H. Kan, and Y. Yamamoto, “A single-photon turnstile device,” Nature 397, 500-503 (February 1999).

[44] C. Santori, M. Pelton, G. S. Solomon, Y. Dale, and Y. Yamamoto, “Triggered single photons from a quantum dot,” Phys. Rev. Lett. 86, 1502-1505 (February 2001).

[45] S. Kako, C. Santori, K. Hoshino, S. Gotzinger, Y. Yamamoto and Y. Arakawa, “A gallium nitride single-photon source operating at 200 K,” Nature Materials 5, 887-892 (November 2006).

[46] M. Pelton C. Santori, J. Vučković, B. Y. Zhang, G. S. Solomon, J. Plant, and Y. Yamamoto, “Efficient source of single photons: A single quantum dot in a micropost microcavity,” Phys. Rev. Lett. 89, 233602 (December 2002).

[47] C. Santori, D. Fattal, J. Vučković, G. S. Solomon, and Y. Yamamoto, “Indistinguishable photons from a single-photon device,” Nature 419, 594-597 (October 2002).

[48] O. Benson, C. Santori, M. Pelton, and Y. Yamamoto, “Regulated and entangled photons from a single quantum dot,” Phys. Rev. Lett. 84, 2513-2516 (March 2000).

[49] I. L. Chuang and Y. Yamamoto, "Simple quantum computer", Phys. Rev. A 52, 3489-3496 (November 1995).

[50] E. Waks, K. Inoue, C. Santori, D. Fattal, J. Vučković, G. S. Solomon, and Y. Yamamoto, “Quantum cryptography with a photon turnstile,” Nature 420, 762 (December 2002).

[51] K. Sanaka, A. Pawlis, T. D. Ladd, K. Lischka, and Y. Yamamoto. “Indistinguishable photons from independent semiconductor nanostructures,” Phys. Rev. Lett. 103, 053601 (July 2009).

**Quantum control of an election spin with light pulses and generation of spin-photon entangled states (2005 – 2012) **

Since the discovery of nuclear magnetic resonance (NMR) and electron spin resonance (ESR) in 1946, coherent manipulation (or rotation) of a spin is always realized by a sequence of pulses of resonant electromagnetic fields. Using this technique, a time duration required to rotate a spin by π/2 or π must be much longer than the inverse of a Zeeman frequency, which is typically 10-100 psec for an electron spin. A spatial resolution of spin manipulation depends on an artificially created magnetic field gradient and is typically 10-100 μm. A new technique we have developed using a laser pulse can control an electron spin with a much shorter time duration of 100 fsec - 1 psec and a much smaller spatial resolution of less than ~1 μm.

Quantum control of an impurity bound electron spin by ultra-fast laser pulses was first demonstrated in 2005 [52]. By using this new technique, we can rotate an election spin by an arbitrary angle with a time duration much shorter than the inverse Zeeman splitting. We also proposed quantum repeaters [53,54] and quantum computers [55] based on this technique of optical pulse controlled electron spins.

The above concept was subsequently extended to a more stable quantum dot electron spin system. Photon anti-bunching effect from a single quantum dot in a strong coupling regime was observed in InAs/GaAs systems [56]. In 2008, we demonstrated complete SU(2) quantum control of a single quantum dot spin with ultra-fast laser pulses for the first time [57]. A decoherence time could be increased to a few microseconds from an original lifetime of a few nanoseconds by using light pulse induced spin echo technique [58]. The same technique was applied to a single quantum dot hole spin [59].

Generation of spin-photon entanglement is an important basic technique for various quantum communication protocols. A standard method to realize this state employs a high-Q cavity which includes a spin. This system is suitable for basic proof-of-principle experiments but does not scale as a practical system. Our scheme based on the spontaneous emission decay of a charged exciton from quantum dots or donor impurities produces a spin-photon entangled state in a massive parallel way, so that the scheme is suitable for constructing a practical system.

Spontaneous emission decay of a charged exciton state in a single quantum dot naturally prepares a spin-photon entangled state, which is useful to realize a scalable quantum repeater system. This concept was first demonstrated using an InAs quantum dot embedded in GaAs matrix in 2012 [60]. Finally, the complete system architecture and performance evaluation of gate-model and fault-tolerant quantum computers based on optical pulse controlled electron spins and topological surface codes were reported in 2012 [61].

[52] K. C. Fu, C. Santori, C. Stanley, M. C. Holland, and Y. Yamamoto, “Coherent population trapping of electron spins in a high-purity n-type GaAs semiconductor,” Phys. Rev. Lett 95, 187405 (October 2005).

[53] P. van Loock, T. D. Ladd, K. Sanaka, F. Yamaguchi, K. Nemoto, W. J. Munro, and Y. Yamamoto, “Hybrid quantum repeater using bright coherent light,” Phys. Rev. Lett. 96, 240501 (June 2006).

[54] T. D. Ladd, P. van Loock, K. Nemoto, W. J. Munro, and Y. Yamamoto, “Hybrid quantum repeater based on dispersive CQED interactions between matter qubits and bright coherent light,” New Journal of Physics 8, 184 (September 2006).

[55] S. M. Clark, K. C. Fu, T. Ladd, and Y. Yamamoto, “Quantum computers based on electron spins controlled by ultrafast off-resonant single optical pulses,” Phys. Rev. Lett. 99, 040501 (July 2007).

[56] D. Press, S. Goetzinger, S. Reitzenstein, C. Hofmann, A. Löffler, M. Kamp, A. Forchel, and Y. Yamamoto, “Photon antibunching from a single quantum dot-microcavity system in the strong coupling regime,” Phys. Rev. Lett. 98, 117402 (March 2007).

[57] D. Press, T. D. Ladd, B. Zhang, and Y. Yamamoto “Complete quantum control of a single quantum dot spin using ultrafast optical pulses,” Nature 456, 218-221 (November 2008).

[58] D. Press, K. De Greve, P. L. McMahon, T. D. Ladd, B. Friess, C. Schneider, M. Kamp, S. Höfling, A. Forchel, and Y. Yamamoto “Ultrafast optical spin echo in a single quantum dot,” Nature Photonics 4, 367-370 (April 2010).

[59] K. De Greve, P. L. McMahon, D. Press, T. D. Ladd, D. Bisping, C. Schneider, M. Kamp, L. Worschech, S. Höﬂing, A. Forchel and Y. Yamamoto, “Ultrafast coherent control and suppressed nuclear feedback of a single quantum dot hole qubit,” Nature Physics 7, 872-878 (August 2011).

[60] K. De Greve, L. Yu, P. L. McMahon, J. S. Pelc, C. M. Natarajan, N. Y. Kim, E. Abe, S. Maier, C. Schneider, M. Kamp, S. Höfling, R. H. Hadfield, A. Forchel, M. M. Fejer, and Y. Yamamoto, “Quantum-dot spin-photon entanglement via frequency downconversion to telecom wavelength,” Nature 491, 421-425, (November 2012).

[61] N. C. Jones, R. Van Meter, A. G. Fowler, P. L. McMahon, J. Kim, T.s Ladd, and Y. Yamamoto, “Layered architecture for quantum computing,” Phys. Rev. X 2, 031007 (July 2012).

**Combinatorial optimization with optical parametric oscillator network (2011 – Present) **

Combinatorial optimization problems are ubiquitous in our modern information society. Classic examples of combinatorial optimization problems include the lead optimization in drug discovery and bio-catalyst development, resource optimization in wireless communications, scheduling and logistics, sparse coding for compressed sensing, portfolio optimization in Fintech and deep machine learning. Such combinatorial optimization problems belong to NP, NP-complete or NP-hard class in complexity theory, so that modern digital computers cannot solve them efficiently as problem sizes increase if a brute-force search is employed to find the exact solutions.

Various heuristic (or approximate) algorithms have been proposed to cope with an exponential increase of computational resources required for a large-scale combinatorial optimization problem. Yamamoto and his colleagues have proposed coherent Ising machine (CIM) based on network of injection-locked lasers as one of those heuristic solvers for combinatorial optimization problems in 2011 [62]. The proof-of-principle experiment was implemented with semiconductor lasers and fiber lases, but those systems suffered from unstable phase. To improve the stability and the noise property of CIM, the network of optical parametric oscillators was studied. Such systems were theoretically studied [63] and experimentally demonstrated [64-67] as physical coherent Ising machines. The advantage of using optical parametric oscillators instead of lasers is their stable bi-phase (0 or π) operation. Using this technique, we can construct a robust coherent Ising machine even in noisy environments. We demonstrated that such a heuristic solver outperforms a state of the art in modern digital computer [68] and quantum annealing machine [69]. We are currently elucidating the quantum principles of this novel computing machine [70].

[62] S. Utsunomiya, K. Takata, and Y. Yamamoto, “Mapping of Ising models onto injection-locked laser systems,” Opt. Express 19, 18091-18108 (September 2011).

[63] Z. Wang, A. Marandi, K. Wen, R. L. Byer, and Y. Yamamoto, “Coherent Ising machine based on degenerate optical parametric oscillators,” Phys. Rev. A 88, 063853 (December 2013).

[64] A. Marandi, Z. Wang, K. Takata, R. L. Byer, and Y. Yamamoto, “Network of time-multiplexed optical parametric oscillators as a coherent Ising machine,” Nature Photonics 8, 937-942 (October 2014).

[65] T. Inagaki, K. Inaba, R. Hamerly, K. Inoue, Y. Yamamoto, and H. Takesue, “Large-scale Ising spin network based on degenerate optical parametric oscillators,” Nature Photonics 10, 415-419 (June 2016).

[66] P. L. McMahon, A. Marandi, Y. Haribara, R. Hamerly, C. Langrock, S. Tamate, T. Inagaki, H. Takesue, S. Utsunomiya, K. Aihara, R. L. Byer, M. M. Fejer, H. Mabuchi, and Y. Yamamoto, “A fully-programmable 100-spin coherent Ising machine with all-to-all connections,” Science 354, 615-617 (October 2016).

[67] T. Inagaki, Y. Haribara, K. Igarashi, T. Sonobe, S. Tamate, T. Honjo, A. Marandi, P. L. McMahon, T. Umeki, K. Enbutsu, O. Tadanaga, H. Takenouchi, K. Aihara, K. Kawarabayashi, K. Inoue, S. Utsunomiya, and H. Takesue, “A coherent Ising machine for 2000-node optimization problems,” Science 354, 603-606 (October 2016).

[68] T. Leleu, Y. Yamamoto, P.L. McMahon, and K. Aihara, “Destabilization of local minima in analog spin systems by correction of amplitude heterogeneity,” Phys. Rev. Lett. 122, 040607 (2019).

[69] R. Hamerly, T. Inagaki, P. L. McMahon, D. Venturelli, A. Marandi, T. Onodera, E. Ng, C. Langrock, K. Inaba, T. Honjo, K. Enbutsu, T. Umeki, R. Kasahara, S. Utsunomiya, S. Kako, K. Kawarabayashi, R. L. Byer, M. M. Fejer, H. Mabuchi, D. Englund, E. Rieffel, H. Takesue, and Y. Yamamoto, “Experimental investigation of performance differences between coherent Ising machines and a quantum annealer,” Science Advances 5, eaau0823 (2019).

[70] Y. Yamamoto, K. Aihara, T. Leleu, K. Kawarabayashi, S. Kako, M. Fejer, K. Inoue and H. Takesue, “Coherent Ising machines - optical neural networks operating at the quantum limit,” npj Quantum Information 3, 49 (December 2017).